Introduction to bifurcation theory

John David Crawford

Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712
and Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 1526(f
The theory of bifurcation from equilibria based on center-manifold reduction and Poincare-Birkhoff nor-
mal forms is reviewed at an introductory level. Both differential equations and maps are discussed, and re-
cent results explaining the symmetry of the normal form are derived. The emphasis is on the simplest gen-
eric bifurcations in one-parameter systems. Two applications are developed in detail: a Hopf bifurcation
occurring in a model of three-wave mode coupling and steady-state bifurcations occurring in the real
Landau-Ginzburg equation. The former provides an example of the importance of degenerate bifurca-
tions in problems with more than one parameter and the latter illustrates new effects introduced into a bi-
furcation problem by a continuous symmetry.

CONTENTS VII. Center-Manifold Reduction 1009

A. Flows 1009
1. Local representation of Wc 1010
2. The Shoshitaishvili theorem 1011
I. Introduction 991
3. Example 1011
A. The basic setup 992
B. The basic question 994 B. Maps; Local representation of Wc 1012
II. Linear Theory 994 C. Working on intervals in parameter space: suspended
A. Flows 994 systems 1012
1. Invariant linear subspaces 994 VIII. Poincare-Birkhoff Normal Forms 1014
2. Hartman-Grobman theorem 996 A. Flows 1014
3. Loss of hyperbolicity and local bifurcation 996 1. Generalities 1014
B. Maps 997 2. Steady-state bifurcation on E 1016
1. Invariant linear subspaces 997 3. Hopf bifurcation on R2 1016
2. Hyperbolicity, Hartman-Grobman, and local bi- 4. Normal-form symmetry 1017
furcation 997 B. Maps 1019
III. Nonlinear Theory: Overview 998 1. Generalities 1019
IV. Persistence of Equilibria 999 2. Period-doubling bifurcation on E 1 1020
A. Implicit function theorem 999 3. Hopf bifurcation on E 2 1020
B. Applications to equilibria 999 IX. Applications 1021
1. Flows 999 A. Hopf bifurcation in a three-wave interaction 1022
2. Maps 1000 1. Linear analysis 1022
V. Normal-Form Dynamics 1000 2. Approximating the center manifold 1024
A. Flows 1000 3. Determining the normal form 1024
1. Steady-state bifurcation: simple eigenvalue at B. Steady-state bifurcation in the Ginzburg-Landau
zero 1000 equation 1026
a. Saddle-node bifurcation: the typical case 1000 1. Bifurcation from A = 0 1028
b. Transcritical bifurcation: exchange of stabili- a. g = 0 1029
ty 1001 b. Q¥=0 1029
c. Pitchfork bifurcation: reflection symmetry 1002 2. A digression on phase dynamics 1030
2. Hopf bifurcation: a single conjugate pair of 3. Bifurcation from the pure modes 1031
imaginary eigenvalues 1003 a. Symmetry 1031
B. Maps 1004 b. Linear stability for a = 0 1031
1. Steady-state bifurcation: simple eigenvalue at c. Center-manifold reduction for X + « 0 1032
+1 1004 X. Omitted Topics 1033
a. Saddle-node bifurcation 1005 Acknowledgments 1034
b. Transcritical bifurcation 1005 Index 1034
c. Pitchfork bifurcation 1005 References 1035
2. Period-doubling bifurcation: a simple eigenvalue
at — 1 1005
3. Hopf bifurcation: simple complex-conjugate
pair at |A,| = 1 1006 I. INTRODUCTION
C. Final remarks 1007
VI. Invariant Manifolds for Equilibria 1007 Bifurcation theory is a subject with classical
A. Flows 1008 mathematical origins, for example, in the work of L.
B. Maps 1009 Euler (1744); however, the modern development of the
subject starts with Poincare and the qualitative theory of
differential equations. In recent years, this theory has
* Revised and expanded version of lectures delivered at the In- undergone a tremendous development with an infusion of
stitute for Fusion Studies at the University of Texas in 1989. new ideas and methods from dynamical systems theory,
P e r m a n e n t address. singularity theory, group theory, and computer-assisted

Reviews of Modern Physics, Vol. 63, No. 4, October 1991 Copyright ©1991 The American Physical Society 991
992 John David Crawford: Introduction to bifurcation theory

studies of dynamics. As a result, it is difficult to draw the symmetric systems and Hamiltonian systems are not con-
boundaries of the theory with any confidence. The char- sidered, with the exception of pitchfork bifurcation for
acterization offered twenty years ago by Arnold (1972) at reflection-symmetric systems. A precise mathematical
least reflects how broad the subject has become: description of generic can be given at the expense of in-
The word bifurcation, meaning some sort of branching troducing a number of technical definitions (Ruelle,
process, is widely used to describe any situation in which 1989). The heuristic idea is simply that, when a
the qualitative, topological picture of the object we are parametrized system of equations exhibits a generic bi-
studying alters with a change of the parameters on which furcation, if we perturb the system slightly then the bifur-
the object depends. The objects in question can be ex- cation will still occur in the perturbed system. One says
tremely diverse: for example, real or complex curves or that such a bifurcation is robust. Bifurcations that are
surfaces, functions or maps, manifolds or fibrations, vec- robust in this sense for systems depending on a single pa-
tor fields, differential or integral equations. rameter are referred to as codimension-one bifurcations.
In this review the "objects in question" will be dynami- More generally, a codimension-n bifurcation can occur
cal systems in the form of differential equations and robustly in systems with n parameters but not in systems
difference equations. In the sciences such dynamical sys- with only n — 1 parameters. 2
tems commonly arise when one formulates equations of The aim is to provide an accessible introduction for
motion to model a physical system. The setting for these physicists who are not expert in dynamical systems
equations is the phase space or state space of the system. theory, and an effort has been made to minimize the
A point x in phase space corresponds to a possible state mathematical prerequisites. Consequently I begin with a
for the system, and in the case of a differential equation summary of linear theory in Sec. II that includes the
the solution with initial condition x defines a curve in Hartman-Grobman theorem to underscore the link be-
phase space passing through x. The collective represen- tween linear instability and nonlinear bifurcation; this
tation of these curves for all points in phase space summary is supplemented in Sec. IV by an analysis of the
comprises the phase portrait. This portrait provides a persistence of equilibria using the implicit function
global qualitative picture of the dynamics, and this pic- theorem. The center-manifold-normal-form approach is
ture depends on any parameters that enter the equations outlined in Sec. I l l and developed in Sees. V - V I I I .
of motion or boundary conditions. Two applications of the theory are considered in Sec.
If one varies these parameters the phase portrait may IX. These illustrate the calculations required to reduce a
deform slightly without altering its qualitative (i.e., topo- specific bifurcation to normal form. In addition the ex-
logical) features, or sometimes the dynamics may be amples offer a glimpse of several important and more ad-
modified significantly, producing a qualitative change in vanced topics: new bifurcations that arise when there is
the phase portrait. Bifurcation theory studies these qual- more than one parameter, center-manifold reduction for
itative changes in the phase portrait, e.g., the appearance infinite-dimensional systems, e.g., partial differential
or disappearance of equilibria, periodic orbits, or more equations, and the effect of symmetry on a bifurcation.
complicated features such as strange attractors. The Finally in Sec. X a brief survey of some topics omitted
methods and results of bifurcation theory are fundamen- from this review is included for completeness and to pro-
tal to an understanding of nonlinear dynamical systems, vide some contact with current research areas in bifurca-
and the theory can potentially be applied to any area of tion theory. Our subject is very broad, and there is much
nonlinear physics. activity by mathematicians, scientists, and engineers; the
In Sees. I I - V I I I , we present a set of core results and literature is enormous and widely scattered. This intro-
methods in local bifurcation theory for systems that de- duction does not attempt to assemble a comprehensive
pend on a single parameter /x. Here local bifurcation bibliography; the material of Sees. I I - V I I I can be found
theory refers to bifurcations from equilibria where the in many places, and in most cases the cited references are
phenomena of interest occur in the neighborhood of a chosen simply because I have found them helpful. More
single point. This restriction overlooks an extensive extensive bibliographies can be found in the references.
literature on global bifurcations where in some sense
qualitative changes in the phase portrait occur that are
A. The basic setup
not captured by looking near a single point. Wiggins
(1988) provides an introduction to this aspect of the sub-
It is advantageous to express different systems in a
ject. 1 In addition, we shall concentrate on those bifurca-
standard form so that the theory can be developed in a
tions encountered in typical or "generic" systems. Thus
uniform way. As an example consider the second-order

*It is worth emphasizing that the division between local and

2
global bifurcations introduced here should not be taken too The geometric connotations of codimension can be made pre-
seriously. A detailed investigation of a global bifurcation often cise, but we do not require this development here (Arnold,
uncovers a rich spectrum of accompanying local bifurcations; 1988a). Roughly speaking, the set of systems associated with a
similarly a local bifurcation of sufficient complexity can imply codimension-n bifurcation corresponds to a surface of codimen-
the occurrence of global bifurcations. sion n.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 993

oscillator equation /(0,0)=0 (1.3b)

y+y+y+y3=0; (1.1a) Note that given a fixed-point solution (^o^o) o n e c a n a*~

ways move it to the origin by a change of coordinates, so
by defining xt =y and x2=y, we can rewrite this evolu- the representation in Eq. (1,3) is quite general.
tion equation as a first-order system in two dimensions, The theory we develop for maps (1.2b) is useful in a
x
2 variety of circumstances. Two particularly important ap-
d_ xx
(Lib) plications are to bifurcations from periodic orbits of
dt x2 x
~2 x
\ '
x
\ differential equations and in the related context of bifur-
cations in systems that are periodically forced. Let xT(t)
Clearly if higher-order derivatives in t had appeared in denote a periodic solution to Eq. (1.2a) with period r, i.e.,
Eq. (1.1a), we could still have obtained a first-order sys- xr(t)=xr(t + r ) ; the dynamics near xT(t) can be ana-
tem by simply enlarging the dimension, e.g., defining lyzed by constructing the Poincare return map. Let X
JC3 ~y; similarly, if the equations of motion had involved denote an n — 1 dimensional plane in Wn which intersects
dependent variables in addition to y(t), these could also xr(t) at the point p (see Fig. 1). To define the return map
have been incorporated by enlarging the dimension ap- / , consider a point aG2 near/?, and solve Eq. (1.2a) us-
propriately. As this example suggests, there is great gen- ing a as an initial condition. For J sufficiently near p9
erality in considering dynamical processes defined by the trajectory from a will intersect 2 at some new point
first-order systems: a'; this intersection defines the action of the map / o n a ,
:
V(fi,x ] lii (1.2a) =/(CT). (1.4)
depending on a parameter /n and describing motion in an This definition is sensible for all points on 2 in an ap-
n-dimensional phase space E w . When formulated in this propriate neighborhood of p. Notice that p is a fixed
way a differential equation is identified with a vector field point for f,f(p)—p, since xT is a periodic orbit.
V(fi,x) on E w ; conversely, given a vector field one can al- In the second application, a periodic modulation is ap-
ways define an associated differential equation. 3 plied to the system in Eq. (1.2a) so that V(JLI9X) is re-
We shall also consider a second type of dynamics that placed by
represents the evolution of a system at discrete time in-
tervals. In this case, the motion is described by a map, x = V(fi,x,t) , xGE",^GE (1.5a)
xj + l=f(ix,Xj) , x€Rn , ]u€I , (1.2b) and
where j'= 0,1,2, . . . is the index labeling successive V(fx,x ,t)= V(fi, x,t + T) (1.5b)
points on the trajectory. There are close connections be-
where r is the period of the modulation. In this cir-
tween the dynamical systems defined by maps and vector
cumstance it is convenient to introduce the "stroboscop-
fields. For example, in Eq. (1.2a), we may also think of
solutions as trajectories: an initial condition x(0) ies map / by, in effect, recording the state of the system
uniquely determines a solution x (t), and the correspond- only once during each period of the modulation. More
ing curve in K" (parametrized by t) is the trajectory of precisely, fix a definite time t0 and then choose any initial
x(0). More abstractly, the association x(Q)—>x(t) condition x 0 €E E n . Let x(t;t0) denote the solution with
defines a mapping the initial condition x{tQ;t0)=x0, and define / b y

<pt-Mn->mn (1.2c) xj + l=f(Xj) , j =0,1,2, . . . (1.6)

where <f>t(x(0)):=x(t). This mapping is called the flow where Xj=x(t0-\~jr;t0). The qualitative properties of
determined by Eq. (1.2a). the map fix) in Eq. (1.6) are independent of the specific
In each case, the dynamics is allowed to depend on an choice t0 used in the definition. Furthermore, fixed
adjustable parameter /i, and the origin (fi,x ) = ( 0 , 0 ) is as-
sumed to be an equilibrium or fixed point for the motion,
F(0,0)=0 > (1.3a) n-l
5>IR
or

xT(t)
3
One often wishes to consider phase spaces more general than
Mrt, for example, finite-dimensional manifolds such as tori or
spheres. However, in these cases the dynamics on a neighbor-
hood of a fixed point can be described by the models we consid-
er by introducing a local coordinate system. FIG. 1. Poincare return map for a periodic orbit.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

994 John David Crawford: Introduction to bifurcation theory

points (1.3a) for the unmodulated system typically persist o

x
as fixed points for the map (1.6), at least for weak modu- r
X 2 X<y
lation. 4
(2.4)
dt

B. The basic question

if the spectrum of DV(0,0) includes complex-conjugate
According to Eq. (1.2), at / z = 0 there is an equilibrium pairs of eigenvalues, then the corresponding new coordi-
state at x = 0. The basic question in local bifurcation nate components xj will also be complex (Hirsch and
theory is Smale, 1974). The general solution x'(t) is obviously
Xa
x\(0)e
What can happen in phase space near x = 0 when
there are variations in /n about fi=01 x2(0)eV
*'(*) = (2.5)
The Hartman-Grobman theorem, described in the next Knt
x^0)e
section, effectively reduces this question to an analysis of
a narrower issue: If Rekj < 0, then as t —> oo, the x/ component decays to
As fi is varied near ^ = 0 , what happens near x—0 if zero; conversely, ReA,, > 0 implies exponentially rapid
the stability of the equilibrium changes? growth of x't.

Before addressing this question, which involves the non- 1. Invariant linear subspaces
linear terms of Eq. (1.2) in an essential way, it is neces-
sary to develop the theory of linear stability.
For each eigenvalue X of DV(0,0), there is an associat-
ed subspace of E w — t h e eigenspace Ek. For simplicity we
II. LINEAR THEORY assume D F ( 0,0) is diagonalizable; then our definition of
Ek depends only on whether X is real or complex. The
A. Flows

At x = 0 the Taylor expansion of Eq. (1.2a) begins,

Im X
x = V(fx,0)+DxV(fi,Q)-x+O(x2) , (2.1)
where DxV(fi,0) represents the square matrix with ele-
ments
ReX

(D^^O))^—(/i,0), (2.2)

and &(x2) indicates higher-order terms that are at least (a)

quadratic in the components of x. When the context is
clear we shall omit the subscript x and write DV{JU,,Q) or
simply DV. At /x = 0 the constant term in Eq. (2.1) van-
ishes, and near x = 0 w e study the linearized system,
x=DV(0,0hx (2.3)
ignoring momentarily the effects of the nonlinear terms.
In the typical situation the eigenvalues of DV( 0,0) are
nondegenerate 5 and this matrix can be diagonalized by a
linear change of coordinates x —>x'. This allows Eq. (2.3)
to be reexpressed as

4 FIG. 2. Example of invariant subspaces and manifolds for a

More precisely, this is true for hyperbolic fixed points, as fixed point, (a) Linear spectrum showing stable modes, neutral
defined in Sec. ILA.2, and follows from the averaging theorem modes, and unstable modes for an equilibrium x = 0 in a flow;
(Guckenheimer and Holmes, 1986). (b) invariant linear subspaces; for the spectrum in (a) we would
5
A degenerate eigenvalue is one for which there are two or have dim Es=3> J5 C =4, dim Eu—3; (c) invariant nonlinear
more linearly independent eigenvectors or generalized eigenvec- manifolds; for the spectrum in (a) we would have dim W s =3,
tors. dim W c =4,dim Wu = 3.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 995

case of a real eigenvalue is most familiar. When X is real, real matrix if vt-hiv2 is the eigenvector for X9 the
EK is simply the subspace spanned by the eigenvectors, complex-conjugated vector i^ — iv2 is an eigenvector for
X. The eigenspace Ex in this case is spanned by the real
X€R, Ex==iv(=Rn\(DV(0,0)-XIhv=0} . (2.6a)
and imaginary parts of the eigenvectors for X9 e.g.,
If A, is nondegenerate, then we have dim Ek = 1. vx and v2. Noting that both vx and v2 satisfy
When k is complex, then the eigenvectors are also (DV(0,0)-XI)(DV(0,0)~XI)-v=0, we replace Eq.
complex; furthermore, since DV(0,0) is assumed to be a (2.6a) with

XmM , Ei ={vGRn\(DV(0,0)-XI)(DV(0y0)-XI)-v 0) . (2.6b)

Now if X is nondegenerate we have dim Ek=2. Ec=$pan{v\v&Ex and ReA,=0} . (2.7c)

When DV(Q,0) has eigenvalues that are degenerate,
this construction for Ek is satisfactory provided DV(0,0) These subspaces span the phase space, W.n—ES®EC®EU,
is diagonalizable. When Z>F(0,0) cannot be diagonal- and they are invariant: if x(Q)EzEa9 a=s9c9u, then the
ized, then the definitions in Eq. (2.6) must be extended to trajectory x(t) of Eq. (2.3) with this initial condition
include not only eigenvectors but generalized eigenvec- satisfies x(t)£iEa. For Es and Eu the dynamics has a
tors as well (Arnold, 1973; Hirsch and Smale, 1974). simple asymptotic description: if x(t)€EEs> then as
A n eigenvalue X corresponds to a " m o d e " of the sys- t —>- -f oo the trajectory converges to the equilibrium; if
tem that is stable, unstable, or neutral, depending on x(t)E.Eu, then the trajectory converges to the equilibri-
whether ReA < 0, ReA, > 0, or R e A = 0 , respectively [Fig. um as t —> — oo. These features are illustrated in Fig.
2(a)]. We divide the eigenvectors (and generalized eigen- 2(b).
vectors) of D F ( 0,0) into three sets according to these A n equilibrium at x = 0 is asymptotically stable if there
possibilities and form the stable subspace Es, unstable exists a neighborhood of initial conditions, 0 < | x ( 0 ) | < e ,
subspace E M, and center subspace Ec: such that for all x (0) in this neighborhood
Es=$v2LXi{v\v^Ek and R e X < 0 ] (2.7a)
(i) the trajectory x (t) satisfies \x(t)\ < £ for t > 0, and
u (ii) | x U ) | - > 0 a s * - + o o .
E -=sp2iti{v\v^Ek and R e X > 0 ] (2.7b)

Im

Re

(a)

(b) T (b) I

FIG. 4. Asymptotic stability of x=0. Such stability for the

FIG. 3. A stable linear spectrum for a fixed point of (a) a flow linear system (a) implies that x = 0 is asymptotically stable for
and (b) a map. the nonlinear system (b).

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

996 John David Crawford: Introduction to bifurcation theory

For the linear system (2.3), the equilibrium x = 0 is Then there exists a homeomorphism6
asymptotically stable if and only if Re(A) < 0 for each ei-
genvalue A of D F ( 0 , 0 ) . In other words, the spectrum
must lie within the left half-plane of the complex A plane and a neighborhood Uofx = 0 where7
[see Fig. 3(a)].
This criterion is particularly valuable because one can <f>t(x) = y~lo$toq/(x) (2.8)
prove that if x = 0 is asymptotically stable for Eq. (2.3),
then it will also be asymptotically stable for the original for all (x,t) such that x G C / and <f>t(x)GU.
nonlinear system (1.2a) (Hirsch and Smale, 1974). In Fig.
4(b) we show a schematic phase portrait for a two- For a proof see Hartman (1982). Note that ¥ ( x ) and
dimensional system with two fixed points on the x x axis. its inverse cannot in general be assumed differentiable.
If we imagine linearizing about the stable equilibrium at In the terminology of dynamical systems, Eq. (2.8) defines
the origin, then the resulting 2 X 2 matrix will have a a topological conjugacy (locally) between the linear flow
complex-conjugate pair of eigenvalues (A,, A,) satisfying and the nonlinear flow; this is a precise statement that
R e A = R e A < 0 . The phase portrait for the linearized sys- the nonlinear dynamics near x = 0 is qualitatively the
tem is shown in Fig. 4(a); the equilibrium x = 0 is obvi- same as the linear dynamics. In particular, if there are
ously asymptotically stable in Fig. 4(a) for arbitrarily no unstable directions so that W(x) belongs to E\ then
large initial conditions. In the nonlinear phase portrait <fito q/(x)-+Q as t —> oo for the linear flow and Eq. (2.8) im-
Fig. 4(b) x = 0 is also asymptotically stable, but the plies that (f*t(x)—>0 as t—>oo as well; i.e., linear asymp-
neighborhood, 0 < \x(0)| < E, of stable initial conditions is totic stability implies nonlinear asymptotic stability.
not arbitrarily large; it must not intersect the trajectories
which are asymptotically drawn to the unstable fixed
point on the negative x x axis. The linear test for asymp-
totical stability provides no information regarding the 3. Loss of hyperbolieity and local bifurcation
size of the neighborhood in the nonlinear system where
the conclusion of stability holds. The Hartman-Grobman theorem implies that any
qualitative change or bifurcation in the local nonlinear
dynamics must be reflected in the linear dynamics. If
2. Hartman-Grobman theorem x = 0 is hyperbolic, then the linearized dynamics is quali-
tatively characterized by the expanding and contracting
The qualitative relation between (2.3) and (1.2a) pro- flows on Eu and E\ respectively; this qualitative struc-
vided by the property of asymptotic stability is only appl- ture remains fixed unless the equilibrium loses its hyper-
icable when all the eigenvectors are stable, i.e., when Eu bolieity. For this loss to occur, the eigenvalues of the sta-
and Ec are empty, but this instance does not exhaust the bility matrix D F m u s t shift so as to touch the imaginary
information about the nonlinear problem that is available axis.
from the linearized dynamics. Even if the equilibrium is In Sec. IV we shall show that when a fixed point is hy-
not asymptotically stable, there are general theorems perbolic, if ji is varied slightly near p—0, then the fixed
describing in what sense the qualitative features of Eq. point must persist, although its precise location in the
(2.3) faithfully reflect the full nonlinear flow (1.2a) near phase space may shift. In this event the eigenvalues of
x = 0 . For example, near a hyperbolic equilibrium, i.e., a the associated linear stability matrix DV depend on /i,
fixed point with no eigenvalues on the imaginary axis, and as the parameter value changes, it may happen that
there exists a change of coordinates that transforms the an eigenvalue A(JU) approaches the imaginary axis. The
nonlinear flow into the linear flow locally. Thus, even system is said to be critical when R e ( A ) = 0 , and the cor-
when there are unstable directions, the linearized dynam- responding parameter value /J>—fic belongs to the bifur-
ics remains a qualitatively accurate description of the cation set This loss of hyperbolieity occurs in one of two
nonlinear dynamics. The Hartman-Grobman theorem ways, which we distinguish by the appearance of the
provides a precise statement of this idea. There is a gen- spectrum at criticality 8 :
eralization of this theorem due to Shoshitaishvili that
treats the nonhyperbolic case when Ec is not empty; this
is discussed in Sec. VII.A.2.
6
A homeomorphism is a continuous change of coordinates
Theorem I I . l (Hartman-Grobman). Let x = 0 be a hyper- whose inverse is also continuous.
7
bolic equilibrium for Eq. (1.2a) at some fixed value of [i, Here the composition of functions fix) and g(x) is written
fog(x)=f(g(x)).
§t denote the flow of (1.2a), and <f>t denote the flow for the 8
corresponding linear system: A loss of hyperbolieity can readily involve more complicated
scenarios if there are multiple parameters or if the problem has
some special structure, e.g., if the equations are Hamiltonian or
x=DV([i,0)-x . have symmetry.

Rev. Mod. Phys., Vol. 63, Ho. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 997

(1)A simple real eigenvalue at A»=0. We shall refer to

H 0
this type of critical spectrum as steady-state bifurcation
[see Fig. 5(a)]. The nonlinear behavior produced by x
2 0 k2
steady-state bifurcation may take several forms, which (2.11)
we discuss in Sec. V. Most typical is saddle-node bifurca- 0
tion, but in applications one also encounters transcritical Ji + i
bifurcation and pitchfork bifurcation. with solution
(2) A simple conjugate pair of eigenvalues satisfying
ReA,=ReX=0; see Fig. 5(b). This type of instability is k{ 0 0 x\
commonly referred to as Hopf bifurcation [although the
name does not reflect earlier work of Poincare and An- *2 o M X 2

dronov (Arnold, 1988a)]. (2.12)

0 K
B. Maps
If \Xt\ < 1, then as jf->oo, the x/ component decays ex-
The corresponding linear theory for a map may be dis- ponentially; if | ktr | > 1, then the x/ component will grow.
cussed in a similar fashion. The expansion of Eq. (1.2b) 1. Invariant linear subspaces
at x = 0 .
For the linearized map (2.10) the eigenspace Ex for
xj + l=f(ii,Q)+Dxf(fi,Ohxj + 0(x2) , (2.9)
D/(0,0) are defined as in Eq. (2.6) for the previous case
by replacing DV(0,0) with D/(0,0). The invariant linear
leads to the linearized system
subspaces Ea,a=s,u,c, are defined as in Eq. (2.7), replac-
ing ReA, by |A,| — 1 to reflect the appropriate stability cri-
xj + l=Df(Q,0)-Xj (2.10) teria,

at ju=0. As before, we diagonalized Df (0,0) by chang- Es=&pan{v\vGEx and |A,|<1} , (2.13a)

ing coordinates x —*x' = (x \ ,x \,. . . , x^) and obtain u
E ^$ptm{v\vGEx and |A,|>1} , (2.13b)
Ec=sp&n{v\vE.Ek and |A,| = 1} . (2.13c)
n s c u
As before, we have R =E ®E ®E f and the stable and
unstable subspaces have simple asymptotic dynamics as
Im
j _> + oo and j ~> — oo, respectively.
The definition of asymptotic stability given earlier ap-
plies to fixed points of maps provided x (t) is replaced by
Xj. For the linear dynamics (2.10), the equilibrium x = 0
will be asymptotically stable if and only if the spectrum
Re of I>/(0,0) lies within the unit circle in the complex X
plane, i.e., |A,/| < 1 for each eigenvalue [see Fig. 3(b)]. It
can be shown that if x = 0 is asymptotically stable for Eq.
(2.10) then the same conclusion holds for the full non-
linear dynamics (1.2b). In addition, for return maps (cf.
(a) Fig. 1), whose fixed points correspond to periodic orbits,
the stability of a fixed point reflects the stability of the
Im
corresponding periodic orbit. [When the differential
equation is linearized about the periodic orbit, the result-
ing linear equation may be analyzed using Floquet
theory; the stability of the periodic orbit is determined
from the spectrum of Floquet multipliers (Jordan and
Re Smith, 1987). The eigenvalues of the return map linear-
ized at the fixed point correspond to the Floquet multi-
pliers of the periodic orbit.]

2. Hyperbolicity, Hartman-Grobman,
(b) and local bifurcation

F I G . 5. Basic instabilities for an equilibrium in a flow: (a) As for flows, a fixed point is said to be hyperbolic if the
steady-state bifurcation and (b) Hopf bifurcation. center subspace (2.13c) is empty, and there is a

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

998 John David Crawford: Introduction to bifurcation theory

Hartman-Grobman theorem relating the linearized dy- (3) A simple real eigenvalue at k= — 1; see Fig. 6(c).
namics to the local nonlinear dynamics: if, at JU. = 0,X = 0 This case is novel, as it does not have an analog in the
is a hyperbolic fixed point, then there exists a homeomor- earlier discussion of flows. This instability is generally
phism ^ and a local neighborhood U of x = 0 where termed period-doubling bifurcation, although the names
flip bifurcation and subharmonic bifurcation are also
/(0,Jc) = ^" 1 (i>/(0,0)-vi/(x)) (2.14) used.
for x such that XELU and f(0,x)E:U.
If x = 0 is a hyperbolic fixed point for / ( / x , x ) at fi — Q, This completes our summary of linear stability theory
then as /a is varied about zero this equilibrium will shift and the forms of instability one typically expects to en-
its location, but it will persist (see Sec. IV). The eigenval- counter when a single parameter is varied. Characteriz-
ues of Df will be functions of /z, and a variation in /a will ing an instability by the form of the linear spectrum at
cause them to move in the complex plane. If an eigenval- criticality is more than a convenience; it is very advanta-
ue reaches the unit circle, then the fixed point is no geous to organize the theory (and one's understanding) in
longer hyperbolic and a bifurcation can occur. this way. The most important reason for this is that the
The possibilities may be classified by the form of the linear spectrum determines the normal form. Precisely
linear spectrum when the condition \k( 1=7^1 fails: what this means will be explained in Sec. VIII.

(1) A simple real eigenvalue at X= 1; see Fig. 6(a). This HI. NONLINEAR THEORY: OVERVIEW
type of instability is quite similar to the A, = 0 case for
flows and is referred to as a steady-state bifurcation for Suppose an asymptotically stable equilibrium is per-
maps. As in the case of flows, we find the saddle-node, turbed by varying an external parameter JJL> and at a criti-
transcritical, and pitchfork bifurcations as examples of cal value /x= i u c the equilibrium develops a neutral mode
steady-state bifurcation. (ReX = 0 for flows; | A,| = 1 for maps). At fic hyperbolicity
(2) A simple conjugate pair of eigenvalues (A,, X) where is lost, and we must study what happens to the system as
k — el27rd; see Fig. 6(b). We shall refer to this case as Hopf jit is varied about fxc.
bifurcation for maps to emphasize similarities with Hopf For all of the basic instabilities described in Sec. II,
bifurcation in flows. this issue can be investigated using the techniques of
center-manifold reduction and normal-form theory. In
brief outline, this approach has several steps:

(1) Reduction: identify the neutral mode (or modes) at

fji=fic and restrict the dynamical system to the appropri-
ate center manifold;
(2) Normalization: if possible, put this reduced
dynamical system into a simpler form by applying near-
identity coordinate changes. This yields the normal form
for the bifurcation;
(3) Unfolding: describe the effects of varying \i away
from fic by introducing small linear, and possibly non-
linear, terms into the normal form;
(4) Study the bifurcations described by the unfolded
normal form. In this analysis, one truncates the unfolded
system at some order and considers the resulting system.
Once the truncated system is understood, the effect of re-
storing the higher-order terms can be discussed. 9

The virtue of step one is that it reduces the dimension

of the problem without any loss of essential information
concerning the bifurcation. The advantages of the
simplification offered in the second step are often decisive
in being able to solve the problem. Furthermore, the re-

9
In sufficiently complicated bifurcations, these effects can be
significant and highly nontrivial. However, for most of the bi-
FIG. 6. Basic instabilities for an equilibrium in a map: (a) furcations considered in this review, these higher-order terms
steady-state bifurcation; (b) Hopf bifurcation; (c) period- do not produce any qualitative changes. The one exception is
doubling bifurcation. Hopf bifurcation in maps, discussed in Sec. V.B.3.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 999

suiting simplified representation of the dynamics pro- such thatX(0) = 0and

vides a universal, low-dimensional model for the given bi-
furcation. G(/z,X(/x))=0 , IXELM (4.4)
This approach allows the general qualitative features In words the theorem says the following. Given
of a bifurcation to be distinguished from specific quanti- G(fj,,x) we assume that the zero set, i.e., the set of (/i,x)
tative aspects that will inevitably vary between different such that G(fi,x)=0, contains at least one point (0,0);
realizations of the bifurcation. The dimension of the re- see Fig. 7(a). If, in addition, the matrix
duced system and the structure of the appropriate nor-
mal form may be determined without requiring explicit dGt
U>,G(0,0))y = - g ^ ( 0 , 0 ) U = l, (4.5)
evaluation of the coefficients in the normal form. Thus
the variety of phenomena associated with a bifurcation
can be described in a theory that is model independent. has a nonzero determinant, then we can solve the equa-
When this general theory is applied to a particular insta- tion G (fi,x)=0 uniquely for x as a function of /a, at least
bility the normal-form coefficients can be calculated from for values of/x sufficiently near / x = 0 . This means that,
near (JU,X) = ( 0 , 0 ) , the zero set of G(/LL,X) consists of a
the specific physical model under consideration. The
single arc or branch as shown in Fig. 7(b).
possibility of determining the normal form without need-
ing to derive the coefficients is often a considerable ad-
vantage. B. Applications to equilibria
In Sec. V, we present the normal forms for the bifurca-
tions enumerated in Sec. II. Then the basic theory un-
1. Flows
derlying the center-manifold reduction is discussed in
Sees. VI and VII. Finally, in Sec. VIII, we develop the
For Eqs. (1.2a) and (1.3a), we choose G(fiyX)==V(fi9x ).
theory of Poincare-BirkhofF normal forms and indicate
Then
how to derive the normal forms previously introduced in
Sec. V. det[DxG(0,0)]=det[DV(Q,0)] ; (4.6)
In the next section, we consider a preliminary issue
that it is useful to discuss before taking up the program this implies that condition (4.2b) will be satisfied if and
outlined above. The question is basic: can the given equi- only if A,=0 is not an eigenvalue for DV(0,0). It then
librium solution simply disappear when /u is varied? For follows that small changes in JLL will not destroy the equi-
both flows and maps, there are simple conditions on the librium solution as long as zero is not an eigenvalue of
linear spectrum that are sufficient to guarantee the per- the linear stability matrix for the equilibrium. The solu-
sistence of an equilibrium. tion must persist and lie on a local branch of such solu-
tions, X(/LL), as required by the implicit function theorem.
Two further conclusions may be drawn. First, Hopf
IV. PERSISTENCE OF EQUILIBRIA bifurcation cannot alter the number of equilibrium solu-
tions, since the only eigenvalues of DV on the imaginary

A. Implicit function theorem

The implicit function theorem provides necessary con- [Rn

ditions for an equilibrium of a flow or a map to disappear
as // varies. Equivalently these conditions can be restated
as sufficient conditions for the equilibrium to persist.
The following version of the theorem is adequate for our
discussion; a proof may be found in Spivak (1965).
(a)

Theorem IV.l. Let G(JLL,X ) be a C 1 function on R X E " ,

G:RXRn-+Rn , (4.1)
.X(/x)
such that
G(0,0)=0 (4.2a)
and
det[DxG(0,Q)]¥*0 . (4.2b)
Then there exists a unique differentiate function X(^i) FIG. 7. A unique solution branch from the implicit function
defined on a neighborhood M C M offi = 0, theorem. Given < J ( 0 , 0 ) = 0 and det[DxG(0,0)]=£0 at a point
(a), the local structure of the solution set for G (fj,,x)=0 is a sin-
X:M-^Rn , (4.3) gle branch (b).

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1000 John David Crawford: Introduction to bifurcation theory

axis form a conjugate pair [Fig. 5(b)]. Second, the condi- x = V([i,x) , xGR , JUGE , (5.1a)
tion dctlDV^O fails at a steady-state bifurcation, since
which will satisfy the following two conditions at critical-
by definition there is always an eigenvalue at zero. Thus
ity:
in general we cannot expect a unique branch of equilibria
through (fi,x) — (0,0) if this solution corresponds to a K(0,0) = 0 > (5.1b)
fixed point at criticality for steady-state bifurcation.
•(0,0)=0 (5.1c)
2. Maps dx
Center-manifold theory tells us that Eq. (5.1a) should be
For Eqs. (1.2b) and (1.3b), we take G(fi,x)=f(iu,,x) one dimensional. Furthermore, the reduction to one di-
—x, so that G(0,0) = 0 and
mension will preserve Eq. (1.3a) and the occurrence of a
DxG(0,0)=Df(0,0)-I (4.7) zero eigenvalue; hence Eqs. (5.1b) and (5.1c), respectively.
Expanding (5.1a) at (JLL,X) = (0,0), we find
where / is the identity matrix on K". With this choice, if
G(/J,,X ) = 0 then x is a fixed point for the map at parame- dV d2V x2 32V
x = ~ ( 0 , 0 ) / i + ^z ( 0 , 0 ) ^ + - ^ ( 0 , 0 ) / z x
ter value fi. For the solution (/n,x ) = (0,0), condition dfi dx 2 dfiox
(4.2b) will be met if and only if the linear stability matrix d3V x3
Df (0,0) does not have an eigenvalue at k= + 1 . Provid- + ^>,0)f
d/LLZ
(5.2)
ed A,= l is not an eigenvalue, the implicit function
theorem implies (0,0) lies on an isolated branch of equi- For this instability, the vector field at criticality,
librium solutions.
crV x2
For the three basic instabilities illustrated in Fig. 6,
only steady-state bifurcation involves an eigenvalue at
dxz 2 a,>l0'0,^+
+ 1. Neither period-doubling nor Hopf bifurcation can cannot be significantly simplified by making coordinate
alter the number of equilibrium solutions. In the context changes (cf. Sec. VIII); we shall obtain normal forms by
of Poincare return maps for periodic orbits, these results making truncations and rescalings. There are three situ-
on persistence of equilibria show that the periodic orbit ations that arise most often in applications.
can always be followed through a period-doubling or
Hopf bifurcation. The question of following periodic or- a. Saddle-node bifurcation: the typical case
bits through parameter space in a global sense has also
been studied (Mallet-Paret and Yorke, 1982; Yorke and Equations (5.1a)-(5.1c) define a steady-state bifurca-
Alligood, 1983). tion; without further assumptions we typically ("generi-
cally") expect
V. NORMAL-FORM DYNAMICS
(0,0)^=0 (5.3a)
In this section we analyze very simple equations that 3/i
describe the local dynamics associated with the linear in- and
stabilities of Sec. II. Remarkably, these simple examples
are in fact quite general; to appreciate this generality re- d2V
(0,0)^0 (5.3b)
quires the material on center manifolds and normal-form dx2
theory developed in later sections. Let us first analyze to hold. In this case Eq. (5.2) may be rewritten as
the dynamics of these simple models and then establish
their generality. We shall consider the various bifurca- jc = | ^ ( O , O ) M [ l + 0(/x,x)] + ^ ( O ,O)^-[l+0(iu,x)] ,
tions in the same order they were listed in Sec. II. In the ofLL dxz 2
following it is convenient to assume that criticality for an (5.4)
instability occurs at / i = 0 .
where 0(fx,x) indicates terms at least first order in fi or
A. Flows x. For example,
1. Steady-state bifurcation: d2V 3V
simple eigenvalue at zero (0,0)/ (0,0)
dfidx
For a simple zero eigenvalue 10 as illustrated in Fig. 5(a) is one such term in the first bracket in Eq. (5.4). Near
the center-manifold reduction yields a system of the form (jti,x)«(0,0) we can neglect these 0(/j,,x) terms relative
to unity and then define rescaled variables (fi,x ) ,

10 M=' V> (5.5a)

An eigenvalue is simple if it is nondegenerate; for a real ei- 3V d2V
genvalue the associated eigenspace (2.6a) is then one dimension- f H 0 , 0 ) ~ ( 0z , 0 )
al. d[i dx

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1001

x = X , (5.5b)
d2V
(0,0)
3x2
to obtain the normal form 11
(a)
x =€ip, + e2X2= V{p,,x) , (5.6)

where

e, = sgn | ^ ( 0 , 0 )
-i—M - f - /*
d2V
e2=sgn (0,0)
dx' (c) (d)

Obviously at j S = 0 , x=0 is an equilibrium in Eq. (5.6),

and this equilibrium has a zero eigenvalue. What hap-
pens near (fl,x ) = (0,0) depends on (eue2); there are four FIG. 8. Diagrams for saddle-node bifurcation with normal
possibilities. Consider 6 1 = £ 2 ~ + 1 (the other three cases form (5.6): (a) £1 = e 2 = l , (b) €l = €2= — ly (c) — e1 = e 2 = l , (d)
can be analyzed similarly). Then the equilibria in Eq. €\ — ~ e2 = 1. Solid branches are stable; dashed branches are un-
stable.
(5.6) satisfy Jl+x 2 = 0. This describes a parabola in the
(x,fl) plane as shown in Fig. 8(a). At a fixed value of
p < 0 , there are two equilibria x±(p,) = which
coalesce as p increases to criticality. The upper branch
x + (ju) is unstable, and the lower branch x_(pi) is asymp- theorem cannot guarantee a unique branch of equilibria
totically stable. This is indicated by the arrows in Fig. passing through (0,0).
8(a) and can be checked by linearizing Eq. (5.6) about The results for the remaining three cases, €l=—€2=l,
x±(fi). Let x=x±(fi)+y±. Then (for e2— + 1) €{ = — 62— — 1, and €l = €2—~l, are also shown in Fig. 8.
These diagrams in the (p,,x ) plane are simple examples of
bifurcation diagrams.
y±-
dv ([i,x±)y± = [2x±(/j,)]y (5.7)
dx ±>

the eigenvalue [2x±(p,)] is positive (unstable) for x+ and

b. Transcriticalbifurcation: exchange
negative (stable) for x _ .
of stability
In two dimensions, a fixed point with one stable and
one unstable eigenvector is referred to as a saddle; if the
In applications, it may happen that an asymptotically
fixed point has two real negative eigenvalues it is a stable
stable equilibrium loses stability through a steady-state
node (Arnold, 1973). When a parameter is varied, so that
bifurcation, but the equilibrium solution itself survives.
such fixed points are brought together, then the resulting
In this case saddle-node bifurcation, which characteristi-
merger can be described by the one-dimensional model
cally destroys (or creates) equilibria, does not occur.
(5.6); the bifurcation is named for this prototypical exam-
When the equilibrium survives, we may denote it by X(fi)
pie. 12
such that X ( 0 ) = 0 and
Note that for p < 0 there are two equilibria, but for
p, > 0 there are none. This is consistent with the fact that V(JLL,X(/J,))=0 , /i£R (5.8)
Eq. (4.2b) fails at (fi,x ) = (0,0) and the implicit function
replaces Eq. (5.1b). Let us make the /^-dependent change
of variables x—Xi^+x' and then drop the primes.
[This amounts to setting X(fi) = 0.] Then Eq. (5.8) be-
comes
n
I n the terminology of Sec. IV, the normal form is actually
X — e2x 2, and €{fi is an unfolding term. I often overlook this V(fLL90)=0 , (5.9)
distinction in the following and simply refer to the unfolded
normal form as the normal form. for an appropriately redefined V(/LL,X ). Since Eq. (5.9)
12
More generally, the one-dimensional model (5.6) describes a implies
much wider class of bifurcations, in which two fixed points are
either created or destroyed. In higher dimensions it is not al- (0,0) = 0 , n = l,2> (5.10)
dfin
ways the case that one equilibrium is stable and the other unsta-
ble; both may be unstable. Neither is it necessarily true that the if we now make a Taylor expansion around (JU,,X ) = (0,0),
eigenvalues not involved in the bifurcation must be real. then Eq. (5.2) is replaced by

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1002 John David Crawford: Introduction to bifurcation theory

d2V d2V x2 331 y = (-€xti)y . (5.15)

<0 0, + 0, + 0)
*--^ ' '" i>' T !>' ir + Thus x=0 and x=xb(fi) have opposite stabilities; at
(5.11) fi—Q these equilibria collide and their stabilities are "ex-
changed." The precise form of the resulting bifurcation
Without further assumptions, we shall typically find diagram depends on €j and €2\ the four possibilities are
d2V shown in Fig. 9.
(0,0)^0 , (5.12a)
dfidx
d2V c. Pitchfork bifurcation: reflection symmetry
(0,0)^0 , (5.12b)
dx:
This version of steady-state bifurcation arises formally
where Eq. (5.12a) replaces (5.3a). Now, proceeding ex- when Eq. (5.9) holds as in transcritical bifurcation but
actly as in the discussion of saddle-node bifurcation, we (5.12b) fails and is replaced by the assumption
truncate and rescale variables to obtain a normal form,
° -(0,0)^0 . (5.16)
x~x(elfJL-\-e2x ] (5.13) dx:
where A natural context for these assumptions is V(\i,x ) having
a reflection symmetry, i.e.,
d2V
ei=$gn (0,0) F ( / x , x ) = F ( / i , — x) (5.17)
dfidx
Obviously, this symmetry implies Eq. (5.9), and forces
and Eq. (5.12b) to fail. Replacing (5.12b) by (5.16), we may
d2V rewrite (5.11) as
e 2 = sgn (0,0)
dx: d2V
{0,0)px[l + O(fJifx)]
dx 3/x
Note that x = 0 is an equilibrium for all ju, but at p = 0
v3
the eigenvalue ejl is zero. When sgn(6 1 p)= —1( + 1 ) the 33F
(5.18)
equilibrium * = 0 is stable (unstable). The second factor + dx*3 ( O , O )3!~ r [ l + 0 ( j u , x ) ] .

on the right-hand side of Eq. (5.13) yields a second

Now truncating higher-order terms and rescaling vari-
branch of equilibria, Xb(fL)9
ables appropriately leads to the normal form
£i_
xb(fi)~~ M (5.14) x =zx[€lfx + e2x 2] , (5.19)
*2
where
The stability of xb is found by linearizing Eq. (5.13)
x —xb~\-y to find d2V d3V
6i = sgn (0,0) £ 2 = sgn (0,0)
9/i,3x dx:

t -M --
ix t f i
t \ |
1 r'
LL
\
^
\, (a) (b)
J
t \
(a) (b)

U U
(c) (d)

(c) (d)

FIG. 9. Diagrams for transcritical bifurcation with normal FIG. 10. Diagrams for pitchfork bifurcation with normal form
form (5.13): (a) £1 = £ 2 = 1 , (b) e1 = e 2 = - l , (c) -€l = e2=l, (d) (5.19): (a) 6 1 = - 6 2 = 1 , (b) -€l = €2=l, (c) e1 = e 2 = - l , (d)
€\ = — e2 — 1. Solid branches are stable; dashed branches are un- 61 = 6 2 =1. solid branches are stable; dashed branches are un-
stable. stable.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1003

bitrarily in the sense that it need not respect any special

assumptions such as Eqs. (5.17), (5.9), or (5.1b). For tran-
scritical bifurcation, when €=£0 one expects the bifurca-
tion diagram to be modified in one of two ways [see Fig.
11(a)]. In one case the perturbed diagram contains two
saddle-node bifurcations; in the other case there are no
bifurcations at all. With pitchfork bifurcation there are
6 =0 four possibilities expected for the perturbed diagram,
€ tO Fig. 1Kb). There is one important new feature: the pos-
(a)
sibility of finding hysteresis in the bifurcations of the per-
turbed pitchfork. This effect can be understood intuitive-
ly by noting that when 6 = 0 the outer branches of the
pitchfork meet the middle branch with an angle of exact-
ly 90°. A small perturbation will split and join the
branches as shown and also perturb this 90° angle slight-
ly. This latter effect leads to the appearance of hys-
teresis.

€ = 0
2. Hopf bifurcation: a single conjugate pair
of imaginary eigenvalues

The normal form is two dimensional and in polar coor-

(b) <£ /0 dinates (r,6) may be written as

FIG. 11. Perturbing nongeneric diagrams: (a) transcritical bi- r — r y(fi)- --yr+a^ + Oir5) , (5.22a)
furcation; (b) pitchfork bifurcation.
e==co(jLt)+^bj(^)r 2j (5.22b)

The analysis of Eq. (5.19) differs from transcritical in that where y(fi)±ico(fi) is the complex-conjugate pair of ei-
the second factor in (5.19) contributes two branches of genvalues that are assumed to satisfy
equilibria,
r(o)=o, ^(o)^o, (5.23a)
X±(fx) = ±V-(€1/€2)fi (5.20)
ILL,( 0 ) > 0 (5.23b)
which only exist for s g n t e j / i / ^ ) ^ ~~ 1- The stability of dfx
the solutions may be worked out as before, and the four
The conditions (5.23) simply mean that the conjugate
possibilities are illustrated in Fig. 10. The bifurcation di-
pair crosses the imaginary axis at ju, = 0 in a nondegen-
agrams resemble pitchforks in the (ju,x) plane, hence the
erate way.
name.
A characteristic feature of Eq. (5.22) is the absence of 6
We conclude this discussion of steady-state bifurcation
on the right-hand side. This means that the dynamics of
by indicating how perturbations of transcritical or pitch-
the normal form is invariant with respect to the group of
fork bifurcation can restore the expected "generic" be-
rotations of the phase 6. In the literature, this in variance
havior, i.e., saddle-node bifurcation. 13 Suppose V(fi,x)
is called the Sl phase-shift symmetry,14 and it allows the
describes a transcritical or pitchfork bifurcation at
dynamics of Eq. (5.22a) to be analyzed independently
(JU,X) = ( 0 , 0 ) . We can perturb V(fi9x) by including a
from (5.22b).
small term Vx(fi9x ) in the dynamics,
For (5.22a), we assume that at criticality (^ = 0) the cu-
x = V([j,9x) + €Vl(n,x ), (5.21) bic coefficient does not vanish,

where 0 < e « 1. The perturbation Vx may be chosen ar- fl,(0)=^0 (5.24)

13
In the presence of such perturbations the transcritical or
14
pitchfork bifurcation is said to be imperfect. A rigorous and The phase shifts in 0 are described mathematically by the ro-
systematic theory of such imperfect bifurcations can be tation group SO{2) or, equivalently, as the action of the circle
developed using the techniques of singularity theory (Golubit- group Sl. It is conventional to use the latter terminology for
sky and Schaeffer, 1985). the Hopf normal-form symmetry.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1004 John David Crawford: Introduction to bifurcation theory

then the solutions to dr /dt = 0 near r = 0 are determined F(0,0)=0 (5.27b)

by the sign of a^O) (see Fig. 12). Consider a ^ O ) < 0 for
and
example—from Eq. (5.22a) the radial equilibria satisfy
dV
r(y( i u) + a 1 (/x)r 2 )? *0 (5.25) (0,0) = 0 (5.27c)
dx
and there are two branches: r=0 and follow from (5.26b) and (5.26c), respectively. This prob-
The latter solution exists only for lem corresponds to finding the branches of equilibria in a
y(/i)>0 since rH must be real. When Eq. (5.22b) is taken steady-state bifurcation for flows, i.e., Eqs. (5.27) are
into account we see that this new solution in fact de- equivalent to Eqs. (5.1). Consequently, insofar as the
scribes a periodic orbit of amplitude rH and frequency branches of equilibria are concerned, we have precisely
fi)H«fl)(iu) + 2j 0 =i6 7 -(/x)r^. The plot of r vs r in Fig. 12(a) the cases already studied.
makes it clear that the periodic orbit is asymptotically
stable; this can be checked analytically by linearizing
(5.22a) about r = rH and determining the linear eigenval-
ue. The bifurcation diagram is also drawn in Fig. 12(a);
since the new branch of solutions is found in the direc-
tion of increasing /x, above the threshold for instability of
the equilibrium, the bifurcation of rH is said to be super-
critical.
The analysis for ax{§)>0 is similar but the results are
slightly different. Now the rH solution is found only for .<0
y(fx)<0 or jU<0. In this case the branch of periodic
solutions is subcritical and unstable 15 ; see Fig. 12(b).
Hopf bifurcation is a richer phenomenon than steady- R<
state bifurcation in the sense that it leads to time-
dependent nonlinear behavior. In an experiment, a su-
percritical Hopf bifurcation manifests itself in the spon- -H-
taneous onset of oscillatory behavior. Often this oscilla-
tion corresponds to the appearance of a wave in the sys-
tem.
H-
B. Maps

1. Steady-state bifurcation:
simple eigenvalue at + 1

The normal form is one dimensional,

xj + 1=f{fi9Xj) , juGE , x G R , (5.26a)
for y ' = 0 , l , 2 , . . . , where
JJL<0 fJL>0
/(0,0) = 0 , (5.26b)

(0,0)= + l (5.26c) \RC

dx
Let V([JL9X )~f(fiyx)—x. Then to find fixed points for /
we need to solve •H* /0

K(|i,x)=0 (5.27a)
Note that
•H-

15 (b)
There is no consensus in the literature as to how the terms
supercritical and subcritical should be defined in general, al-
though all conventions agree with my usage in this context. For FIG. 12. Radial dynamics and diagrams for Hopf bifurcation
a didactic discussion advocating one sensible set of definitions with normal form (5.22): (a) supercritical bifurcation a^O) <0;
see Tuckerman and Barkley (1990). (b) subcritical bifurcation al(0)>0.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1005

a. Saddle-node bifurcation d2V

(0,0)=0 (5.33a)
dx2
As before, this occurs if
and
dV d2V
^(0,0)^0 and - ^ -2 ( 0 , 0 ) ^ 0 (5.28) d2V d3V
ofi dx ( 0 , 0 ) ^ 0 , - 2dx- ^3 . ( 0 , 0 ) ^ 0 . (5.33b)
dfidx *~'~''
From Eq. (5.28) and our previous discussion of saddle-
node bifurcation for flows, we are led to the normal form From Eq. (5.19) we obtain the normal form,

Xj^i^€lfl-i-Xj-}-€2x:j=f(fi,Xj) (5.29) Xj + l=:X(l+€iJi + €2%j) • (5.34)

for this bifurcation in the rescaled variables (5.5) with The analysis of the branches of fixed points and their sta-
bilities yield the same bifurcation diagrams as in the
|Ao,0) = aV,-(0,0) pitchfork bifurcation for flows (Fig. 10).
£i=sgn sgn
Ofl dx'
Since the analysis of branches of fixed points for Eq. 2. Period-doubling bifurcation:
a simple eigenvalue at — 1
(5.29) is equivalent to finding equilibria for Eq. (5.6), we
need only check the stability of x + (p) = ± V — p . The
In Sec. IV we proved that this instability does not
linear eigenvalue at x± is simply
change the number of fixed-point solutions, thus any
branches of solutions bifurcating from the equilibrium
- M j 3 , * ± ) = 1 +2e2X±(pL) (5.30) will necessarily have different dynamical properties. The
dx
normal form is one dimensional and has a reflection sym-
from Eq. (5.29), hence x±(pL) is stable (unstable) if metry,
€2x±(jl) is negative (positive). Thus the stability assign-
ments for the branches of equilibria turn out to be the xj + l=f(fi,Xj) , ^GR , x £ M , (5.35a)
same as in the bifurcation diagrams for flows (see Fig. 8).
/<*i,0) = 0 , (5.35b)
The interpretation of these diagrams depends on how
we interpret the map. If we imagine that the saddle-node (5.35c)
(0,0)=-l ,
bifurcation occurs in a Poincare return map for a period- dx
ic orbit in a flow, then the branches of solutions di- -f(fi,x)=f([i,—x) . (5.35d)
agrammed in Fig. 8 correspond to mergers of periodic or-
bits. In writing Eq. (5.35b), we have made use of the fact that
the branch of fixed points X(fi) through (fi,x ) = (0,0)
b. Transcortical bifurcation must persist and have assumed a coordinate shift which
places the branch at the origin. With these properties,
This bifurcation occurs if Eq. (5.28) is replaced by the Taylor expansion of f{fi,x) at the fixed point x=0
dV takes the form
•(0,0) = 0 (5.31a)
dfi
/ ( / i , x ) = W / i ) x + a 1 ( i u ) x 3 + a 2 (/x)jc 5 + Wjc 7 ) , (5.36)
and
where A,(0)= — 1 . The trick is to notice that the twice-
d2V d2V iterated map, f2(fi,x)=f(fi,f(fi,x))9 is undergoing a
(0,0)^0, (0,0)^0 . (5.31b)
dx dfi dx2 steady-state bifurcation, which is a pitchfork because of
the reflection symmetry (5.35d) of the normal form. Fol-
From our previous discussion of the normal form (5.13)
lowing our discussion of pitchfork bifurcation, we take
for flows, we obtain
V(fi,x)=f2(fi,x)—x and check the prerequisite condi-
Xj + l=Xj(l+€1[J, + €2Xj )E=f(fl,Xj) (5.32) tions (5.31a), (5.33) using (5.35) and (5.36):

as the normal form in this case. The bifurcation dia-

- | ^ ( 0 , 0 ) = -|£(0,0) 1 + - | £ ( 0 , 0 ) = 0 , (5.37a)
grams for the branches of fixed points are shown in Fig. Ofl Ofl dx
9, and the stability assignments in Fig. 9 are also correct,
since the linear eigenvalues for x=0 and x=xb in Eq.
(5.32) are {\-\-€{pi) and (1— eji), respectively. At p = 0
!^(0,0)
3xz
= dx
f^(0,0)|Ao,0)
z
ox
1 +ox4^(0,0) =0
the two branches of fixed points merge and exchange sta-
bility. (5.37b)
and
c. Pitchfork bifurcation
/^(0 ; 0) = 2-f£(0,0)/f- ( 0 , 0 ) = - 2 - dfi
3^(0),
This case occurs if Eq. (5.31a) holds while (5.31b) is re- Ofl OX OX Ofl OX

placed by (5.37c)

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1006 John David Crawford: Introduction to bifurcation theory

2 1 bits with approximately twice the period of the original

d3V
ax3
(0,0) = ox
1^(0,0)44(0,0)
dx3 1 + M.(0,0) orbit (see Fig. 13). This leads to the terminology period-
dx doubling bifurcation.
= -12a,(0) , (5.37d)
3. Hopf bifurcation: simple complex-conjugate
respectively. Thus to satisfy Eq. (5.33) we need only as-
pair at |A| = 1
sume (dk/d p)(0)¥=0 and a ^ O J ^ O in Eqs. (5.37c) and
(5.37d); each of these two conditions is compatible with
The normal form in this case is two dimensional; how-
Eqs. (5.35) and will typically be satisfied. The normal
ever, its structure depends on subtleties not evident in the
form for the pitchfork in f2(fi,x ) is
examples of steady-state bifurcation or period-doubling
bifurcations. Denote the complex eigenvalue by
x
j + 1 : = X / (l+e 1 /X + £ 2 ^y) >
where Mja) = ( l + a ( f i ) ) e / 2 i r * 1 + * ( ' t ) ) , (5.39a)
where
6! = sgn e 2 = sgn(— ax(0)) ,
dfji 0<0<±, (5.39b)
2
with the bifurcation diagrams for fixed points of f (fi,x ) da
a(0) = M 0 ) = 0 and -==-(0)>0 (5.39c)
shown in Fig. 10. dja
These diagrams for f2(juL,x) show three branches of
If the eigenvalue at criticality X(0) = el27re satisfies the
fixed points x=0 and x=x±(ft), and we now consider
nonresonance conditions,
the implications for the original map f(fi,x ). Obviously,
the x = 0 branch is the fixed point for f(fi,x ), whose sta- W 0 ) ¥ l and W0)Vl , (5.40)
bility is lost at £4=0. The X±(fL) branches of the pitch-
fork for fHfiyX) cannot be fixed points for / ( / x , x ), since then in polar variables (r,tp) the normal form for the bi-
the implicit function theorem guarantees that x = 0 is the furcation is
unique branch through (/x,x ) = (0,0). Therefore r / + 1 = ( l + a ( / i ) ) r y [ l + f l 1 ( A i ) r ? + ©(r / 4 )] , (5.41a)
(x + , x _ ) must represent a new bifurcating branch of
2
two-cycles for f(fi,x). More precisely, denoting x± as ^ + 1 = ^ + 2 i r 0 ( l + 6 ( / i ) ) + 6 1 (^)r / + © ( r / ) . (5.41b)
x± in the original variables of Eq. (5.35), we must have
At this order in rj9 the right-hand side is independent of
X_=/(/i,X+) (5.38a) ^, a feature analogous to the phase-shift symmetry en-
countered in the normal form for Hopf bifurcation in a
X+=f(fl,X_) (5.38b) flow. In Sec. VIII, we shall show that this xp indepen-
dence depends on the nonresonance conditions (5.40). If
The conclusion that f(^i,x) must interchange x + and
these conditions are relaxed then ^-dependent terms will
x _ can be understood as follows. The fixed-point equa-
appear in Eq. (5.41); when Eq. (5.40) holds, the depen-
tion f2(fi,x+ )=x+ implies that the image of x + ,
dence on xfj will first occur in terms that are indicated as
x\=f(ti,x+), will also be a nonzero fixed point for
0 ( r 4 ) i n E q . (5.41).
/ 2 ( / i , x ) , i..e, x+=f2{fi,x'+). But we know that the
For small r, we neglect the higher-order terms in Eq.
pitchfork bifurcation for f2 produces only two nonzero
(5.41) and then solve the radial dynamics separately from
branches of fixed points, so x+ must coincide with x _ ;
the phase evolution. For this tactic to succeed, the cubic
hence Eq. (5.38a) follows. Moreover, the reflection sym-
term in Eq. (5.41a) must not vanish at criticality, i.e., we
metry of f(fi,x ) requires JC_ = ~x+ when the dynamics
require
is represented by the normal form. 16 The stability of the
two-cycle (x + yx_) is determined by the stability of x +
(or x _ ) as fixed points for f2 and is correctly indicated in
Fig. 10.
If we consider the bifurcation from the perspective
that Eq. (5.35) describes an instability of a fixed point in
the return map for a periodic orbit, then the bifurcating
two-cycle represents a bifurcating branch of periodic or-

16
In fact, the reflection symmetry of the period-doubling nor-
mal form implies that all new branches of two-cycles can be cal-
culated by solving /(/x,x) = — x; it is not necessary to consider
explicitly the second iterate of / (cf. Crawford, Knobloch, and
Riecke 1990). FIG. 13. Period-doubling bifurcation in a Poincare return map.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1007

<M0)^0; (5.42) (1990) for an introductory discussion.

Finally, we consider this bifurcation from the perspec-
then Eq. (5.41a) describes a pitchfork bifurcation at /x = 0.
tive that Eq. (5.41) describes an instability of a periodic
Only the positive bifurcating branch
orbit as viewed in the return map to a Poincare section.
1/2 In this setting the invariant circle that appears in the sec-
= a (5.43)
rH tion corresponds to a two-dimensional invariant torus in
the flow; see Fig. 14.
is relevant, since r must be non-negative. In combination
with Eq. (5.41b) the rH branch describes a circle of radius C. Final remarks
rH that is mapped into itself by (5.41), i.e., the circle is in-
variant under iteration of the dynamics (5.41). If normal forms are to be generally useful, we must
This branch of invariant circles may be either super- show that the bifurcation analysis of an arbitrary high-
critical or subcritical depending on the sign of a /a x in dimensional system can be reduced to a simple normal
Eq. (5.43). With the eigenvalue in (5.39a) assumed to be form. It is not obvious that we should be able to get as
leaving the unit circle (5.39c), we have much information in one or two dimensions as we can in
several, nor is it obvious that we will be able, even in low
( 0 ) dimensions, to find coordinates in which our dynamical
^
sgn = sgn + 0(/i 2 ) system is so simple.
fli(0)
The reduction in dimensionality is accomplished by
= sgn(/ia 1 (0)) (5.44) observing that the interesting dynamics near a bifurca-
tion occurs on a low-dimensional subset of phase space
near / x = 0 . Therefore, if ax{0)<0, the invariant circle is called the center manifold.11 The dimension of this center
found when ji>0 ( supercritical), and if ax{Q)>0, then manifold determines the dimension of the normal form.
the branch bifurcates when fi < 0 (subcritical). Using Eq. The simple structure of the normal form is established by
(5.41a), one can show that the supercritical branch is the theory of Poincare-BirkhofF normal forms.
stable and the subcritical branch will be unstable. Fur-
thermore, one can prove that these invariant circles per-
sist and have the properties just described if the &(r4) VI. INVARIANT MANIFOLDS FOR EQUILIBRIA
terms in Eq. (5.41a) are restored (Ruelle and Takens,
1971; Lanford, 1973). However Eq. (5.41b) is much less A mathematically precise definition of manifolds and
satisfactory as a description of the dynamics on the in- related geometric ideas may be found in many places, for
variant circle. According to (5.41b), the circle dynamics example Chillingworth (1976), or Guillemin and Pollack
is simply a fixed rotation by (1974). Intuitively, a c?-dimensional manifold in Rn
should be visualized as a smooth surface forming a subset
A ^ « 2 i r t 0 + M/z)) + &1(iK)r£ + 0 ( r £ ; (5.45) of Rrt. For example, a closed loop in R 2 and the surface
of a doughnut in R 3 are one- and two-dimensional mani-
In the theory of maps of the circle (Guckenheimer and
folds, respectively.
Holmes, 1986; Arnold, 1988a), it is well known that such
a uniform rotation is unstable if subjected to small per- Suppose M denotes a manifold in the phase space R n of
turbations. Indeed, with the inclusion of small i/;- a dynamical system Eq. (1.2a) or (1.2b). Let m E M be an
dependent perturbations present in the 0(r4) terms of arbitrary point on the manifold, and let Om denote the
Eq. (5.41b), we expect phenomena such as mode locking trajectory of the dynamical system through m, i.e.,
to occur in the dynamics on the circle; see Rasband jc(0) = m for Eq. (1.2a) and xj=0=m for Eq. (1.2b). If
Om GM for all m£:M, then M is an invariant manifold
for the dynamical system. More concisely, an invariant
manifold is a surface that is carried into itself by the dy-
namics.
If M C R n is an invariant manifold, then the full dy-
namics on R" implies the existence of a distinct auto-
nomous dynamical system defined on M alone, which can

17
Liapunov-Schmidt reduction is an alternative procedure for
reducing the dimension of the problem. An introduction to this
technique may be found in Golubitsky and SchaefFer (1985); the
connection between center-manifold reduction and Liapunov-
Schmidt reduction has been explored by Chossat and Golubit-
FIG. 14. Hopf bifurcation in a Poincare return map. sky (1987) and Marsden (1979).

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1008 John David Crawford: Introduction to bifurcation theory

in principle be studied independently. For example, if a x = 0 . The unstable and center manifolds may be similar-
map (1.2b) admits an invariant circle, then the dynamics ly defined by replacing Es with Eu and Ec, respectively.
on this circle is described by a one-dimensional map of We shall denote these manifolds by Ws, Wu> and Wc, see
the circle to itself, e.g., Fig. 2(c).
The stable and unstable manifolds are unique. Fur-
Oj + ^fOj) mod(27r) , (6.1)
thermore, trajectories in these manifolds have some sim-
where the angle 0 labels points on the circle. The in vari- ple dynamical properties. If x (r)G Ws, then x (r)—>0 as
ance of the circle implies that f(9) will not depend on r - > + oo; if x(t)GWu, then x(t)-+0 as t-+—<x>. This
the other phase-space coordinates. Thus Eq. (6.1) de- asymptotic behavior is indicated schematically in Fig.
scribes an autonomous one-dimensional dynamical sys- 2(c).
tem embedded in the dynamics (1.2b) on a larger phase The properties of center manifolds are somewhat more
space. subtle (Lanford, 1973; Carr, 1981; Sijbrand, 1985). In
Individual trajectories provide very simple examples of general, the center manifold is not unique; we give an ex-
invariant manifolds. In a flow, an equilibrium and a ample of this nonuniqueness below. There is no general
periodic orbit are invariant manifolds with zero and one characterization of the dynamics on Wc, not even asymp-
dimension, respectively. Much less trivial examples are totically as 111 —• oo. Nevertheless center manifolds play
the stable, center, and unstable manifolds associated with a distinguished role in bifurcation theory because of two
equilibria. 18 We first consider flows; the manifolds for important properties. We discuss these properties here,
maps are quite similar and they are discussed briefly in and in Sec. VII we state a generalization of the
subsection VLB. Hartman-Grobman theorem that justifies our discussion.
For a center manifold Wc, there exists a neighborhood
U of x = 0 such that
A. Flows
(i) if x(0)E.U has forward trajectory x(t) in U9 i.e.,
For a flow (1.2a), (1.3a), x(t)€zU for all t > 0 , then as t—*oo the trajectory x(t)
converges to Wc;
x = V(/i9x) , (6.2) (ii) if x ( 0 ) E U has a trajectory in U, i.e., x(t)E. U for
— oo < t < oo, then x (0)€E Wc and by invariance the en-
the stable, center, and unstable manifolds for an equilibri-
tire trajectory must lie in Wc.
um are generalizations of the invariant linear subspaces
E\ Ec, and Eu that arise in the linearized dynamics
One does not know in general how large U will be, only
x=DV(090hx . (6.3) that such a neighborhood exists; the situation is illustrat-
ed in Fig. 15.
These subspaces were described in Sec. II.A [cf. Eq. The first property (i) is sometimes referred to as local
(2.7)]; hereafter we denote their dimensions by ns,nc> and attractivity. Notice that there is no claim here that a typi-
nuy respectively.
For the linear system (6.3), the subspaces (2.7) are in
fact invariant manifolds. However, they are atypical,
since these manifolds are also linear vector spaces; this
special additional property reflects the linearity of Eq.
(6.3). When the nonlinear terms in Eq. (6.2) are restored,
the invariant manifolds just constructed for the linear
system are perturbed but they persist. Their qualitative
features also persist, except that the vector-space struc-
ture is lost. Intuitively, the nonlinear effects deform the
invariant linear vector spaces into invariant nonlinear
manifolds.
For an equilibrium x = 0, we have the following
definition. A stable manifold is an invariant manifold of
dimension ns that contains x = 0 and is tangent to Es at

18
There is an extensive mathematical theory of invariant mani-
folds with application to sets far more complex than the equili-
bria considered here. For a relatively introductory discussion
see Irwin (1980) and Lanford (1983); other standard mathemati-
cal references include Hirsch, Pugh, and Shub (1977) and Shub FIG. 15. Neighborhood U within which Wc is locally attract-
(1987). ing.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1009

cal initial condition will satisfy the required hypothesis; VII. CENTER-MANIFOLD REDUCTION
in particular, if there is an unstable manifold then most
points will be pushed away from Wc, Local attractivity For the various bifurcations introduced in Sec. II, the
holds only for points x(0)€zU whose orbits remain goal is to detect and analyze new branches of solutions,
sufficiently close to * = 0 for all future times. e.g., fixed points and periodic orbits. This analysis
The second property is a special case of the first and should determine their existence, their dynamics, and
provides sufficient conditions for a trajectory to lie in Wc. their stability. It is important to note that these branches
In particular, property (ii) implies that invariant sets of emerge from the given equilibrium in a continuous
any type, e.g., equilibria, periodic orbits, invariant 2-tori, fashion as [i varies near zero. For fi sufficiently small, the
must lie in Wc if they are contained in U. For this reason distance from the original equilibrium to the new solu-
one may restrict attention to the flow on Wc when analyz- tion can be made arbitrarily small. Therefore these
ing a local bifurcation; this restriction provides a setting of small-amplitude (recurrent) solutions will fall within the
lower dimension with no loss of generality. We return to neighborhood of local attractivity for Wc; hence they are
this point in Sec. VII. contained in the center manifold. This conclusion is
There is an interesting way to reformulate property (ii) correct, but the argument just given ignores a subtlety:
so that it refers only to the forward trajectory. A point the bifurcation analysis requires that we work on an in-
A: (0) is recurrent if, for any T > 0 and any £ > 0, there ex- terval in parameter space about ^ = 0, but our locally at-
ists a time t0>T such that \x(t0)—x(0)\ < e . In other tracting center manifold is defined at only a single point
words, the recurrent trajectory returns arbitrarily close fi = 0 when the system is critical. (Indeed, for saddle-
to JC(0) over and over again—forever: node bifurcation one does not even have an equilibrium
when ju is slightly supercritical.) This awkward
(ii)' if x ( 0 ) G U is recurrent and the forward trajectory discrepancy can be finessed by formally applying center-
x(t) is contained in (7, then local attractivity [property manifold reduction to the "suspended system" for Eqs.
(i)] implies x(0)G.Wc. (1.2a) and (1.2b). This extension is described in Sec.
VII.C below, and it establishes the existence of a locally
Thus one can say that the center manifold captures all lo- attracting submanifold on a full neighborhood of ^ = 0.
cal recurrence. 1 9 For the moment we shall accept the conclusion that all
continuously bifurcating branches of solutions will lie in
B. Maps an appropriately defined center manifold. Since the
center manifold is invariant, the dynamics on the mani-
The invariant manifolds for an equilibrium (1.3b) of a fold is autonomous. That is, one has an independent
map (1.2b) may be described in very similar terms. We dynamical system of dimension dimWc=nc, which de-
indicate only the necessary modifications in the discus- scribes exactly the trajectories of points on Wc. In par-
sion for flows. ticular, this reduced dynamical system describes all local
The linearized map for Eq. (1.2b), bifurcations in Wc. Our goal is to derive the equations
for this reduced dynamical system, at least approximate-
xj + 1=Df(090)'Xj ,
ly.
determines invariant linear subspaces E\ Ec, Eu that
were described in Sec. II.B. One defines the stable (Ws)> A. Flows
center (Wc), and unstable (Wu) manifolds relative to these
subspaces just as for flows. The manifold Wa(a=s,c,u) In general, the nonlinearity of a center manifold
is an invariant manifold of dimension na which is tangent prevents us from obtaining an exact analytic description
a
toE atx=0. of its dynamics. However, near the equilibrium x = 0 , it
In addition, the discussion of the properties of these is possible to accomplish this task with sufficient accura-
manifolds for flows applies to the case of maps as well, cy to obtain useful results.
with the obvious modification of replacing continuous At criticality (fi=0) for an instability, the spectrum of
time by discrete iteration. D F ( 0 , 0 ) is contained in the left half-plane (ReA,<0) ex-
cept for the critical modes whose eigenvalues satisfy
ReA = 0. Our method of deriving the center-manifold dy-
19
Dynamical systems theory utilizes various notions of re- namics does not require the absence of unstable modes,
current behavior. In addition to the recurrent points, there is however, and we shall describe the procedure without as-
the larger set of nonwandering points. A point x(0) is a suming Eu is empty. Thus consider DV(0,0) with a spec-
wandering point if there exists some neighborhood V of x(0) trum like that illustrated in Fig. 2(a), and write Eq. (1.2a)
such that for t sufficiently large the trajectory x{t) never as
reenters V. A point that is not a wandering point is a
nonwandering point; all recurrent points are nonwandering. ^-=DV(0,0)-x+N(x) (7.1)
The local nonwandering points in the neighborhood U are in dt
the center manifold. for /x = 0, where N(x) denotes the nonlinear terms.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1010 John David Crawford: Introduction to bifurcation theory

Without loss of generality we can choose variables Im

x{E:Ec, X2ELES<$EU, such that x=(xux2) and Eq. (7.1)
becomes
Re
A xx = A *xx +Nl(xl,x2) , (7.2a)
dt
A X2 ==B 'X2 \ N2\X 1 >-*-2 ' > (7.2b)
dt
where A is an ncXnc matrix with all eigenvalues on the
imaginary axis, B is an {ns+nu )X(ns-\~nu) matrix with
all eigenvalues off the imaginary axis, and iVi,iV2 are the
resulting nonlinear terms in ( x 1 , x 2 ) variables

Nx:Rn- NyMn-*Es®Eu .
(x„h(x,))

1. Local representation of W°

A center manifold associated with Ec will pass through

x=0, and at x=0 the manifold will be tangent to Ec.
This tangency means that near x = 0 one can describe Wc
as the graph of a function h(xx),

h:Ec-+Eu®Es , h(xl)=x2 , (7.3a)

FIG. 16. Geometry of the graph representation. When there
where for xx sufficiently small the point x={xl9h{xl)) are no unstable modes, a center manifold Wc may be represent-
belongs to Wc. Since x =0 is in the center manifold, we ed as the graph of a map h(x{) from Ec to Es. For the linear
spectrum shown in (a) we have the situation illustrated in (b)
require
where dim Ec—2 and dim Es— 1.
A(0)=0 , (7.3b)
20
and the tangency condition at x — 0 implies
second expression for x 2,
Dx A(0) = 0 . (7.3c)
dxi
--B-h{x\)+N2(x\Mx\)) (7.6)
The geometric interpretation of this representation is il- dt
lustrated in Fig. 16 for the particular three-dimensional
which must be the same as Eq. (7.5) if the trajectory
example ns = 1, nc =2, and Eu empty.
remains on Wc. Combining Eqs. (7.5) and (7.6) yields the
The invariance of Wc implies an equation for h(xl).
desired equation for h (x{):
Let xc(t) = (xc1(t),x2(t)) denote a trajectory of Eq. (7.1)
that belongs to Wc and has sufficiently small amplitude Dh(xlHA-xl+Nl(xl,h(xl))]
that we may write
=B-h(xl)+N2Ul,h{xi)) . (7.7)
xc2(t) = h(xcl(t)) . (7.4)
A solution to this equation that also satisfies Eqs. (7.3b)
This implies immediately that and (7.3c) determines a center manifold near x = 0 .
The dynamics on the center manifold h(x1) follows
dx^ dx\
=Dh(x\{t))- from Eqs. (7.2a) and (7.4):
dt dt
d
=Dh(x l(t))'[A-x i+Nl(x<i,h(x<i))]
c <
(7.5) X ! = A X ! +NX (X !, h (X j ) ) (7.8)
dt
from Eqs. (7.2a) and (7.4). However, (7.2b) provides a By replacing x2 with h (xY) in Eq. (7.2a) we have decou-
pled (7.2a) from (7.2b); thus Eq. (7.8) describes an auto-
nomous flow in nc dimensions. These two results, (7.7)
20
This follows from the observation that tangent vectors to Wc and (7.8), are the crucial (exact) equations required to
at (0,0) must have the form (xu0). Let s(e)~(xi(e),x2(€)) reduce a bifurcation problem to the center manifold.
denote an arc lying in the center manifold and The "invariance equation" (7.7) is in general a non-
passing through (0,0) when 6=0. Then for small 6, x2(€) linear partial differential equation for h(xx) and cannot
= h(xx{e))y and the tangent vector £(0) can be written be solved in closed form except in special cases. Howev-
s(0) = (xl(0),Dx hlOhx^O)); hence DX h(0)-x1{0) = 0. Since er, we can solve Eq. (7.7) approximately by representing
x 2 (0) is arbitrary, we must require Eq. (7.3c). h (x j) as a formal power series,

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1011

<f>(xX) = 2 0//<*l)f(*l)y

+ 2 *ijklxihlxi)j(xih+-- (7.9)

where (x i),- denotes the ith component of x x and the

coefficients <^-, 0 ^ , etc. are (ns -f wu )-dimensional
column vectors. It can be shown that if 0(JCJ) satisfies
0(0) = 0, Z>JCi0(O)=O and solves Eq. (7.7) to 0 ( J C { ), i.e.,
FIG. 17. Geometric illustration of Shoshitaishvili's theorem.
D<f){xxy[A^xxJrNx(xx,<j>{xx))) The local change of coordinates ^ in Shoshitaishvili's theorem
=B-4>(xx)+N2{xx,4>(xx)) + Q(x{) , (7.10) maps the flow (f>t of the original system onto a simpler flow ft
for which nonlinear effects are confined to the dynamics on the
then center manifold.

h(xx) = <f>(xx) + 0(xpx) as X j - ^ 0 (7.11)

eters. This theorem is also discussed in Arnold (1988a)
(Carr, 1981). It is a straightforward calculation to insert and Vanderbauwhede (1989).
<f>(xx) from Eq. (7.9) into (7.7) and solve for the Heuristically, the change of coordinates ^ "straightens
coefficients to any desired order. Examples of this calcu- out" the nonlinear manifolds of Eq. (7.1) locally; note
lation are provided in Sec. VII.A.3 and Sec. IX. that for Eq. (7.13) the invariant manifolds coincide with
the linear invariant subspaces (see Fig. 17). In addition,
2. The Shoshitaishvili theorem the flow (f>t transverse to the center manifold is linear.
A useful feature of this theorem is the information it
In Sec. II, the Hartman-Grobman theorem for hyper- provides on the stability of solutions in the center mani-
bolic equilibria demonstrated that local bifurcations re- fold. The first equation in (7.13) describes stability rela-
quired a loss of hyperbolicity. In the present notation, tive to perturbations within the center manifold, and the
the theorem applies when there are no eigenvalues on the second equation characterizes stability relative to pertur-
imaginary axis and Eq. (7.2a) is absent; then the flow of bations transverse to the center manifold. Thus for the
Eq. (7.2b) can be mapped onto the flow of original nonlinear problem (7.1) the stability to transverse
perturbations can be inferred from the eigenvalues of the
matrix B in Eq. (7.13).
The properties (i) and (ii) of center manifolds discussed
in Sec. VI.A follow from the equivalence in Eq. (7.15).
on a neighborhood of x 2 = 0. Consider the decoupled system (7.13) and suppose
For a nonhyperbolic equilibrium, the theorem was gen- x (0)€E U is an initial condition whose forward trajectory
eralized by Shoshitaishvili to allow for the effect of the x(t) remains in U. Since x(0) = [xx(0)yx2(0)], there are
critical modes (7.2a); in effect, Eq. (7.12) must be supple- two cases: if JC2( 0)9^=0 then x(0) must lie in the stable
mented by the center-manifold dynamics (7.8). manifold, otherwise the component in the unstable mani-
fold would grow without bound, forcing x(t) to leave U;
Theorem VII.l. Let <f>t(x) denote the flow of Eq. (7J) if x 2 (-0) = 0 then X{0)ELWC. In either case, the forward
and <f>t(xx,x2) denote the flow for the decoupled system trajectory will converge to Wc as t —> oo. For the second
dxi dx0 property, we assume that the entire trajectory x(t)
—L = A-xx+Nx(xx,h(xx))y —^=£x 2 . (7.13) remains in U for — oo <t < oo. Now if X 2 ( 0 ) : T ^ 0 there
at at must be components of x(0) in either Ws or Wu (or
both), which grow without bound as M—•oo. Therefore
Then there exists a homeomorphism6 the assumption x(t)€z Ufor all t requires x 2 ( 0 ) = 0, which
¥:]R"-*R r t (7.14) implies X{0)ELWC for Eq. (7.13). Because of vp, these
properties for (7.13) will also hold for the center manifold
and a neighborhood Uofx = 0 where of (7.15) described locally by h (xx).
1
<f>t(x) = ^- o^to^(x) (7.15)
3. Example
for all (x,t) such that x E U and <f>t(x)E. U.
This result was proved by Pliss (1964) in the cir- Consider the two-dimensional flow
cumstance that there are no unstable modes (nu=0).
X 0 0 X
Shoshitaishvili (1972, 1975) generalized Pliss's result to = +
allow for both unstable modes and dependence on param- y. 0 -1 y x2

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1012 John David Crawford: Introduction to bifurcation theory

whose equilibrium (x,y) = {0,0) determines Es and EC as there is no dependence on h (x) because x in Eq. (7.12) is
s independent of y.
E ={(x,y)\x=0} , (7.17a)
Ec={(x,y)\y=0} . (7.17b) B. Maps; Local representation of Wc
Note that in this example the stable manifold Ws coin-
cides with Es because x does not depend on y. The The reduction procedure for a map is wholly analo-
center manifold has a graph representation y =h(x) near gous to that just described for flows. With the splitting
(x,y) = (0,0), and the invariance equation (7.7) for this ex- of x=(y,z) where y€zEc and z£zEs@Euy the dynamics
ample is the ordinary differential equation (1.2b) becomes

dh r 3l'
yj+i^A-yj + YiyjtZj) (7.24a)
-h(x)+x2 . (7.18)
Zj + i=B'Zj+Z(yj,Zj) (7.24b)
We first calculate the asymptotic description <f>(x) as in
in a manner equivalent to Eq. (7.2).
Eqs. (7.9) and (7.10),
A center manifold for (j>,z) = (0,0) may be locally
(f>(x) = (f>2X2-+-(f)3X3-\-<f)4X*+ •* • , represented by a graph z=h(y) as in Eqs. (7.3a), (7.3b),
and (7.3c). The invariance of Wc implies that h (y) must
and obtain (f>2—ly(f>3 = 0y(f)4 = 29 so that satisfy
h(x)=x2 + 2x4+0(x5) . (7.19)
h(A-y + Y(y,h(y)))=B-h{y)+Z(y,h(y)) (7.25)
It turns out that in this example Eq. (7.18) can be by the same reasoning used before. Combining the solu-
solved exactly by the method of variation of parameters. tion to Eq. (7.25) with (7.24a) yields
Dropping x2 in (7.18), we obtain the solution to the
homogeneous problem yj + i = A-yj + Y{yj9hlyj)), (7.26)
c
h0{x)=cle-U2xl . (7.20) which describes the dynamics on W near y = 0 . In prac-
tice, the solution to Eq. (7.25) is obtained approximately
Then setting h (x)= A(x)h0(x) in (7.18) yields using power series (7.9) as before.
z
A/2x
dA
(7.21)
dx CiJC C. Working on intervals in parameter space:
suspended systems
with the solution
Let /x = 0 be the critical parameter value for an equilib-
cxA(x) = c+±f -dy . (7.22)
1 y rium undergoing either steady-state or Hopf bifurcation.
At /i = 0, there is a locally attracting center manifold Wc
Hence the solution to Eq. (7.18) is
that contains all small-amplitude equilibria and periodic
z I/JC Z e yn
orbits; these solutions can be detected by analyzing a
h{x) =_ e - l / 2 x C+- -dy (7.23) low-dimensional system on Wc. Unfortunately, the
• / ,

small-amplitude solutions of interest do not usually exist

at fi = 0; or rather, they have "zero amplitude" at criti-
The prefactor e~x/2x enforces h(x)—>0 as x-+Q and
cality. These new bifurcating solutions become distinct
h'(x)—>0 as x—>0. Note that Eq. (7.23) contains an arbi-
from the original equilibrium only for nonzero ILL, and
trary constant. Hence the solution is not unique, and in
when /x^O we have no center subspace Ec and thus no
fact the equilibrium (x9y) = (0,0) has an uncountably
center manifold to justify studying the reduced dynami-
infinite number of distinct center manifolds. However,
cal system Eq. (7.8) or Eq. (7.26) [see Fig. 18(a)].
the term ce ~ l /2x causing the lack of uniqueness vanishes
Ruelle and Takens (1971) pointed out that the reduc-
to all orders at the origin, so these manifolds all have the
tion was justified not only at /x = 0, but in fact on a neigh-
same power-series representation (7.19). One can show
that this circumstance is generally the case (Sijbrand, borhood of criticality /xG( — /z 0 ,/x 0 ), in parameter space.
1985): when the center manifold is not unique the Indeed, the notion of locally attractivity defined in Sec. V
differences are too small to be detected in the asymptotic implies the existence of such a neighborhood; the pro-
description (7.9). Thus in practice one does not worry cedure of Ruelle and Takens is to apply center-manifold
about possible non-uniqueness, since it will not affect reduction to the "suspended system." This trick works
practical calculations based on the power-series represen- equally well for flows and maps; we consider only the ar-
tation of the center manifold. gument for flows.
Finally, for this example the center-manifold dynamics It is convenient to split the variables in Eq. (1.2a) as
(7.8) is was done in subsection VILA above. Let Rn=zEc®X
c
where E is the center subspace associated with the bifur-
. 3
cation at /x = 0 and X is the subspace spanned by the

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1013

remaining eigenvectors. We choose variables x=(xlfx2)

^ = 0 (7.27c')
such that x{ ELEC and X2ELX\ then Eq. (1.2a) becomes dt
dxx and formally apply center-manifold theory to the equilib-
•Vi(/J,,xux2) (7.27a) rium (/n,xl,x2) = (0,0,0) of Eq. (7.27'). Since Eqs. (7.27)
~dt~
and (7.27') are obviously equivalent, the only virtue of
dx2
V2(fi,xux2) . (7.27b) this exercise is that certain features of Eqs. (7.27) are
dt made explicit. Note that when Eq. (7.27c') is appended,
At criticality, this splitting coincides with Eq. (7.2): the linear spectrum of Eqs. (7.27') at fi — 0 now includes a
zero eigenvalue not found in the spectrum of Eqs. (7.27)
^ ( 0 , 0 , 0 ) = F 2 (0,0,0) = 0 , (7.28a) at jU = 0; hence the center subspace EC for (7.27') at
(/i,x 1 ,jc 2 ) = (0,0,0) is larger than Ec for Eqs. (7.27), and
Z) X 2 F 1 (0,0,0)=i> J C i K 2 (0,0 > 0)=0 , (7.28b)
similarly the center manifold Wc for Eqs. (7.27') is larger
DXiVl(0,0,0)=A , (7.28c) than the center manifold Wc in (7.27) for ( x 1 , x 2 ) = (0,0)
at JJL—0. More precisely, we have WcDWc and
DX2V2(0,0f0)=B . (7.28d) d i m ^ c = d i m F F c + l . Since (fj,,xl) provide coordinates
on Ec, we can describe Wc as a graph: x 2 = A(/i,x 1 ),
Instead of applying center-manifold reduction to Eq. where h satisfies
(7.27) at /u=0, we form the suspended system for (7.27):
Dx h(jj,,xl)-Vl(fi,xl,h)=V2(/j,yxuh) (7.29a)
dxx
Vx{iiyxXyx2) , (7.27a') subject to
~~dt~
dx2 A(0,0)=0, (7.29b)
V2(fi,xux2) , (/x,x1,x2)GRXRn , (7.27b')
dt
Dx M 0 , 0 ) = 0 , (7.29c)

(0,0)=0 (7.29d)

The crucial observation is that local recurrent points

belong to Wc for /i near 0 (rather than simply at
/z = 0). Let & C l x r denote^ a neighborhood of
(/Lt,xl,x2) — (0,0,0) within which Wc is locally attracting;
the intersection of U with the fi axis defines an open set
containing /x —0. Within this open set we can find a
value of \iy denoted ^ 0 , such that the interval (— jiz0,/z0) on
the /a axis is contained in U [see Fig. 18(b)]. When
u~ -^.^ / - •

N. jT ^V
fi'EL( ~/x 0 ,/x 0 ), it follows that a given point (/LL'9X\yx2) be-
/ longs to U provided x \ and x 2 are sufficiently small. If
i
f i
such a point is recurrent, then (ii\x\yx'2)E. Wc. Fur-
\-Hos Mo/
/ thermore, since /2 = 0, the point (JLL',X\,X2) is recurrent
//
\y/
for Eqs. (7.27') if and only if (x \ ,x 2 ) is recurrent for Eqs.
s /
E '
»_ ^' (7.27). Hence, if ju'G( ~ ^ 0 , / i 0 ) , all local recurrent points
for Eqs. (7.27) belong to W c .
In addition, since / i = 0 , the center manifold Wc is foli-
ated by invariant submanifolds W £, obtained by taking a
slice of Wc at a fixed value of /z. When /i = 0, the sub-
manifold W^£=o coincides with the original center mani-
fold Wc of Eqs. (7.27), and each of these slices is of the
same dimension, d i m ^ ^ d i m f f ^ . The geometry of the
suspended system is most easily illustrated when the
equilibrium at ( x 1 , x 2 ) = (0,0) happens to persist as fi
varies near [i — O (as in Hopf bifurcation). In this case we
can modify the definition of xx in Eqs. (7.27) so that
FIG. 18. Illustration of the suspended system: (a) the center xxE.Ek where Ex is the eigenspace associated with the
manifold for the original system; (b) the enlarged center sub-
space E c and a neighborhood of local attractivity U for Wc critical eigenvalue A,, i.e., EX = EC when /x —0. Now Eq.
{Wc is not shown); (c) schematic appearance of Wc when the (7.29b) becomes A(/x,0)=0, and the manifold WC is
fixed point is not destroyed by the bifurcation. The original tangent to the subspace defined by Ex and the ILL axis as
center manifold Wc is recovered by slicing Wc at /n=0. shown in Fig. 18(c).

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1014 John David Crawford: Introduction to bifurcation theory

Finally, the dynamics on W^ is given by x'—>x that remove as many nonlinear terms as possible.
This task is accomplished in an iterative fashion. First
dxx
= V1(fi,xl9h(ijL,xl)) , /xE(-/x0,/i0) . (7.30) we remove V{2)(jLt9xf), then Vi3)(fi9x')9 and so forth. The
dt entire procedure can be understood by attempting to con-
Thus on a neighborhood of criticality center-manifold struct, if possible, the coordinate change to remove
reduction gives us the autonomous low-dimensional Vik\fi,x'), k > 2. Consider then the coordinate change
dynamical system (7.30); to rewrite (7.30) in normal form
x=Wx')=x' + 4>{k)(x') (8.3a)
requires the methods of the next section.
with inverse

VIII. POINCARE-BIRKHOFF NORMAL FORMS x' = <I>-l(x)=x-<f>{k)(x) + 0(x2k-1) , (8.3b)

where
At the conclusion of Sec. II, we remarked that the
linear spectrum determines the normal form. More pre- (f>{k):Rnc^Rnc
cisely, we shall show that the type of spectrum at critical-
ity determines which nonlinear terms are essential and is a homogeneous polynomial map of degree k; i.e., for
remain after inessential nonlinearities have been removed oGE,
by a smooth near-identity change of coordinates. We as-
fiK){ax) = K( k
kjXk)
a p \x (8.4)
sume that the problem has already been reduced to the
{k)
appropriate center manifold, and accordingly the specific Aside from Eq. (8.4), we regard <f> as unknown and try
dynamical systems we consider are one or two dimen- to determine the choice of c/)ik) that removes V{k) in (8.1).
sional. From Eqs. (8.1) and (8.3a) one has (suppressing the
For the bifurcations analyzed in Sec. V, normal-form dependence on fi)
theory is most interesting for Hopf bifurcation and dx die'
period-doubling bifurcation. For steady-state bifurca- ^=D<S>(x')^=D<P(<Z>-l(x))V(<Z>-l(x)) . (8.5)
tion, the lowest-order nonlinear terms are in fact essen- dt dt
tial, and no particular simplification results from per-
forming normal-form transformations of the type con- Now using the expansions
sidered here. For this reason, after developing the nor- V(<Z>-l(x))=V(x-(f>{k)(x) + &(x2k-1))
mal form procedure we work out the application to Hopf
bifurcation and period-doubling as examples. Finally we = V(x)-DV(l)(x)-<f>{k)(x) + 0(xk + l) ,
describe some recent theoretical work that explains why
(8.6a)
normal forms often have greater symmetry than the orig-
inal dynamical system. D*(*""1(jc))=/+Z)^(*)(*""1(jc))
A. Flows =I+D(f>{k)(x) + 0(x2k~2) , (8.6b)
{k)
we rewrite Eq. (8.5) keeping terms involving cf> up to
1. Generalities
&(xk)9
Center-manifold reduction yields a flow dx
dt V(x)-L((f>{k)) + 0(xk +l
), (8.7a)
dx' Kl K (2), l)
V(n,x')=V Kp,x')+V (v,x') where
dt
x'GR c
, L{^k)){x)=DV{l\x)^{k\x)-D(f){k\xyV{x\x) . (8.7b)
+ . . . +VKK)(VL,X')+ •
(8.1) Our notation is chosen to emphasize that the new terms
where nc=dimE c
and V \n,x') {k
represents all terms in of &(xk) in Eq. (8.7a) are linear in <f){k) and have the form
of a linear operator L acting on <f>{k). Note that L,
the Taylor expansion of V(fi,x') of order k in x'. For ex-
defined by Eq. (8.7b), depends only on the linear term
ample, at a Hopf bifurcation nc~2 and
V{l)(x) of the original flow.21
y(/x) ca(fi)
K U W )= — coifi) y(fi)
(8.2) 21
In connection with Eqs. (8.7b) and (8.8) there are a variety of
characterizations in the literature. The linear operator is sim-
is the appropriate first-order term. For simplicity we ply related to the usual Lie bracket of the two vectorfields(f>{k)
have assumed there is no constant term on the right in and V(l\ i.e., L(<f>{k))=-[V{l),<f>{k)]. Arnold (1988a) refers to
Eq. (8.1); this need not be true for steady-state bifurca- Eq. (8.8) as the homological equation associated with the linear
tion, but as already mentioned the application to steady- operator DV{1)(x). Guckenheimer and Holmes (1986) write the
state bifurcation is not of great interest. Lie bracket as adF (1) (0 U) ), since this vector field is induced
Given Eq. (8.1) our goal is to simplify V(/n,x') by per- when Vn) acts on vector fields through the adjoint representa-
forming near-identity nonlinear coordinate changes tion (cf. Olver, 1986).

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1015

To remove all terms of &(xk) in Eq. (8.7a) we must To make this interpretation precise, we go back to Eq.
solve (8.4) and define ftik)(Rn),

Vik\x)-L(d>ik))=0 (8.8)

for <)>{k)(x). Formally, this is easy, ft{k)(Rn)=[<f>:Rn-+Rn\<f>(ax) = ak<f>(x) for all oGR) ,
(8.10)
<f>{k\x)=L~HVik)(x)) , (8.9)
the space of all homogeneous polynomial maps on Rn of
but our solution is only sensible if L ~l is well defined. degree k. For fixed k and n, 3i{k)(Rn) is a finite-
The task of finding L~l, if it exists, is a problem in dimensional linear vector space. The vector-space struc-
finite-dimensional linear algebra. That is, L in Eq. (8.7b) ture is obvious, and an example serves to make the finite
may be viewed as a finite-dimensional matrix, and L ~ l is dimensionality clear. Consider fti2)(R2) with coordinates
well defined if and only if detL^O. (x^)GR 2 ; then any 0U,j;)ej¥ ( 2 ) (K 2 ) may be written

2
ax2 + bxy + cy;
<f>(x,y)-- dx2 + exy+fy2

\x2 xy y2 0 0 0
a + Z> 0 + C 0 +d +e +f y2\ (8.11)
[o x2 xy

Obviously, 3i{2){R2) is six dimensional, and one possible

choice of basis is given in Eq. (8.11). L(pk)(x)) = aj<l>< {k)
« l(x)
vl- <f>(K)(x)
/=i
In terms of the spaces !H{k\ Eq. (8.4) asserts
<f>ik)(=ft{k)(Rnc) and (8.7) implies
PK)(x) (8.12)
{k) nc {k) nc 7= 1
L:ft (R )-+ft (R ) ;
thus L is a linear transformation acting on a finite-
dimensional vector space. Once a basis for 31{k)(R c) is hence vectors <f>{k)(x) of this form are eigenvectors, and
chosen, then L can be written down in matrix form. the eigenvalues have the form [^y~"X7 = i a / <J 7]- If w e
Any convenient basis may be selected, since det L is in- can satisfy the condition
dependent of this choice. Recall that detL^O means
:
that <f>{k) in Eq. (8.9) is well defined and the resulting X«i C 7 7 (8.13)
7= 1
change of variables (8.3a) will remove all terms of order k
in Eq. (8.7a). More generally, however, one finds that for any choice of j and a then L has a zero eigenvalue.
detL depends on k and on whether or not the system is One can see by inspection that when critical eigenvalues
critical; at criticality there will be values of k such that (Recr=0) occur there are always choices of a which satis-
detZ, = 0 because L has at least one zero eigenvalue. fy the "resonance condition" Eq. (8.13). We analyze the
Since L is given in terms of DF (1) (0), it is reasonable to case of imaginary eigenvalues in Sec. VIII.A.3 below.
investigate what the condition detL = 0 implies about the In the presence of such zero eigenvalues, the range of
eigenvalues of DV{1)(0); this is most easily done if we as- L, denoted L(ftik)(R c )), is a proper subspace of
sume that DVn\0) can be diagonalized with eigenvalues 3f{k){R c ), and we can specify a complementing subspace
(al,a2, . . . >(Tn ). Once Eq. (8.7a) has been written in C{k) so that
coordinates that diagonalize Z)F (1) (0), then the eigenvec-
tors of L are easily found. Let <f>ik)(x) have only a single 31{k\Rnc)=L(3t{k\Rnc)) + C (k) (8.14)
nonzero component [cf. Eq. (8.11)], (^ (fe) (^)) / =S/ / 0j A:) (x),
which we take to be a A:-degree monomial: ^){k\x)—xa Once C{k) is chosen (and the choice is not unique) then
where j' = 1, . . . , nc labels the component and the multi- the kth order terms V{k) in Eq. (8.7a) may be split ac-
index22 a is arbitrary except \a\ =k. Then applying L to cordingly: v{k)=Vl.k)+V{ck) with V{rk)<=L(3i{k)(Rnc))
<f>{k\x) gives { k) {k)
and V c E:C . The component in the range can be re-
moved,

(KK))
22 A<*) = -1 •<A:)\
(8.15)
In this notation, a = (ai,a2,... ,a M ) denotes an «-tuple of
non-negative integers and xa=EXiX2 * - xnn. In addition, we
define notation | a | = a 1 - f - a 2 + • • • +an and, for future refer- leaving behind the "essential" nonlinear terms at order k
ence, a! = a1!a2- ' ' ' an\. namely V{ck). In this way <£(2) is first specified, then (f>{3),

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1016 John David Crawford: Introduction to bifurcation theory

and so forth so that one generates a power series23

y co
representing the desired normal-form transformation to Z,<^*') =
-co y
all orders:
(k)
x=&(x')=x' + 4>U)(x') + (l> K )
fc(3)/
* (x')+ ' (8.16) Wxk) *4>\
dx dy yx-\-coy
yx —coy
(8.18)
The normal form resulting from this procedure has the
dx dy
structure
dx As noted in Sec. VIII.A.l, to determine the eigenvalues
V{1)(x)+V{c2)(x)+V{ci\x) + • (8.17) of L it is more convenient to use the complex coordinates
dt
{ZyD that diagonalize DV{x){0), i.e.,
if at order k, d e t L ^ O , then of course V{ck)~Q.
There is an important subtlety in this procedure. The
series Eq. (8.16), representing the transformation ®(x') z 1 / X X
— 1 =S- (8.19)
required to put the original flow Eq. (8.1) into normal z — I y y
form to all orders, typically diverges. Thus, while we can
describe which terms can be removed at any given order, so that
the change of variables required to remove them to all or- 1 1
ders does not generally exist. In practical applications 5 - = l
—/ /
one implements the transformation to normal form only
up to some finite order, and this finite-order approxima- and
tion to the original flow Eq. (8.1) is studied. The possible
effects of the neglected higher-order terms can then be r CO
1 _
y — ico 0
considered in reaching final conclusions. — co r \s 0 y-\-ico

In terms of (z,z~), the same vector (f>{k)E.y^ik)(M2) is reex-

pressed
2. Steady-state bifurcation on R
\</>{zk)(z,z)

* " [*<*>U,z)
For steady-state bifurcation with a simple zero eigen-
(f>{xk)(x(z,z),y(z,z))
value, nc = \ in Eq. (8.1), and V(fi,x') has the form de-
scribed in Eq. (5.2). If we try to simplify Eq. (5.4) by ap- =s- ^}(x(z,z),^(z,z))
plying the coordinate change Eq. (8.35) to remove the x2 k :„
term, then the required change of variables is singular at and the action of L on </> is
criticality (/i = 0). For this reason, the method of KM
Poincare-Birkhoff normal forms is not particularly useful \y — ico 0
,<*h =
L(0<*') 0 y-hico k(k)
in this case. A similar limitation holds for steady-state
bifurcation in maps.

dz dz (y-ico)z
3. Hopf bifurcation on R 2 (8.20)
{y-\-ico)z

Generically for Hopf bifurcation, ftc—2 in Eq. (8.1), dz dz

and we take V{l)(fi,x') as given by Eq. (8.2). Let By inspection from Eq. (8.20) we see that the eigenvec-
(x,j;)GR 2 denote the coordinates, then ^{k)^3^k)(R2) tors have the form ((/>z,0) or (0,<£_), so we introduce the
has the form vectors

\zlzk~l
p*}(x9y) = K)
Vy (xyy) J ' [ 0
/=0,l,...,/c , (8.21)
0
{k)
and in these variables L((f) ) is expressed as zlzk~l

which are eigenvectors of L and also provide a basis for

yi{k)m2). From Eq. (8.20) we calculate the eigenvalues
23
Because of the iterative process used to construct <E>, the full
series in Eq. (8.16) is not of the form x,-\-^k>l<f>ik)(x'). L{&») = ti£l\n)&l) , (8.22a)

Rev. M o d . Phys., V o l . 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1017

where ance of Eq. (8.26) under 0 - > # + <£, illustrates this point.
( l) Note that this symmetry was not assumed to hold for the
k ^ (fx) = (l-k)Y(fi)-ico(fi)(k-2l±l) . (8.22b)
original vector field Eq. (8.1); rather, it is introduced by
Since det L = 0 implies at least one zero eigenvalue, the normal-form transformation Eq. (8.16). As already
and a zero eigenvalue in Eq. (8.22b) requires that real and discussed, the normal-form procedure is formal in the
imaginary parts vanish separately, we must satisfy sense that Eq. (8.16) may not converge if carried to all or-
ders. When the series diverges, then a symmetry intro-
(l-k)y(fi)=0 (8.23a) duced by Eq. (8.16) describes only an approximate prop-
(k-2l±l)(o(fi)=0 (8.23b) erty of Eq. (8.1), even though it is exact for the normal
form.
{ !)
to obtain X ^ = 0. Because k > 2 , Eq. (8.23a) fails unless In the case of Hopf bifurcation we constructed the nor-
y ( / i ) = 0 , which requires that we are at criticality ytx=0 mal form Eq. (8.16) first and then noted the phase-shift
[recall Eq. (5.23)]. At fi=0,co(0)=£0, so Eq. (8.23b) re- symmetry. This order can be reversed; the theory of nor-
quires k— 2 / ± l = 0 . Since 2/ + 1 is odd, for k even we mal forms can be formulated by identifying the relevant
will never satisfy Eq. (8.22b), and for k odd there are ex- symmetry first and defining the normal form by its sym-
actly two null eigenvectors at criticality, metry. The advantages of this second approach were
noted by Belitskii (1978, 1981), Cushman and Sanders
Uk9(k+\)/2) —
:\z\k~l (1986), and Elphick et ah. (1987). The results of Elphick
et al (1987) are clearly discussed in Golubitsky, Stewart,
fc=3,5,7,.. . . (8.24) and Schaeffer (1988), whose presentation is summarized
t(k,(k-l)/2) — here.
\k-\ The key result is that the complementing subspace Cik)
z\z
in Eq. (8.14) may be defined by a symmetry T that is
These two vectors are a natural basis for the complement determined by the linearization at criticality; i.e.,
C(/c)totherangeofZ,, D F ( 0 , 0 ) . More precisely, let M=DV(0y0) and MT=
T
(transpose of M). Then M generates a one-parameter
C (fc) = s p a n [ ^ ( f c + 1)/2)
,^^-1)/2)}fc=3,5,7. . . .
group of transformations with the obvious multiplication
The implication for Eq. (8.1), written in complex coor- rule
dinates Eq. (8.19), is a normal form with all even non-
exp(slMT)exp(s2MT) = exp[(s1 +s2)MT] .
linear terms removed,
.*\*\V The closure of this one-parameter group defines the
y — ico 0
normal-form symmetry 2 4
0 y + ico +2 (8.25)
7=1 r={exp(5Mi)U< (8.27)
l
Rewriting Eq. (8.25) in polar variables, z = re , yields
Let Jl^XW1) denote the subspace of j ¥ U ) ( E w ) compris-
2y ing those maps with T symmetry, i.e., those
r(M)+2>;'' ' (8.26a)
7= 1 Vik)(x)Gftik)(Rn) such that

e = o(ti)+2,bjr2J , (8.26b) V[k)(Qxp(sMI)^x) = cxp(sMI)^V{k)(x) (8.28)

7= 1
for all s G R .
where a 7 = R e ( a y ) and bj = — I m ( a y ) . This is precisely We shall prove that fi^ may be taken as the comple-
the normal form introduced in Eq. (5.22). ment C{k) to the range of L so that Eq. (8.14) becomes

4. Normal-form symmetry
wk)(R c /(*), k)mc))®^f]mc).
)=L(^ (8.29)

In words, this splitting implies that the normal-form

Although normal forms may have fewer nonlinear transformation Eq. (8.3a) can remove all A:th-order terms
terms, the discussion above does not explain why this except those with T symmetry.
should simplify the nonlinear analysis. For example, the The argument relies on a clever definition of inner
one-dimensional logistic map has only one nonlinear product on ¥£ik)(Rn). This definition is based on the fol-
term and the Lorenz equations have only two nonlinear lowing product for monomials: for x GM", let xa and x@
terms, yet the immense dynamical complexity of these denote two monomials in multi-index notation and define
two systems is well known.
There is a more intrinsic explanation for the practical
utility of normal-form theory: normal forms can have
greater symmetry than the original system, and this
24
makes them simpler and therefore useful. The phase- By defining T as a closed group of matrices we ensure that it
shift symmetry of the Hopf normal form, i.e., the covari- is a Lie group.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1018 John David Crawford: Introduction to bifurcation theory

which can be conveniently rewritten as dM(AT-xf

[xa,(AT-xf]--
dxa x=0
[xa,x*] = (8.30)
dxa x=0

This bracket extends to polynomials in the obvious way: dx x=0

letp(x):=^apaxa Sindq(x) = ^f3qpx^, Then a p
••[{A'x) ,x ] ; (8.40)
[p(x),q(x)] = ^paq/3[xa,x13]
in the second step the change of variables y = A -x and
the chain rule,
dlalg(x)
Z
HPa (8.31)
dxa x=0

Finally, given < ^ £ j # u ) ( m , we define their inner prod- were used to justify the substitution 25
uct by
$a\AT*xf
<4>,il>)=2,[<t>j(x)9it;j(x)] , (8.32) dxa dy

where <£y- and ^ are the yth components of (/> and ^, re- With Eqs. (8.39) and (8.40), the second identity (8.37) fol-
lows directly.
spectively.
These identities are applied by choosing
At criticality, the operator L in Eq. (8.7b) becomes
AT=exp(— sM) in Eq. (8.36) and A T=exp(sM) in Eq.
LM(<f>)(x)=M-<f>(x)-D<f>(x)-M-x , (8.33) (8.37) to obtain

(<!>(x),e-sMit>(esM-x)) = (e -sM1 <f>{e

±( „sMs
-x),xfr(x)) .
where LM has been written with a subscript to emphasize
the dependence on M. Given the inner product (8.32), we (8.41)
define the adjoint of LM as that operator satisfying
By differentiating Eq. (8.41) with respect to s at s — 0, we
(Llf<f>^) = (<(>9LMif>) (8.34) finally arrive at
for all <f> and jp in ^ U ) ( R M ) . We shall determine LJ, as ((/>(x),LM(tf;)(x)) = (LM7{(f>)(x),xlj(x)) , (8.42)
T T
Lif((f))(x)==M -<l)(x)-D(f>(x)-M -x which establishes Eq. (8.35).
The argument leading to Eq. (8.29) can now be summa-
L„M*x) (8.35)
rized. The vector space Jitik)(Rn) is first written as the
by applying two identities, direct sum of the kernel of L^ and the orthogonal com-
plement of the kernel:
(4>(x),AT-J>{x)) = {A-4>{x),xl>(x)) , (8.36)
j¥U)(R") = (kerLj/)ie(kerL^) ; (8.43)
{<f>(x), i{>( A T-x ) ) = < <f>{ A x),xf>(x)) , (8.37) then the Fredholm alternative 26 for LM implies
(kerLl)i=L(^(*)(R'1)) and Eq. (8.35) implies
that hold for any linear transformation A :R"—>-R". The
kerL^f —kerL T. Thus Eq. (8.43) may be reexpressed as
first identity follows immediately from Eqs. (8.31) and M
(8.32): {k
ft \R ) n
= L{ft{k\Rn))®(kerL1iyfT) . (8.44)
M1

Finally, with the aid of the identity

<<f>(x),AT-4>(x))= 2 [<l>j(x),(AT)ji<l>i(x)]
^(esMr^(e-sM^x)) = esMTLMj{cf>)(e-sM^x) , (8.45)
= 2 [Au+jMMx)]
we can identify k e r L M r with 9^\tLn). If ^ G J ^ R " ) ,
= {A-4>(x),i>(x)) . (8.38)

For the second identity, we express the j'th component of

25
<f> as 4>j{x) = ^ia(pjaxa [and similarly for tf>j(x)], so that Here
the left-hand side of Eq. (8.37) becomes a a
\ a2
8 8 8
A A A
1 A
dyl{ 2U^ n 3yln
T a
((f>(x),tlj{A -x))='^Jl(f>jyaxf>jJx ,(A -xf} T
. (8.39) dy ^ l
2

(summation on repeated indices).

26
Then with Eq. (8.31) we have See Stakgold (1979), pp. 321-323.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1019

then the left-hand side of Eq. (8.45) vanishes, which im- fKK)(x)-L(pK))(x)(k)\ =0 , (8.51)
plies LMj<f)=z0; hence <J)EL\LQTLMT. Conversely, if -1
by constructing L . When d e t L = 0 , there are zero ei-
<^EkerL M r, then the right-hand side is zero and the left-
genvalues, and some nonlinear terms cannot be eliminat-
hand side must be independent of s. This implies ed. As for vector-field normal forms, if we assume coor-
T
^sM1 >4>(e -sM . x)~<f>(x) , (8.46) dinates can be found which diagonalize Df{1)(0), then
the vectors (f>(k)(x) having a single monomial compo-
since <f>(x) is the value at s = 0 ; hence ^ e ^ ^ t R " ) . Thus nent <f>(jk)(x)=xa will be eigenvectors for L. Let
kerLMT=ft{r)(Wln), and Eq. (8.29) is established. (al,a2, . . . >crn ) denote the eigenvalues of Dfil)(0).
Note that when M can be diagonalized then we may Then we find
assume MT— —M and consequently k e r L M r = k e r L M . In
this case, the definition of T can be based directly on M; L((f>{k))(x) = aj(l>(k)(x)~aa(f>{k)(x)
it is not necessary to use the transpose. = [aj-aa](t>{k\x) (8.52)
In our example of Hopf bifurcation, the linearization
at criticality gives from Eq. (8.50b), where aa = alla22 ' ' ' crn c; hence for
0 co(0) maps the resonance condition required for a zero eigen-
M1 (8.47) value is
~<o(0) 0
(8.53)
so that an element of T has the form
for some choice of j and a.
cos(sco(0)) — sin{sco(0)) When zero eigenvalues occur, then the nonlinear terms
expisM )— (8.48) that cannot be removed may be characterized by their
sin(sco(0)) cos(sco(0))
symmetry. Let M=Df(l)(0) denote the linear map at
As expected, this identifies the normal-form symmetry criticality [cf. Eq. (2.10)] and define the group generated
for Hopf bifurcation as rotations in 0 or F=Sl. Note byMr,
that for steady-state bifurcations Z)F(0,0) = 0, so the as-
sociated r in Eq. (8.27) is trivial, consisting only of the r={(Mr)rt|n=integer) , (8.54)
identity matrix. This explains why Poincare-Birkhoff so that ft^HR") now denotes elements of ft (R ) with ik) n
normal-form methods do not significantly simplify the symmetry (8.54); i.e., (f>(x)^^)(mn) requires
analysis of a steady-state bifurcation. T T
M -<f>(x) = (/)(M -x). With T and 3i^ redefined in this
way the proof that 5¥ U ) (K c ) may be expressed as

B. Maps j ¥ U ) ( R c)=L(ft(fc)(R c
))<BWf}(R c
) (8.55)

is quite similar to the argument leading to Eq. (8.29).

1. Generalities With
ik)(
On the center manifold we find a map that may be LM((/>(k))(x)=M'(l>[K}(x)'-(t> (k) hiM(
(M'X (8.56)
written denoting the operator L [cf. Eq. (8.50b)] at criticality, the
{l i2) identities (8.36) and (8.37) imply L^=L T* Therefore
xj+l=f(xj)=f \xj)+f (x})+'- ,
(kerLl)L=LM(ft{k)) and k e r L ^ = k e r L'M^1 r hold as be-
:E (8.49) fore, and we obtain
in a notation modeled on Eq. (8.1). We suppress explicit ftik\Rnn=LM(ftik)(Rnc))®kerL (8.57)
parameter dependence and ignore constant terms as be- M1
fore. The goal remains the same: remove f{k\x')> if pos- by the same reasoning that led to Eq. (8.44). It is only
sible, using the change of coordinates Eq. (8.3). In the necessary to check that k e r L M r = ^ / ^ : ) ( R n c ) still holds.
new variables (unprimed) we find
This follows by noting that ^ G k e r L ^ ^ i f and only if
xj + l=f(<Z>-l(xj)) + (f>{k)(f(<S>-l(Xj)))
MT-(j){x) = (j){MT-x) , (8.58)
(k) k + l
=f(xj)-L((f> )(xj) + &(x ) , (8.50a)
which in turn is also necessary and sufficient for
where now L is defined by
The splitting (8.55) has the same significance here as in
L(<f>{k))(x)=Df{l\0)^(f>ik\x)-(f>{k)(f{l)(x)) . (8.50b)
the vector-field case: only when T defines a nontrivial
Note that Eq. (8.50b) differs crucially from (8.7b) in the symmetry should we expect the Poincare-Birkhoff nor-
second term; nevertheless we are again seeking to solve mal form to be simpler than the original map. In addi-
an equation of the same form, tion, the normal form for the original map (8.49) will be

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1020 John David Crawford: Introduction to bifurcation theory

f(x)=f{l\x)+f<2\x)+f{c3)(x) + ••• (8.59) Since A,(0)= — 1, eigenvalue A,(l— A,*""1) will vanish at
criticality when k is odd; thus terms of even degree can
where f{ck\x) e ^ r f c ) ( R"C ). be removed, and only odd terms,
-§(*) = £ ( - x ) ,
2. Period-doubling bifurcation on R1
will remain in the normal form. If we consider the ex-
Typically nc = \ for a period-doubling bifurcation, and pected symmetry T in Eq. (8.54), then
Eq. (8.49) is a map in one dimension with M=Dxfil)(0,0)=-I so r = Z 2 ( - I ) , the two-element
group on R generated by —/. Thus we are again led to
f{1)([i,x) = k(n)x , (8.60) the conclusion that for period-doubling the normal form,
where
Xj + l=k(fx)Xj[\+axxf+a2xf + 0(x6)] , (8.62)
wo)=-i, 4^(°)<°.
for Eq. (8.49) will have a reflection symmetry as claimed
The space ^/ (/c) (K) is one dimensional for all k, and the in Eqs. (5.34) and (5.35).
single basis vector,

gkXx)=xk 9
i s ' a n eigenvector for L:7£ik)(R)-+Hik)(R); from Eq. 3. Hopf bifurcation on :
(8.50b) we find
As for flows, one expects nc=2 for Hopf bifurcation,
L(£{k)(x)) = k(tiKl-k(fi)k-l)£{k)(x) . (8.61) and with coordinates (x,y) on R 2 we have for 0 < 6 < \

[l+a(ju)]cos2ir0(l+&(/*)) -[l+a(/i)]Mn2w0(l+Mji))
/ ( 1 W)= [l+a(jL4)]sin27T0(l+&(**)) [l+a(ja)]cos2ir0(l+&(/*))
(8.63)

in Eq. (8.32), where a (fj,),b(/j,) satisfy the assumptions in Eq. (5.39). At criticality, a (0) = 6 ( 0 ) = 0 , so the expected sym-
metry (8.54) will be generated by

coslrrO —sinlwd
M = smlird coslirO (8.64)

the rotation matrix for the angle 0 determined by the critical eigenvalues.
As before it is convenient to introduce complex coordinates (8.19) so that (8.63) becomes
Uii) 0
sDr^s'1^ o MJT) (8.65)

where A.(|x) = [ l + f l ( | i ) > / 2 ^ I 1 + 6 ( ' i ) ] . From Eq. (8.50b) we obtain

k 0 (f>ik)(z,z) <f>{zk)(kz,kz)
(k)\ —
L(<p*>) (8.66)
0 k 4 f e ) (z,z) <f>ik)(kzykz)

The eigenvectors of L are again given by Eq. (8.21), and (8.65) yields
(8.67a)
where

(8.67b)

By inspection detZ^O unless fi = 0, in which case (a) 6 irrational.

^,/)(0) = 0ifandonlyif To satisfy Eq. (8.68) requires
exp[-i2ir0(Jfc-2/±l)] = l . (8.68) e(k-2l±l) = m , (8.69)
The solutions (k,l) to Eq. (8.68) vary depending on with m an integer, and when 0 is irrational we must have
whether 6 is irrational or rational. k— 2 / ± 1 = 0 . This leads back to the null eigenvectors

Rev. Mod. Phys.f Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1021

(8.24) found for the Hopf normal form for flows. The re- When 0 is rational, the symmetry T of the normal
sulting normal form in this case is form is reduced t o T=Zq, the discrete subgroup of S1
generated by rotation through 2TT/#. For the cases of
(8.70) "strong resonance," q = 3 and q—4, we are thus led t o
MAx)+2«Jf/l
/=i study maps that are covariant under rotations by 27r/3
and TT/2, respectively, and the structure of the bifurca-
In polar variables, z = re'% we have
tion is much richer (Arnold, 1988a). [In particular, for
q=4, there are at least 48 different local phase portraits
r / + 1 = [l+a(/x)]r/ 1 +* =2i ^ (8.71a) possible (Arnold, 1989).]

^ +1 = ^+2ir0[l+6(Ai)]+2Vf» (8.71b) IX. APPLICATIONS

2 = 1

where aiz='R&ai and helmet;. This agrees with Eq. The normal-form equations provide the most elemen-
(5.41) in Sec. V. tary examples of the bifurcations we have considered.
The fact that the dynamics of the amplitude (8.71a) However, in practice lengthy calculations may be neces-
decouples from the phase (8.71b) reflects the symmetry sary to extract the relevant normal-form coefficients from
r . For 6 irrational, the matrices, the initial equations expressed in physical variables. I n
this section we analyze bifurcations in two equations that
cos27ra 0 — sin2777* 0 illustrate both the power of center-manifold reduction
Mn-- (8.72) and the computations required to obtain detailed predic-
smlnn 0 COSITTH 6
tions for specific problems. In addition, each of these ap-
for all integers n> provide a dense subset of the group of plications illustrates new features of the theory that can
rotations in the phase. Thus T=Sl is precisely this rota- arise when one encounters equations that have symmetry
tion group and corresponds to the phase-shift symmetry or that depend on more than one parameter.
of the normal form (8.71). The first problem considers a simplified model in plas-
ma physics for the three-wave interaction between an un-
(b) 0 rational. stable plasma wave and two damped waves. The ampli-
Let 0=p/q with 0<p/q <\ where the integers p and tude equations for the waves lead us to a Hopf bifurca-
q are relatively prime. 2 7 Now in addition to the solutions tion in a three-dimensional flow that depends on two pa-
k— 2 / ± l = 0 for Eq. (8.69) we have another set of solu- rameters. The calculations required t o obtain the Hopf
tions represented by normal form (5.22) are carried out in detail. Because this
model contains two free parameters, the cubic coefficient
k-2l±\=nq , n=±l,±29. (8.73)
<z, evaluated at criticality [cf. Eq. (5.24)] is a function of
so that pn=m. We are primarily concerned with solu- the remaining parameter. By varying this additional pa-
tions t o Eq. (8.73) that introduce new low-order terms rameter we are able to locate a degenerate bifurcation in
into the normal form (8.71). Examination of different which t h e nondegeneracy condition (5.24) fails, a n d
cases for (8.73) shows that if q = 3 or q=4 then we get higher-order terms in the normal form must be con-
new terms at quadratic and cubic orders, respectively. sidered. This degeneracy allows us to detect and analyze
For q > 5 the new terms in the normal form are at least a secondary saddle-node bifurcation for the Hopf limit
fourth order and can be shown t o have negligible effect cycle.
on the analysis of Sec. V. The low-order "resonant" In the second application, we study steady-state bifur-
terms are as follows: for # = 3, cations in the (real) Ginzburg-Landau equation. This
analysis illustrates center-manifold reduction for bifurca-
z2' 0 tions in infinite dimensions, i.e., for a partial differential
# ° ' ( z , z ) = 0 , £(32)(z,z) = z \ equation. Because the Ginzburg-Landau equation is rela-
tively simple we are able to calculate not only the initial
tull eigenvectors, and for q = 4, bifurcation from the "trivial" equilibrium but also the
z3' 0 |
secondary bifurcations from the resulting "pure-mode"'
&°\z,z) = 0 , £ ( i' 3) Uz) = 3 solutions. These secondary bifurcations are the mecha-
z ] nism for the Eckhaus instability, which plays an impor-
are t h e null eigenvectors. Provided q¥z2>94 [or, tant role in the theory of spatially extended pattern-
equivalently, assuming t h e nonresonance condition forming systems (Eckhaus, 1965). The center-manifold
(5.40)], t h e normal form u p t o third order is given reductions in this case are complicated by the fact that
correctly by Eq. (5.42). the Ginzburg-Landau equation is highly symmetric. In
the simplest case—one dimension and periodic boundary
conditions—the symmetry group is 0 ( 2 ) X 0 ( 2 ) . A l -
27
Two integers are relatively prime if they have no common though one typically expects one-dimensional center
divisor besides 1. manifolds at a steady-state bifurcation, in this example

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1022 John David Crawford: Introduction to bifurcation theory

the initial bifurcation has a four-dimensional center man- dence on (ft, r ) implicit. The divergence of this family is
ifold, and the secondary bifurcations lead to two-
divK=2(l-D . (9.2)
dimensional center manifolds. In each case the symme-
try forces the zero eigenvalue to have multiple eigenvec- 3
For F < 1 the flow expands volumes in R and there are
tors (four and two, respectively), and this multiplicity no stable bounded solutions; for T > 1 the flow contracts
leads to larger center manifolds, volumes (Verhulst, 1990). Since the equations are un-
changed by the shift (ft,j>)—>( —ft, — y), we may assume
A. Hopf bifurcation in a three-wave interaction ft to be non-negative.

Nonlinear plasma theory involves, in part, an analysis 1. Linear analysis

of the interactions between the various waves supported
by the plasma. In this example we examine the satura- K h a s two fixed points. There is a trivial fixed point at
tion of a linearly unstable plasma wave via a "three-wave (x,y,z) — (0,0,0) corresponding to no waves; this solution
interaction" in a simplified model considered originally is unstable, since the high-frequency wave is unstable.
by Vyshkind and Rabinovich (1976), Wersinger et ah There is a nontrivial fixed point at
(1980), and others. Physically, an unstable high-
2
frequency wave is coupled to two damped waves of lower
(*o>.Vo>zo) — •r, -nr , r i+- n 2
frequency. Under suitable conditions an overall balance (2r-i)
results between high-frequency growth and low-
2r-i
frequency decay. This produces a stable equilibrium in (9.3)
the dynamics of the three waves, and the wave ampli- whose stability depends on il and T. If we shift the ori-
tudes are time independent. If, however, the parameters gin to (x0,y0,z0), x=x'+x0, y=y'+y0, z=z'+z0 and
of the interaction are varied to produce less damping or drop the primes, then
less effective coupling, then this stable balance is des-
troyed and some form of time-dependent state emerges. (l+/i)p 1" X \-y2\
d_
For the model considered here, this transition occurs via l-/i 0 y +2 xy
dt
a Hopf bifurcation. In addition, if the damping of the l-/x(l+p 2 ) 0 0 z —xz t
stable modes is decreased sufficiently, one expects on
physical grounds that they may fail to arrest the growth (9.4)
of the unstable mode. In the model, this failure is where \x^2T and p = ft/(/x—1). The eigenvalues X of
marked by a shift from supercritical Hopf bifurcation the linearization at (x,y,z) = (0,0,0) satisfy
(a{ < 0 ) to subcritical Hopf bifurcation (ax > 0 ) . The cal-
culation of a { allows the location of this transition to be X 3 -f-(/x-2)A 2 + [ l + ( l + 2 i u ) p 2 ] A + i u ( / x - l ) ( H - p 2 ) = 0 .
predicted, and the normal-form analysis yields a detailed
(9.5)
understanding of the dynamics near this critical region.
For simplicity we assume the two stable waves have For fi > 2, all coefficients are non-negative and the con-
equal damping rates T and equal amplitudes a2= a3.
The dynamical variables are then (al,a2,(f>), where ax is
the amplitude of the unstable wave and <j)— (j>i (f>2 (/>$ is
the phase difference between the waves. Following Wers-
inger et al. (1980), we introduce the coordinates
(x,y,z) = (a {cos(/>, a isin(f>, a 2), so that the wave interac-
tion is described by

1 l -y2
A -n X

dt
il l 0 y +2 xy (9.1)
0 0 -2r z —xz

where ft=col — co2 o)3 measures the detuning from the

resonance col=co2-\-co3. Both parameters ft and T are
non-negative. For additional background on the plasma
physics ancestry of Eq. (9.1) see Wersinger et al. (1980).
The chaotic dynamics of the model in the regime of large
damping (T—> oo and ft/T fixed) has been analyzed by
Hughes and Proctor (1990). The analysis of the Hopf bi-
furcation in these equations follows Crawford (1983). FIG. 19. Surfaces of constant y in the (ft,D parameter space.
Let V(x,y,z) denote the two-parameter family of vec- The Hopf bifurcation surface is y = 0; for y < 0 the fixed point
tor fields defined by the model (9.1), leaving the depen- (9.3) is stable.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1023

stant term is positive. This implies that any real root y(y2-3co2) + (ti-2)(y2-co2)
must be negative; in particular, ^ = 0 cannot occur in this
region of parameter space. If eigenvalues with ReA, = 0 + [l+p2(l+2/i)]y+^-l)(l+p2) = 0 (9.7)
occur, they must form a conjugate pair ±ico. Thus, in
the regions of parameter space where the stability of the Although (il,fi) are the physical parameters, y and JJL
fixed point changes, there will be a negative real eigenval- are more convenient, as y directly measures the distance
ue and a conjugate pair. From the characteristic polyno- in parameter space from criticality for a Hopf bifurca-
mial, a complex root, y + ico> satisfies tion, i.e., y = 0 . We may express the dependence of Q, on
(y,ia) by solving Eq. (9.6) for co2, eliminating co2 from Eq.
3 y 2 - ^ 2 + 2(/i-2)y + l+p2(l+2/i) = 0 , (9.6) (9.7), and solving forp 2 :

9
2_ (i-2r)/x2-2[i+4r(r-D]M+2[i-r(4r
2
2
-8y+5)] (9.8)
/i -2(l-2y)/x-2(l-y)
I
Now given parameters (y,ju) we can determine p and where
hence Q, from £l=p(fi — 1). The (fl,fi) parameter space
-yp(l+p)
for ( 1 > 0 , p,>0 corresponds to y < 0 . 5 and fi>0, as
shown in Fig. 19. The curve y = 0 determines the Hopf v = co A(y)
bifurcation surface where a complex-conjugate pair of ei- lp/x(l+p2)(l+/x)
genvalues reaches the imaginary axis. f—p(l+Ax)'
The center-manifold reduction for this bifurcation re-
u^to \—2co
quires that we determine the two-dimensional center sub-
space. For the eigenvalues (kXyk2,k3) w e have eigenvec- I °
c
tors (vl9v2,v3): span E at y = 0 . The linear transformation

-ktpd+fl) S = (vr vt vx) (9.11)

Vi = -A(k() 1,2,3 (9.9) puts the linear problem in block-diagonal form:
2
p/LL(l+p )(l+fl) y co 0
where A (X) = k — X+p(l+p ). 2 2
For the real eigenvalue —co y 0 -S~lLS , (9.12)
kx, the eigenvector is real; for the conjugate pair A,2 = A,3 0 0 Xx
we have
where L is the matrix in Eq. (9.4) and S~l is given by
^2,3 = ^ ± ^ 1 (9.10)

aco(\—2y) afico co[A(kl) + Xl(l-2y)]

P 2
-a[co +A(kx)-A(p)] a/3(Xx-y) kxA(y)-y A(Xx)-Xxco2 (9.13)
detS
2
aco(2y — \) afico co[co — A(y) — y ( l — 2 y ) ]

with dz' _
dt
x^'+Riix'y^') (9.16)
a=/x(l+p2) , 0=p(l+j*) ,
where
detS = al32co[A(Xx)-A(y) + co2 + (Xx-y)(l-2y)]
R^x'y,^)' \-y2)
Next we implement the linear change of variables i
R2(x',y',z') = 2S~ - xy (9.17)
x \x R3(x',y',z') —xz
y' \=s-1 \y (9.14)
with (x,y,z) expressed in terms of (x',y',z') using Eq.
z' [z (9.14). For convenience in our discussion below, the re-
in Eq. (9.4), to express the vector field in the standard sult of fully expanding the right-hand side of Eq. (9.17)
form of Eq. (7.2): will be denoted

d_ x' y co x' Ri(x',y',z') Ri(x',y',z')=Rnx,2+Rily'2+Rnx'y'

+ R2(x',y',z')
(9.15)
dt \y' \ -co y \y' \ +Ri4x'z' + Ri5y'z'+Ri(z'2 (9.18)

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1024 John David Crawford: Introduction to bifurcation theory

for each component i —1,2,3. The coefficients R^ are where terms of fourth degree and higher have been omit-
readily worked out, but we shall not require the detailed ted.
expressions, which tend to be unwieldy, e.g.,
3. Determining the normal form
Ru-^^-[(l-2y)(co2-A(y))2
The quadratic terms in Eq. (9.25) may be removed by a
+p2r(co2-A(r)) near-identity normal-form transformation to new vari-
+02y(A(Xl)+kl(l-2y))] . (9.19) ables (x,y) = (x',y') + (f){2)(x'yyr) with inverse
(x',y') = (x,y)-<f){2)(x,y) + 0(3) [cf. Eq. (8.3)]. From
Eqs. (9.25) and (8.8), the equation for 0 ( 2 ) is
2. Approximating the center manifold \Rnx2+Rl2y2+ Rl3xy'
fc<2> ) = [R x2+R y2+R xy (9.26)
Near (x',y',z') = (0,0,0) we represent the center mani- 2i 22 23

fold by a function /i:R2—>-R describing the z' coordinate where L((f>{2)) is defined in Eq. (8.16). Following the dis-
of the manifold, i.e., z' = h(x',y'). This function satisfies cussion in VII.A.2, we solve Eq. (9.26) by rewriting it rel-
[cf. Eq. (7.7)] ative to the basis [£±'l)] defined in Eq. (8.19). Thus
dh
jlyx'+coy'+R^x'y^)] Rnx2 + Rl2y2+R13xy'
dx
R2lx2 + R22y2+R23xy • 2[* < + ,n l ( +' /, +* ( - ,/) l < - ,/) ]
/=0
+ ^7[~cox' + yy' + R2(x',y',h)}
6y (9.27)
'-kxh(x\y') + Rz(x',y',h) (9.20) with

with
R +' — j(RU~ ^ 12 " ~ ^ 2 3 ) + 7 " ^ 13+^21 ~ ^22)
/j(0,0) = 0 , -|^7(0,0)= ^ - ( 0 , 0 ) = 0 .
ox oy -R (2,2) (9.28a)
f
An asymptotic solution for h(x',y') near (x\y ) = (0,0)
R{£l)=±(Rn+Rl2)+±(R2l+R22) (2,1)
= R{l> (9.28b)
has the form
h(x',y') = hlx'2 + h2y'2 + h3x'y'+ • (9.21) (2,2)
R¥ =
(R] -R12 +R23) + — (-R21 ~ R 1 3 ~ ~ ^ 2 2 )
where terms in (x',y') of third degree or higher have
been dropped. A straightforward evaluation of the quad- =i? ( 3 0) (9.28c)
ratic coefficients yields
and
2^(1*32-1*3!) + (2y-A, 1 )jR 3 3
(9.22)
3
(2co)2 + (2y-Xl)2 ^>=2[^ / ) l ( +' / ) +^ / ) l ( -' / ) ]; (9.29)

coh3+R31
(9.23) hence from Eqs. (8.22) and (9.26)

-coh3+R32 (2,1)—. (9.30)

(9.24)
*! k^l)
h =
2 <,., l •

for / = 0 , 1 , 2 . This change of coordinates must now be

Given h(x'9y')f the two-dimensional vector field on the carried out in Eq. (9.25) to obtain the transformed vector
center manifold follows directly from Eqs. (9.15), (9.18),
field up to terms of fourth degree:
and (9.21),

X y co X
= -co y
''x' y co x' Rx{x'y,h) y y
+
ii

y' -co y y'. R2(x',y',h)

Rux+Rlsy
y co x' Rnx'2+Rl2y'2 + Ruxy
+ (h1x2 + h2y2-±-h3xy
+ R24x+R25y
ii

-co y y' R2lx'2+R22y'2+R2Zx'y'

Rux' + RX5y' Rux2+Rny2+Ruxy
2 2 {2)
-H/t1x' + /*2j;' + /i3Jcy -D<f> (x,y) + 0(4) .
R24x'-\-R25y' + .. . R2iX2+R22y2+R23xy

(9.31)
(9.25)

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1025

Here we see the additional terms of third degree generat- is no stable solution in the neighborhood of r = 0 when 7
ed by the nonlinear coordinate change removing the is positive; in fact, numerical studies indicate that the
quadratic terms. The final task is to consider the terms wave amplitudes grow without bound.
of third degree in Eq. (9.31) relative to the basis {§±J)} These conclusions indicate that the stable Hopf period-
and determine the coefficient ax of £+' 2 ) [cf. Eq. (8.25)]. ic orbit must be destroyed in a separate bifurcation in the
This calculation yields parameter neighborhood of (/z=/x c , 7 = 0 ) , since there is
no periodic orbit in the neighborhood of the fixed point
for fi <fic, 0 < 7 « 1. Thus in parameter space the curve
-2(4>%2)R%1) +4>%0)R^2)) , (9.32) or bifurcation surface at 7 = 0 corresponding to Hopf bi-
furcation must intersect at least one additional such
{ 2)
where R +' is the component of the "original" cubic curve at (jj,=fic, 7 = 0 ) . The instability of r = 0 at this
terms in Eq. (9.25) along the basis vector £+ , 2 ) , point is termed a degenerate Hopf bifurcation because the
nondegeneracy condition (5.24) fails, and the discovery of
R{Z>2)=}[3(hxRu+h2R25) + h2RH
additional bifurcations at this point illustrates the value
+ h3Rl5+hlR25+hiR24] of analyzing such degenerate cases. This particular de-
generacy is one of the simplest examples of a
+ j[3(hlR24-h2Rl4) + h2R24 + h3R25 codimension-two bifurcation, meaning that to locate it we
must simultaneously adjust two independent parameters
-hxRX5~h3RX4] . (9.33) /1 and 7 .
The comprehensive analysis of degenerate Hopf bifur-
We now have the normal form for this bifurcation to cation by Golubitsky and Langford (1981) shows that in
leading nonlinear order [cf. Eq. (8.26)]: this case there is only one additional bifurcation surface
r = yr + Re(ax)r3 + 0(r5) , (9.34a) that intersects the Hopf surface. This second surface
marks parameters values at which the stable Hopf
2
G=co-Im(al)r + O(r*) . (9.34b) periodic orbit merges with an unstable periodic orbit and
both disappear. In the return map for the Hopf orbit,
The dependence of ax on parameters is complicated, and this merger is a saddle-node bifurcation which annihi-
the behavior of the cubic coefficient R e a j along the criti- lates two fixed points. For this reason, this second sur-
cal curve 7 = 0 in parameter space is best examined nu- face may be referred to as the saddle-node (SN) surface; it
merically. The graph of ax ^ R e a j vs fi for 7 = 0 in Fig. was discovered numerically by Meunier et al. (1982).
20 indicates a region of supercritical bifurcation a! < 0 To determine how the SN surface approaches the Hopf
and a region of subcritical bifurcation ax>0, with the surface requires an analysis that includes both periodic
transition ax = 0 occurring at /xc ~ 3 . 2 9 . Thus for damp- orbits. Since the SN surface intersects the Hopf surface
ing rates greater than \ic the instability will saturate at at 7 = 0 , the saddle-node bifurcation occurs for arbitrari-
r = rH in a small stable oscillation of the wave amplitudes ly small positive values of 7 . This means that the two or-
[cf. Eq. (5.25)]. For f*<fic, the analysis implies that there bits can merge while the Hopf orbit is still in a very small
neighborhood of r = 0 . Under these circumstances the
local attractivity of Wc near r = 0 will not permit a
periodic orbit that is not in fact contained in Wc. Hence
both periodic orbits must lie in Wc and their merger is a
_ ' feature of the center-manifold dynamics (9.34). Since the
- phase-shift symmetry decouples 6 from r, the radial equa-
: \ H tion (9.34a) is adequate to describe the saddle-node bifur-
cation provided the fifth-order term a2r5 is included.
- At criticality for the saddle-node bifurcation the linear
- stability of the Hopf orbit is lost, but the orbit still exists.
Near 7 = 0 , the SN surface is determined by these two
_ f(a,) ~~
facts. The existence of the Hopf orbit at criticality means
*" that r = rH is still a solution to dr/dt=0, which implies
f(a2)

y+ai4+a2r£=0. (9.35)
: 1
-. 1 i 1 1 1 1 1 i - i 1 1 11 1 l 1 1 t 1 1 1 1 1 1 In addition, linearizing Eq. (9.34a) about the Hopf orbit
3 4 5
determines the orbit's linear stability within Wc, setting
rj = r — rH we find

FIG. 20. At critically (7=0) the normal-form coefficients ^-=(y + 3axr2+5a24)V + 0(V2) .
ax =Re(al) and a2=Re(a2) in Eq. (9.34a) are plotted against /a
using the function /(x)=sgn(x)log( 1.0-h |JC| ). Linear stability of 17=0 changes when

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1026 John David Crawford: Introduction to bifurcation theory

y + 3a! Tfj + 5a2rji — 0 (9.36) obtain the normal form through fifth order. The details
of this are not of interest here; the resulting expression
Equations (9.35) and (9.36) suffice to determine the SN for a 2 as a function of/x for 7 = 0 is also plotted in Fig.
surface at small y. Subtracting (9.35) from (9.36) yields
20. At the degenerate Hopf point (a{ = 0) we find a2 > 0
r]i{2a1r]I-\-ax ) = 0, hence rjj > 0 requires
and conclude that the saddle-node surface branches to
the right.
<0 (9.37) The results of this bifurcation analysis may be tested
numerically. Figure 22 shows the Hopf bifurcation to a
for a valid solution rjj~—ax/2a2 to exist. Substituting stable oscillation for fi>fic. As /x decreases at fixed
this solution into Eq. (9.35) or (9.36) yields y ==0.01, the stable periodic orbit loses stability near
JJL~ 3.55. This transition appears in Fig. 23 and reveals
4y = —- . (9.38) the dramatic effect of the saddle-node bifurcation.

Taken together relations (9.37) and (9.38) locate the SN B. Steady-state bifurcation
bifurcation surface for 0 < y « 1. There are two cases, in the Ginzburg-Landau equation
depending on the sign of a2 at fyi=/ic,y = 0). From the
point of degeneracy the saddle-node surface branches to For the complex-valued function A(X,T) we consider
the right (ax < 0 ) if a2 > 0 and to the left (a{>0) ifa2<0. the Ginzburg-Landau (GL) equation in one space dimen-
These cases are indicated in Fig. 21. sion,
The actual calculation of a2 is a straightforward exten-
sion of the calculation of ax. The calculation of h{x,y)
must be carried to fourth order so that Eq. (9.25) can be
extended to include fifth-order terms. Then second-,
third-, and fourth-degree terms need to be removed to 7=-i.o ;
^=6.0 ;

III
"^1
:
IIMA A A ^ ~ - -
1/^ :
a0 > 0
SN surface

Hopf surface

a0 < 0

SN surface

Hopf surface

FIG. 21. For the degenerate Hopf bifurcation corresponding to FIG. 22. Evolution of x (t) vs t for Eq. (9.1) from an initial con-
ai=0 and a2^0 there are two possibilities, depending on the dition (-1.0,0.0,0.5): (a) Withy = - 0 . 1 and /x = 6.0 when the
sign of a2 at criticality. For a2 > 0, the saddle-node (SN) surface fixed point (9.3) is stable; note that the trajectory is initially re-
branches toward negative values of a{. For a2 <0, the SN sur- pelled from the unstable fixed point at the origin, (b) With
face branches toward positive values of ax. The unstable y = 0A and /i = 6.0 when the fixed point is unstable and the
periodic orbit which collides with the stable Hopf orbit is not solution is attracted to the stable Hopf periodic orbit. The final
shown. point on this trajectory segment was ( — 4.486,-2.886,4.499).

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1027

dA d2A determine the wavelength 2ir/q of the vortex flow have

=fiA + -A \A\ (9.39)
dr dx2 been carefully investigated as a particularly simple para-
digm for nonequilibrium pattern formation (Langer,
with real coefficients and with boundary conditions that 1986; Ahlers, 1989).
ensure finite-dimensional center manifolds (Tuckerman Analytic theory often assumes either cylinders of
and Barkley, 1990). This equation arises in a wide infinite length or periodic axial boundary conditions. 2 8
variety of settings; in particular, Eq. (9.39) models the be- Then in linear approximation one finds an eigenfunction
havior of a spatially extended system near criticality for a with axial wave number qc whose (real) eigenvalue ap-
steady-state bifurcation (Collet and Eckmann, 1990; proaches zero as ft tends to ftc from below. Slightly
Manneville, 1990). In fluid dynamics, a well-studied ex- above this_threshold all wave numbers within a band of
ample of such a bifurcation is the appearance of Taylor width Vfx about qc are linearly unstable, where
vortex flow in a Taylor-Couette apparatus (Ahlers et ah, / x ^ ( n — f l c ) / f t c defines the bifurcation parameter.
1986; Ahlers, 1989), where one observes the motion of a However, only those wave numbers within a subband of
fluid confined in the gap between two concentric width fi are actually realized experimentally because of a
cylinders. Taylor's original investigation (1923), in which secondary instability that arises for q values outside the
he rotated the inner cylinder with frequency Q, and fixed subband. This latter instability is known as the Eckhaus
the outer cylinder, established a critical frequency flc instability and it modifies q by adding or subtracting vor-
above which the steady and (nearly) featureless flow de- tex pairs.
velops a pattern of vortices characterized by a well-
The competition between different linearly unstable
defined axial wave number q. The fluid mechanisms that
wavelengths can be studied near criticality (0 <fi « 1) by
developing the fluid equations in an expansion in /x. How
this expansion leads to the G L equation can be briefly
sketched by avoiding the complexity of a realistic model
and assuming that the system is described by a single
:
15.0
: field u (z,t) such that u=0 corresponds to the featureless
7-0.01
equilibrium. For small /x, one defines rescaled space and
10.0
• /LL=3.7 : time variables by x — V^z and r = i u r and seeks solutions
5.0
- of the form

-5.0

-10.0
• mmmmmmmmmmmm
iiuyumtmimi.
|iynVnvHv»M*»»««.»«»..
NWWj
-j
1
u(z,t) = fiu0(zyx ,r)-f- juru x(z9x9r) +
which are independent of the fast time scale t and de-
scribe the slow evolution of the pattern about the basic
(9.40)

-15.0 \ length scale q<Tl. The leading-order balance determines

the form of w 0 ,
10.0 20.0 30.0 40.0 50.0 60.0 70.0
iqcz_
u0(z,x,r)— A(x,r)e +c.c (9.41)
in terms of a complex amplitude function; at higher order
the G L equation (9.39) for A arises as a "solvability"
condition, which must be satisfied to avoid secular behav-
ior. The basic equilibrium A—0 for G L corresponds,
therefore, to the spatially uniform state one observes if
/ z < 0 ; in addition, for [i>0 there are spatially periodic
equilibria ("pure modes")
AQ(x;(f>) = Vp-Q2ei<f>eiQx (9.42)

that describe patterns with wave number q =qc+v/iLLQ.

As [x varies there are bifurcations from ^ 4 = 0 and AQ
that can be studied using center-manifold theory; howev-
er, this analysis is more subtle for two reasons. First, the
G L equation (9.39) is highly symmetric. The group of
symmetries is generated by reflections and translations in
FIG. 23. Evolution of x (t) vs t for Eq. (9.1) with the final point
given for Fig. 22(b) used as the initial condition: (a) for y = 0.01
and /x = 3.7 when the Hopf periodic orbit is stable; (b) for the
same initial condition with y = 0.01 and /z = 3.5, after the Hopf
28
periodic orbit has been destroyed. No stable orbit remains and There has also been interesting recent work on the necessity
the solution grows without bound. Notice the difference in the of allowing for finite end effects in order to describe some
vertical scale. features of the experiments in long cylinders (Edwards, 1990).

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1028 John David Crawford: Introduction to bifurcation theory

x, complex conjugation, and phase shifts; these opera- that the eigenvalue spectrum becomes discrete. 29 The
tions we denote by K, Td, C, and Re, respectively: periodic boundary condition
A(-<ir)=A(ir) (9.46)
{K-A)(X)=A(-X) , (9.43a)
is a simple choice that enforces the discretization
Q = integer in (9.45) and also respects the full set of sym-
(Td-A)(x)=A(x+d) (9.43b)
metries (9.43), allowing us to observe their effects on the
bifurcations. 30 With periodic boundary conditions the
(C-A)(x)=A(x) , (9.43c) translations Td act as rotations on the periodic coordi-
nate x. Consequently Td and K generate the symmetries
(R0-A){x) = ei0A(x) . (9.43d) of the circle; i.e., the group 0 ( 2 ) . In addition, Re and C
generate a second 0 ( 2 ) action on the phase of A(x),
Thus if A (x,t) is a solution then (y*A){x9t) is also a since these two O (2) actions commute the full symmetry
solution for y—K, Td, C, Rd, or any combination of these of Eq. (9.39) and Eq. (9.46) is 0 ( 2 ) X O ( 2 ) .
operations. For bifurcation problems with symmetry
there exists a generalization of theory presented in Sees.
I I - V I I I that incorporates a variety of group-theoretic 1. Bifurcation from A = 0
techniques. We do not require this generalization for this
example, but we will indicate how the symmetry (9.43) As /x increases through ti — Q2,Q2 = 0,1,4,9, . . . , the
affects the bifurcation analysis. Golubitsky, Stewart, and eigenvalue X—\x — Q2 crosses zero and the linear mode ^
SchaefFer (1988) provide a comprehensive introduction to becomes unstable. Because of the symmetry there is in
equivariant bifurcation theory, and there are also the fact a four-dimensional center manifold associated with
more concise reviews by Stewart (1988), Gaeta (1990), this instability. If we rewrite Eq. (9.44) in terms of real
and Crawford and Knobloch (1991). A second novelty and imaginary parts, xpx. = u(x) + iv(x), then Lipx = kil>x
arises because Eq. (9.39) describes an infinite-dimensional becomes
dynamical system; i.e., it defines a flow on an infinite-
2 u u
n+ dxa:
dimensional phase space—the space of functions A(x). = A. (9.47)
Center-manifold theory can be rigorously extended to V V
partial differential equations, but this generalization is
rather technical for the present discussion (see the recent and for fixed Q¥^0 there are four linearly independent
review by Vanderbauwhede and looss, 1991). However, real-valued eigenfunctions with eigenvalue k=fi — Q2:
if we assume there are center manifolds associated with
the bifurcations in Eq. (9.39), then the corresponding 1>x(x) = cosQx ,
reduction and bifurcation analysis can be carried through
just as for an ordinary differential equation.
The assumption of a finite-dimensional center manifold x/j2(x)=R7r/2^l(x) = cosQx ,
requires a consideration of boundary conditions for Eq. (9.48)
(9.39). This necessity is clear if we analyze the linear sta-
bility of A = 0 . Linearizing Eq. (9.39) defines the opera- if>3(x) = T_7r/2Q-ip1(x) sinQx ,
tor L>

i/;4(x) = T_7r/2QR1T/2-xl;l(x)-- sinQx

A=tiA + :
A=LA (9.44)
dt dx
The three "extra" eigenvectors t^2, ^ 3 , and ip4 are forced
with eigenvectors and eigenvalues given by by symmetry; thus the steady-state bifurcation at fi~Q2
involves a four-dimensional center subspace. The bifur-
if>k(x) = ei2x , k=ii-Q2 > (9.45)

where — oo < Q < oo. For /x < 0 all eigenvalues satisfy

X < 0 , and the uniform state is asymptotically stable; for 29
The contrast between a finite set of critical modes and a con-
JLL>0 there is a continuous band of wave numbers tinuum of critical modes is not merely a matter of technical
0<Q2<{i whose eigenvectors xpk describe linearly unsta- difficulty. In spatially extended systems one finds a rich variety
ble perturbations of A = 0. This continuum prevents us of new phenomena (Brand, 1989; Collet and Eckmann, 1990;
from isolating a finite number of critical modes that Manneville, 1990).
determine the time-asymptotic behavior and represents a 30
Other boundary conditions have been considered in the
serious technical obstacle to center-manifold reduction literature; see for example Hall, 1980; Graham and
(Coullet and Spiegel, 1987). This difficulty does not arise Domaradzki, 1982; Ahlers et al., 1986; Tuckerman and Bark-
when boundary conditions are imposed on A(x) such ley, 1990.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1029

cation at / x = 0 when Q=0 is an exception, with only a As fi varies near / i = 0 , this equation describes the bifur-
two-dimensional center subspace: Ec=span{xplfi/j2} since cation of the pure modes with zero wave number (9.42),
t^1 = t^3 and ^2 = Vv Returning to complex notation and
introducing two complex amplitudes (a,/?) for the critical ^ o = o = v
V (9.56)
modes (9.48), we have Since the unstable subspace is empty at this bifurcation,
iQ iQx these solutions are stable in the directions transverse to
A(x,t) = a(t)e *+P(t)e- +SU,t) , (9.49)
the center manifold. In addition, linearizing Eq. (9.55)
where S(x,t) is orthogonal to the critical modes around a = V/ie'^ shows that these solutions are stable to
±iQx
perturbations in the amplitude | AQ=0\ but that perturba-
f"dxe S(x,t) =0 . (9.50) tions in the phase <f> correspond to a zero eigenvalue or
neutral linear stability. This zero eigenvalue reflects the
Note that the decomposition in Eq. (9.49) corresponds to
fact that Eq. (9.56) describes a continuous family of
the choice of variables in Eq. (7.2), with (a,/?) corre-
equilibria parametrized by the phase (f>; such eigenvalues
sponding to xx and S corresponding t o x 2 . Near A = 0,
are a general feature of bifurcations that break a continu-
the center manifold for Eq. (9.49) may be represented as
ous symmetry. In this case, the 0 ( 2 ) symmetry generat-
the graph of a function h :EC-+ES@EU; i.e.,
ed by R e and C has been broken.
S(x,t) = h(x,a,/3ya,/3) (9.51)
such that b. Q¥^0

h(x,0,0) = 0 (9.52) This bifurcation is only slightly more complicated,

and since the nonlinearity of the center manifold does not
affect the lowest-order nonlinear terms in a and /?. By in-
serting Eq. (9.50) into the GL equation and projecting
-0, 8* =o, with f^^dx e±lQx we obtain a system of equations analo-
da (a,/3) = 0 da (a,0) = O
gous to (7.2):
(9.53)
d = (M-<22)a-//r dx -iQx
-irlTT
-e ~ A A (9.57a)
=0, =0
a/? (a,0)=O 90 (a,£)=0 0=(fi-Q2)p- r ^e'&AlA (9.57b)
c ** — oo 2 77
This exactly parallels the representation for W intro-
duced in Eq. (7.3), modified only by the fact that ES@EU dlS = lii + d2x)S-A\A\2+e,<tKf'j£:e-'<>'A\A\2,
is now infinite dimensional, i.e., the function h depends -ir 277
on the continuous index x. dx
+e-iQxf7r^eiexA\A\2. • (9.57c)
The reduced equations for the center-manifold dynam- -irlrr
ics depend on whether Q=0 or Q¥=0 because the mani-
On the center manifold near ^4=0, 5 is expressible in
fold dimension changes from four to two when Q—0.
terms of h [Eq. (9.51)], which is at least second order in
the critical amplitudes as |a|,|/?|—>0 [cf. Eqs. (9.52) and
a. Q=0 (9.53)]. Hence, on the center manifold,

In this case the decomposition in Eq. (9.49) can be A\A\2 = (aeiQx+pe~iQx)

simplified to X[\a\2+\/3\2 + aPe+i2Qx+Pae-i2Qx] + 0(4) ,
A(x,t) = cc(t)+S(x,t) (9.54a) (9.58)
where where 0 ( 4 ) denotes terms of fourth degree or higher in
f" dxS(xyt)=0 . (9.54b) (a,/?). Since the cubic terms (9.58) do not involve h it
will not be necessary to calculate the leading coefficients
In addition one finds from Eq. (9.39) that if dA /dx = 0 at in its Taylor expansion, combining Eq. (9.58) with
t = 0, then the solution of (9.39) is independent of x for all (9.57a) and (9.57b) yields the center-manifold equations
t; hence the center subspace is invariant under the full to third order:
nonlinear dynamics, which implies that Ec and Wc coin-
d = (/i-Q2)a-(2|^|2H-|a|2)a-f©(5) , (9.59a)
cide. Consequently S(x,t) — 0 for solutions in the center
manifold, and the dynamics on Wc follows immediately
0=(p-Q2)l3-(2\a\2+\l3\2)j3+0(5) . (9.59b)
by setting 5 = 0 and inserting Eq. (9.54a) into (9.39):
In this four-dimensional system there is no longer any
(9.55) coupling to S(x+t)y and the neglected terms on the right-
dt a=fia- hand side are at least fifth order in (a,/?) because the first

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1030 John David Crawford: Introduction to bifurcation theory

nonzero terms in the expansion of h would appear at TABLE I. Dimension of the unstable subspace for ^ = 0 a s a
third order in this case rather than second order. Intro- function of p.
l
ducing polar variables a^p^e and t3=p2e 2, we find Value of p dim Eu
that Eq. (9.59) reduces to a two-dimensional system,
p<0 0
p1 = (Ai-G2)pi-(2pi+p?)p1 + 0(5) , (9.60a) 0<^<1 2
Q 2 < / x < ( e + l ) 2 , g = l,2,3, 2 + 42
P2 = (V-Q2)P2-(2pi+P2)p2 + 0(5) , (9.60b)

^1,2 = 0 > (9.60c)

since the amplitudes pX2 decouple from the phases xpl 2 .

2. A digression on phase dynamics
One can show that this decoupling is a feature of the
symmetry and extends to all orders; this, however, re-
It is instructive to analyze the stability of these equili-
quires an analysis of the full normal form for this bifurca- bria by introducing the phase and amplitude of A and
tion; see Golubitsky, Stewart, and Schaeffer (1988), defining a local time-dependent wave number k ( x , r ) ,
Chapter XVII, Sec. 2 for a relevant discussion.
The bifurcating solutions are obtained by setting A(x,r)=p{x,r)en ix,r) (9.62)
Pi=p2=0 an
d neglecting higher-order terms:
k(x,r
pM^-Q2)-(2p22~P2l)]=o , dx
2 =
(9.61)
p 2 [(i"-e )-(2p?+pi)] o, In these variables, Eq. (9.39) becomes

There are two types of new solutions in addition to the l ^ ^ + J ^ (9.63a)

trivial equilibrium at (0,0). The pure-mode solutions,
(p]=p — Q2, P2 = ° ) a n d ( P i ^ 0 ' p2==tl~Q2)> correspond _3_ 2k dp , dk
to the bifurcation of the states AQ and A _ £ , respective- (9.63b)
dr dx p dx dx
ly, for p — Q2>Q. In addition there is a "mixed"-mode
solution given by P\~P2=P a n c * p 2 ~ y(M~~Q2)- If we restrict our attention to solutions that are slowly
From Eq. (9.60) the linear stabilities within the center varying in x, then for p>k2 we expect p(x,r) to ap-
manifold of each of these states may be calculated. With proach a quasistatic equilibrium
respect to perturbations in the amplitudes (pi>P2)> the
p(x,r)^V/p — k2(x,r) (9.64)
pure modes are stable but the mixed mode is unstable; in
each case there are also two zero eigenvalues correspond- since the d2p/dx2 term in Eq. (9.63a) can be neglected
ing to perturbations in the phases (9.60c). The phase por- and k(x,r) evolves on a time scale set by the slowly vary-
trait Fig. 24 for (pi,p2) summarizes this analysis. ing x dependence. When p is given by Eq. (9.64), the
In addition, for all of these solutions (pure and mixed equation (9.63b) for the wave number reduces to a non-
modes) there can be unstable directions transverse to the linear diffusion equation,
center manifold because of the unstable directions for
^ 4 = 0 . The number of these unstable directions is equal _3fc _8_
D(k)-^-kix) (9.65a)
to the dimension of Eu at criticality (see Table I). We 8r dx dx
shall see that these initially unstable pure modes AQ re-
gain their stability as p increases further above p = Q . with diffusivity D (k) given by
-3k2
D(k)~- (9.65b)

For a pure mode with p = Vp — Q2 and k = Q, consider

the dynamics of a small fluctuation in phase,
k(x,T)=zQ-\-8Q(x,T) [relaxing the periodic boundary
condition (9.46)]. Inserting this ansatz into Eq. (9.65a)
and linearizing in bQ yields a linear diffusion equation for
the fluctuation

^-3Q=D(Q)^j8Q (9.66)
or dxz
If the wavelength of the pure mode q — qc + Q satisfies
FIG. 24. Phase diagram for the flow on the center manifold as-
sociated with bifurcation from A = 0 . The pure modes are the Q2<p/3 , (9.67)
stable fixed points on the pi axis and the p2 axis. The unstable
mixed mode lies on the diagonal. then D{Q)>0 and bQ decays, the pure mode is stable.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1031

The loss of phase stability when Q > JLL/3 is known as the b. Linear stability for a =0
Eckhaus instability; its physical interpretation as nega-
tive phase diffusion was suggested by Pomeau and The linear operator (9.71b) is self-adjoint,
Manneville (1979). Subsequently more general theories
of phase dynamics have been developed (Cross and (jCaua2) = (auXa2) , (9.73)
Newell, 1984; Brand; 1988; Newell et al.y 1990). with respect to the inner product

3. Bifurcation from the pure modes (al,a2)=ifir •^-[al(x)a2(x) + al(x)a2(x)] , (9.74)

** —TT Z77

We now develop the bifurcation theory of this instabil- so we expect real eigenvalues X in the spectrum deter-
ity with the periodic boundary condition (9.46). mined by
Xv=Xv . (9.75)
a. Symmetry
By inspection the eigenfunctions v are of the form
The 0(2)XO(2) symmetry of the A = 0 equilibrium is v(x)=zla)eikx+z2U)e~ikx , fc=0,l,2,3, ..., (9.76)
partially broken in the pure-mode state; to describe the
remaining symmetry define composite "translation" and with complex coefficients (zx,z2) that satisfy
"reflection" transformations Td and K by [k + k2 + 2kQ + (fi-Q2)]zl = -(/LL~Q2)z2 .
(9.77)
Td(Q)=R„QdTd , (9.68a)
[X + k2-2kQ+(fi-Q2)]z2 = -(n-Q2)zl .
K((f>) = R(l>CKR_<f> . (9.68b) The real and imaginary parts of zi9 / = 1,2, satisfy Eq.
These transformations generate a representation of 0 ( 2 ) , (9.77) separately, since X is real; this decoupling is due to
which we denote by 0 ( 2 ) , and the AQ(X;</>) state is in- the symmetry and forces the eigenvalues to have double
variant with respect to this 0 ( 2 ) action: multiplicity when k=£0. More precisely, let (/- 1 ,r 2 )GR 2
be a real solution to Eq. (9.77) for eigenvector
yAQ(x;(f>)= AQ(x;<f>), y = Td(Q), K(<f>) . (9.69)
v(x) = rl(k)eikx + r2(k)e~ikx . (9.78a)
In addition, the discrete group ZQ generated by ITT/Q
spatial translations {Tl7r/Q) is a symmetry of AQ(X \<f>). Then the translated eigenvector
Let a(x,r) be defined by w(x) = (TW2kv)(x) = iria)eikx-ir2a)e~ikx (9.78b)
A(x,r)= AQ(x;(f>)(l+a(x,T)) . (9.70) has imaginary coefficients and is linearly independent of
Then Eq. (9.39) implies v(x); thus X has multiplicity two. [The other symmetry
CK leaves v(x) invariant.] When k = 0 then Eqs. (9.76)
- | - a =JLa -(/n-Q2)N(a,a) , (9.71a) and (9.77) yield only one (linearly independent) solution:
or
v(x)=l , X=-2(ILL-Q2) . (9.79)
where
For a pure mode AQ to exist requires
Xa = ^+2iQ^--(iJL-Q2)a-(fi-Q2)a , (9.71b)
ox ox \AQ\2=fi-Q2>0 , (9.80)

so the k=0 mode (9.79) is always stable. The possibili-

N(a,a) = [a2 + 2\a\2 + a\a\2] . (9.71c)
ties for instability arise from Eq. (9.77) for fc>l.
From Eqs. (9.46) and (9.70) we find that a(x,r) also Without loss of generality let (zl9z2) be real. Then a non-
satisfies periodic boundary conditions on [—77,77], and trivial solution (fj^O, r 2 ^ 0 ) requires X = X+ or X = X_
with (9.71) we can calculate the induced 0 ( 2 ) action on where
a(x,r)
X±=-(k2+ti-Q2)±V(ii-Q2)2 + (2kQ)2 . (9.81)
(Td- A)(x,r)= AQ(x;(f>)(l + (Td^a)(x9r))
(9.72) The X_ eigenvalue is always negative [cf. Eq. (9.80)], but
(K(<f>)-A)(x,T)=AQ(x;(f>)(l + (CK-a)(x,T)) ; A, + satisfies X+ < 0 if and only if
thus the 0 ( 2 ) action on a(x,r) is generated by Td and ii-Q2>2Q2-\k2 . (9.82)
CK. Note that the ZQ action requires only symmetry
with respect to T27r/Q, and this is contained in Eq. (9.72). With periodic boundary conditions, kmin = 1 so the pure
The covariance of Eq. (9.71a) with respect to Td and CK mode A Q will be linearly stable provided
is easily checked; this remaining 0 ( 2 ) symmetry will in- \AQ\2>2Q2-l . (9.83)
troduce nongeneric features into the secondary bifurca-
tions. Stated slightly differently, as fj, increases above JLL = 0, all

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1032 John David Crawford: Introduction to bifurcation theory

pure modes with wave numbers in the band a(x,t) = a(t)v + {x)+p{t)G)+{x) + S(x,t) , (9.86a)

(9.84) where

are stable. Comparing Eqs. (9.67) and (9.84) we see that <u + ,S> = <u; + , S > = 0 . (9.86b)
the finite-length periodic boundary condition is stabiliz-
[Recall that JL is self-adjoint; otherwise the projection in
ing in the sense that at fixed fi there is a wider band of al-
Eq. (9.86b) would require the appropriate adjoint eigen-
lowed Q values (Tuckerman and Barkley, 1990). With
functions.] The center-manifold dynamics for (a,&) is
periodic boundary conditions, t h e Eckhaus instability
more conveniently expressed by defining the complex am-
corresponds to the k = 1 instability that sets in when con-
plitude
dition (9.83) fails. As /x decreases further below the Eck-
haus boundary (9.84) there are additional instabilities of z—a + i/3 (9.87)
higher k values, as shown by Eq. (9.82).
and rewriting Eq. (9.86)

a(x,t)=z{t)vz(x)+z{t)v-{x)+S , (9.88a)
c. Center-manifold reduction forX+^0
vz{x) = ±[v + {x) — iw + (x)] , (9.88b)
For fixed k and Q, as /x decreases below 3Q2 — jk2, the
A,+ eigenvalue (9.81) crosses through zero moving from vz-(x) = ±[v + (x) + iw + {x)] . (9.88c)
the left half-plane into the right half-plane. A t A,+ = 0
With this notation z(t) is analogous t o ^ j in Eq. (7.2a)
there are two critical modes corresponding t o this eigen-
and S(x,t) is analogous to x2 in Eq. (7.2b); the equation
value,
for z follows from Eq. (9.87), which can be written as
v^(x) = rl(k^)eikx + r2(k+)e-ikx , (v + 9a )+i{w + 9a )
(9.85) z—- (9.89)
ikx ikx 2
w + (x) = irl(k+)e -ir2(k+)e- rl(k+) + r2{k+)2
so that the expansion of a(x,t) [cf. Eq. (7.2)] requires two using (v + ,v+) = {w + 9w+)=r\+r\ Differentiating
real amplitudes (a,/3)EiEc, Eq. (9.89) and using Eq. (9.71) yields

( g
z = k+z~ ^ * [(v + ,N(a,a))+i<w + ,ma9a))]

= k+z- ,to (9.90a)

T^iy/_',$'"
-irllT i'-i Wfl » ar)+ ^ (a ' ff) ]
Then from Eqs. (9.88) and (9.71) we obtain

TT dx —i kx
^-=£S-(v-Q2)N(a,a)+ ^~Q\]- \vz(x)f [rlN(a,a) + r2N{aya)}
ot r\+r2 •2rr

" dx ikx
lkx
+ »*<*>/. -ir 2ir e [rlN(a,a) + r2N(a9a)] . (9.90b)

The center manifold near a = 0 is described by

S(x,t) = hix9z(t),zUJ) , (9.91)
where the function h satisfies [cf. Eqs. (7.4) and (7.7)]

(9.92)
dz dz dt s =h

The Taylor expansion of h beings at second order in (zz)>

h(x>z,z) = hl(x)z2 + h2(x)\z\2 + h3(x)z2 + 0(3) , (9.93)
with coefficients ht(x), i = 1,2,3, that are determined by applying Eq. (9.93) to (9.92) and requiring that the second-order
terms balance. This procedure yields the following equations:

£, + 2iQ^-^-Q^2X, h1ix)-(n-Ql)h3(x) = (lj,-Qz)(ri+2rlr2)e ilkx (9.94a)

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1033

£_ + 2 l - e A_ ( A ,_ e 2 ) _ 2 A + h2(x)-(ti-Q2)h2(x)=2({i-Q2){rxr2+r\+r22) , (9.94b)

£_ +2 fcjL- (/4 - fi 2 ) _ 2X+ h3(x)-(fi-Q2)h1(x)=(n-Q2)(ri+2rlr2)e -ilkx (9.94c)

which in turn yield solutions

hl(x) = Vlei2kx , (9.95a)
(H-Q2)(rir2+r2+r22)
h2 = r , (9.95b)
2
H-Q +X+
(9.95c)
2kx
h3(x) = V3e-' ,
where (77j,r]2) satisfy
r?+2r,r2
\(2k)2+AQk + (ix-Q2)+2X+ p-Q2 •(fM-Q2) (9.95d)
fi-Q2 2
(2k) -4Qk + 2
(fi-Q )+2X+ r|+2r,r2
This approximation for h (x,z,z) is adequate to determine the leading nonlinear terms in Eq. (9.90a). On the center
manifold,
a=zvz+zvI + r]lei2kxz2 + h2\z\2+T]3e-i2kxz2+ ••• ,

so the nonlinearity in Eq. (9.90a) yields

•f dx -
II fLe-ikx[rlN(a,a) + r2N(a,a)] = az\z\2+&(z\z\4) , (9.96)
• 2TT

where

a=4h2(ri+rlr2 + r2) + 4rlr2(Vi+r)2) + 2(riyi+r]ir2) + rj(rj+2ri) + ri(ri+2rj) . (9.97)

Thus the steady-state bifurcation from the pure-mode h2 = -(r\+rxr2+r22) , (9.101b)

branch is described by
rsf
1'2
*?3 = (9.101c)
= X+z-^-Q2l°z\z\2 + Oiz\z\<) (9.98) rx+r2
(r\+r\)
With these formulas at criticality, the expression for a
and there are new branches of equilibria (i = 0 ) satisfying (9.97) simplifies considerably,

z2« X
\ , (9.99) a=-3(rl+r2)2(r2l+rl) 9 (A,+ = 0 ) (9.102)

so that Eq. (9.98) may be rewritten

with an arbitrary phase reflecting the translation symme-
try Td that has been broken. z = k+z + 3(fi-Q2)(r1 +r2 )2z\z\2
The sign of a determines whether these new equilibria + 0(k+z\z\2,z\z\4) . (9.103)
occur for A,+ < 0 (subcritical) or A,+ > 0 (supercritical). It
is enough to determine this sign at k+=Q, in which case Since (iu — g 2 ) ( r 1 + r 2 ) 2 > 0 , the new equilibria (9.99) are
the expressions for rx and r2 from Eq. (9.77) simplify: subcritical and unstable.

r^-UQ-k) , r2 = 2Q+k (9.100)

X. OMITTED TOPICS
(up to an overall normalization). In addition, at A,+ = 0
The ideas of center-manifold theory and Poincare-
li-Q2=-2rxr2 Birkhoff normal forms are discussed by many authors.
An introductory account is provided by Rasband (1990);
from Eq. (9.82) and the center-manifold coefficients (9.95) for the reader seeking a more sophisticated treatment,
reduce to both Guckenheimer and Holmes (1986), Chapter 3, and
Arnold (1988a), Chapter 6, are suggested. The recent re-
r\r\r2 view by Vanderbauwhede (1989) provides very detailed
Viz (9.101a)
r,+r0 proofs for the finite-dimensional theory, and a careful re-

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1034 John David Crawford: Introduction to bifurcation theory

view of center manifolds in Banach spaces is provided by ing from the "Kolmogorov school:" Anosov and Arnold
Vanderbauwhede and looss (1991). Additional material (1988), Sinai (1989), Arnold (1988b), and Arnold and No-
on the infinite-dimensional case in particular can be vikov (1990), which provide many references to the So-
found in Marsden and McCracken (1976), Hassard et al. viet literature. In particular, Anosov and Arnold (1988)
(1978), Ruelle (1989), and the encyclopedic volume by treat normal forms and invariant manifold theory, and
Chow and Hale (1982). Finally, there is the monograph Arnold (1988b) discusses Hamiltonian normal forms and
by looss and Joseph (1989), which develops local bifurca- bifurcation.
tion theory without using center manifolds.
In one-parameter systems, the Feigenbaum bifurcation
ACKNOWLEDGMENTS
and global bifurcations involving homoclinic and hetero-
clinic phenomena are important topics outside the scope
I am grateful to P. Morrison for arranging my visit to
of this review. References for the former topic include
the Institute for Fusion Studies and to R. Hazeltine for
Cvitanovic (1984), Collett and Eckmann (1980), Lanford
suggesting this review. C. Kueny, W. Saphir, and B.
(1980), Vul et al. (1984), as well as the original papers by
Shadwick assisted in writing notes for the original lec-
Feigenbaum (1978, 1979, 1980). Global bifurcations,
tures. I would like to thank M. Silber and especially A.
especially Silnikov-type bifurcations and Melnikov
Kaufman for suggesting improvements in the final
theory, are discussed by Guckenheimer and Holmes
manuscript. This work was supported in part by the U.S.
(1986) and Wiggins (1988, 1990). In addition, the paper
Department of Energy Grant No. DE-FG05-80ET-53088
by Glendinning and Sparrow (1984) provides an accessi-
at the IFS and in part at the Institute for Nonlinear Sci-
ble introduction to the Silnikov bifurcation.
ence at The University of California, San Diego by
The recent lecture by Arnold (1989) touches on many
D A R P A Applied and Computational Mathematics Pro-
current research topics, in particular, multiparameter bi-
gram Contract No. F49620-87-C-0117 and by the D A R -
furcation problems and bifurcations in symmetric sys-
P A University Research Initiative Contract N o . N0014-
tems. The examples in Sec. IX were selected in part to il-
86-K-0758.
lustrate the importance of these subjects. An introduc-
tion to codimension-two bifurcations (i.e., bifurcations
typical for two-parameter systems) is provided by Guck- INDEX
enheimer and Holmes (1986), Chapter 7, and Arnold
(1988a), Chapter 6, but much of the work in this area is asymptotic stability 995, 997
scattered in the research literature; Golubitsky and bifurcation 992
Guckenheimer (1986) and Roberts and Stewart (1991) are degenerate 1021, 1025
two recent conference proceedings. Bifurcation theory Hopf 997, 998, 1003, 1006, 1022
for symmetric systems is likewise an actively developing imperfect 1003
subject. In addition to the recent reviews by Stewart period-doubling 998, 1005
(1988), Gaeta (1990), and Crawford and Knobloch (1991), pitchfork 997, 998, 1002, 1005
there are the more extensive treatments in Vander- saddle-node 997, 998, 1000, 1005
bauwhede (1982), Sattinger (1983), and Golubitsky, steady-state 997, 998, 1000, 1004, 1026
Stewart, and Schaeffer (1988). subcritical 1004
Hamiltonian bifurcation theory is an important subject supercritical 1004
that is neglected here altogether. Unfortunately, there transcritical 997, 998, 1001, 1005
does not appear to be a systematic discussion of this bifurcation diagram 1001
theory for nonmathematicians at a level comparable to bifurcation set 996
this review, and the literature is extensive. For bifurca- bifurcation surface 1025
tion from equilibria of flows, Chapter 8 in Abraham and bifurcation theory 992
Marsden (1978) is a possible starting point, in addition to branch of solutions 999
the brief overviews by Meyer (1975, 1986). Up-to-date center manifold 1008, 1009
discussions of Hamiltonian normal-form theory can be local attractivity 1008
found in Bryuno (1988) and van der Meer (1985). This nonuniqueness 1008, 1012
latter monograph treats the so-called Hamiltonian Hopf reduction to 1009
bifurcation in detail. Howard and MacKay (1987) give a center subspace 995, 997
nice discussion of the linear instabilities encountered in codimension 992, 1025
symplectic maps; Golubitsky and Stewart (1987) describe critical system 996
a generic setting for bifurcation in symmetric Hamiltoni- dynamical system 992
an systems. The closely related subject of bifurcation Eckhaus instability 1021, 1031, 1032
theory for reversible systems is showing a rapid develop- eigenspace 995
ment. Recent reviews have been given by Arnold and eigenvalue
Sevryuh (1986) and Roberts and Quispel (1991). degenerate 994
Finally, we mention the authoritative volumes emerg- simple 1000

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: ction to bifurcation theory 1035

fixed point 993 REFERENCES

hyperbolic 996, 997
Floquet theory 997 Abraham, R., and J. Marsden, 1978, Foundations of Mechanics
flow 993 (Benjamin-Cummings, Reading, MA).
generic 992 Ahlers, G., 1989, "Experiments in bifurcations and one-
graph representation 1010, 1012 dimensional patterns in nonlinear systems far from equilibri-
Ginzburg-Landau equation 1027 um,' ' in Lectures in the Sciences of Complexity 1, edited by D.
Hartman-Grobman theorem 996, 997 L. Stein (Addison-Wesley, Redwood City, CA), p. 175.
homeomorphism 996 Ahlers, G., D. Cannell, M. Dominguez-Lerma, and R. Hein-
homological equation 1014 richs, 1986, "Wavenumber selection and Eckhaus instability in
Hopf bifurcation Couette-Taylor flow," Physica D 23, 202.
Anosov, D. V., and V. I. Arnold, 1988, Eds., Dynamical Systems
flows 1003
I: Ordinary Differential Equations and Smooth Dynamical Sys-
maps 1006
tems, Encyclopedia of Mathematical Sciences 1 (Springer,
implicit function theorem 999 New York).
invariance equation 1010, 1012 Arnold, V. I., 1972, "Lectures on bifurcation in versal families,"
invariant circle 1007 Russ. Math. Surveys 27, 54.
invariant manifold 1007 Arnold, V. I., 1973, Ordinary Differential Equations (MIT,
invariant subspace 994, 997 Cambridge, MA).
invariant torus 1007 Arnold, V. I., 1988a, Geometrical Methods in the Theory of Or-
Liapunov-Schmidt reduction 1007 dinary Differential Equations, second edition (Springer, New
Lie bracket 1014 York.
map 993 Arnold, V. I., 1988b, Ed., Dynamical Systems III: Mathemati-
cal Aspects of Classical and Celestial Mechanics, Encyclopedia
mode-locking 1007
of Mathematical Sciences 3 (Springer, New York).
node 1001 Arnold, V. I., 1989, "Bifurcation and singularities in mathemat-
nonwandering point 1009 ics and mechanics," in Theoretical and Applied Mechanics,
normal form 998 Proceedings of the 17th International Congress of Theoretical
dynamics 1000 and Applied Mechanics, Grenoble, 1988, edited by P. Germain,
symmetry 1017, 1019 M. Piau, and D. Caillerie (North-Holland, Elsevier,
unfolding 998 Amsterdam/New York), p. 1.
period-doubling bifurcation 898, 1005 Arnold, V. I., and S. P. Novikov, 1990, Eds., Dynamical Sys-
phase portrait 992 tems IV: Symplectic Geometry and its Applications, Encyclo-
pedia of Mathematical Sciences 4 (Springer, New York).
phase-shift symmetry 1003, 1021
Arnold, V. I., and M. B. Sevryuk, 1986, "Oscillations and bifur-
phase space 992
cations in reversible systems," in Nonlinear Phenomena in
pitchfork bifurcation Plasma Physics and Hydrodynamics, edited by R. Z. Sagdeev
flows 997, 1002 (Mir, Moscow), p. 31.
maps 998, 1005 Belitskii, G. R., 1978, "Equivalence and normal forms of germs
recurrent point 1009 of smooth mappings," Russ. Math. Surveys 33, 107.
reflection symmetry 1002 Belitskii, G. R., 1981, "Normal forms relative to the filtering ac-
return map 993 tion of a group," Trans. Mosc. Math. Soc. 40, 1.
saddle 1001 Brand, H. R., 1988, "Phase dynamics—a review and a perspec-
saddle-node bifurcation tive," in Propagation in Systems Far from Equilibrium, edited
flows 997, 1000 by J. E. Wesfreid, H. R. Brand, P. Manneville, G. Albinet, and
N. Boccara (Springer, New York), p. 206.
maps 998, 1005
Bryuno, A. D., 1988, "The normal form of a Hamiltonian sys-
Shoshitaishvili theorem 1011 tem," Russ. Math. Surveys 43, 25.
stable manifold 1008, 1009 Carr, J., 1981, Applications of Center Manifold Theory (Springer,
stable subspace 995, 997 New York).
steady-state bifurcation Chillingworth, D., 1976, Differentiate Topology with a View to
flows 997, 1000 Applications (Pitman, London).
maps 998, 1004 Chossat, P., and M. Golubitsky, 1987, "Hopf bifurcation in the
stroboscopic map 993 presence of symmetry, center manifold and Liapunov-Schmidt
subcritical 1004 reduction," in Oscillation, Bifurcation and Chaos, CMS Conf.
supercritical 1004 Proc. 8, edited by F. Atkinson, W. Langford, and A. Mingarel-
li (American Mathematical Society, Providence), p. 343.
suspended system 1012
Chow, S. N., and J. K. Hale, 1982, Methods of Bifurcation
topological conjugacy 996
Theory (Springer, New York).
trajectory 993 Collett, P., and J-P. Eckmann, 1980, Iterated Maps on the Inter-
unstable manifold 1008, 1009 val as Dynamical Systems (Birkhauser, Basel).
unstable subspace 995, 997 Collett, P., and J-P. Eckmann, 1990, Instabilities and Fronts in
vector field 993 Extended Systems (Princeton University, Princeton, NJ).

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

1036 John David Crawford: Introduction to bifurcation theory

Coullet, P., and E. Spiegel, 1987, "Evolution equations for ex- Hall, P., 1980, "Centrifugal instabilities of circumferential flows
tended systems," in Energy Stability and Convection, Pitman in finite cylinders: nonlinear theory," Proc. R. Soc. London,
Research Notes 168, edited by G. P. Galdi and B. Straughan Ser. A 372, 317.
(Wiley, New York), p. 22. Hartman, P., 1982, Ordinary Differential Equations (Birkhauser,
Crawford, J. D., 1983, "Hopf bifurcation and plasma instabili- Boston).
ties," Ph.D. thesis (University of California, Berkeley). Hassard, B., N. Kazarinoff, and Y.-H. Wan, 1978, Theory and
Crawford, J. D., and E. Knobloch, 1991, "Symmetry and sym- Applications of Hopf Bifurcation, London Mathematical So-
metry breaking bifurcations in fluid dynamics," Annu. Rev. ciety Lecture Notes 41 (Cambridge University, Cambridge,
Fluid. Mech. 23, 341. England).
Crawford, J. D., E. Knobloch, and H. Riecke, 1990, "Period- Hirsch, M., C. Pugh, and M. Shub, 1977, Invariant Manifolds,
doubling mode interactions with circular symmetry," Physica Springer Lecture Notes in Mathematics No. 583 (Springer,
D 44, 340. New York).
Cross, M. C , and A. C. Newell, 1984, "Convection patterns in Hirsch, M., and S. Smale, 1974, Differential Equations, Dynami-
large aspect ratio systems," Physica D 10, 299. cal Systems, and Linear Algebra (Academic, New York).
Cushman, R., and J. A. Sanders, 1986, "Nilpotent normal forms Howard, J. E., and R. S. MacKay, 1987, "Linear stability of
and representation theory of sl{2,R )," in Multiparameter Bi- symplectic maps," J. Math. Phys, 28, 1036.
furcation Theory, Contemporary Mathematics 56, edited by M. Hughes, D., and M. Proctor, 1990, "Chaos and the effect of
Golubitsky and J. Guckenheimer (American Mathematical So- noise in a model of three-wave mode coupling," Physica D 46,
ciety, Providence), p. 31. 163.
Cvitanovic, P., 1984, Ed., Universality in Chaos (Hilger, Bristol). Iooss, G., and P. Joseph, 1989, Elementary Stability and Bifur-
Eckhaus, W., 1965, Studies in Nonlinear Stability Theory cation Theory (Springer, New York).
(Springer, New York,). Irwin, M. C , 1980, Smooth Dynamical Systems (Academic,
Edwards, W. Stuart, 1990, "Linear spirals in the finite Couette- New York).
Taylor problem," in Instability and Transition Vol. 2, edited by Jordan, D. W., and P. Smith, 1987, Nonlinear Ordinary
M. Y. Hussaini and R. G. Voigt (Springer, New York), p. 408. Differential Equations (Oxford University, New
Elphick, C , E. Tirapegui, M. E. Brachet, P. Coullet, and G. York/London).
Iooss, 1987 "A simple global characterization for normal Kramer, L., and W. Zimmerman, 1985, "On the Eckhaus insta-
forms of singular vector fields," Physica D 29, 95; 32, 488 (Ad- bility for spatially periodic patterns," Physica D 16, 221.
dendum). Lanford, O., 1973, "Bifurcation of periodic solutions into in-
Euler, L., 1744, Additamentum I de Curvis Elasticis, Methodus variant tori: the work of Ruelle and Takens," in Nonlinear
Inveniendi Lineas Curvas Maximi Minimivi Proprietate Gan- Problems in the Physical Sciences and Biology, edited by I.
dentes. In Opera Ommia I (1960), Vol. 24 (Fiissli, Zurich), p. Stakgold, D. D. Joseph, and D. H. Sattinger, Springer Lecture
231. Notes in Mathematics No. 322 (Springer, Berlin/
Feigenbaum, M., 1978 "Quantitative universality for a class of Heidelberg/New York), p. 159.
nonlinear transformations," J. Stat. Phys. 19, 25. Lanford, O., 1980, "Smooth transformations of intervals," in
Feigenbaum, M,, 1979, "The universal metric properties of non- Seminaire Bourbaki 1980/81, Springer Lecture Notes in
linear transformations," J. Stat. Phys. 21, 669. Mathematics No. 901 (Springer, Berlin/Heidelberg/New
Feigenbaum, M., 1980, "Universal behavior in nonlinear sys- York), p. 36.
tems," Los Alamos Sci. 1, 4. Lanford, C , 1983, "Introduction to the mathematical theory of
Gaeta, G., 1990, "Bifurcation and symmetry breaking," Phys. dynamical systems," in Chaotic Behavior of Deterministic Sys-
Rep. 189, 1. tems, edited by G. Iooss, R. Helleman, and R. Stora (North-
Glendinning, P., and C. Sparrow, 1984, "Local and global be- Holland, Amsterdam), p. 3.
havior near homoclinic orbits," J. Stat. Phys. 35, 645. Langer, J. S., 1986, "Lectures in the Theory of Pattern Forma-
Golubitsky, M., and J. Guckenheimer, 1986, Eds., Multiparam- tion," in Chance and Matter, edited by J. Souletie, J. Van-
eter Bifurcation Theory, Contemp. Math. 56 (American nimenus, and R. Stora (North-Holland, Amsterdam), p. 629.
Mathematical Society, Providence). Mallet-Paret, J., and J. Yorke, 1982, "Snakes: oriented families
Golubitsky, M., and W. Langford, 1981, "Classification and un- of periodic orbits, their sources, sinks, and continuation," J.
foldings of degenerate Hopf bifurcations," J. DifF. Eqns. 41, Diff. Eqns. 43,419.
375. Manneville, P., 1990, Dissipative Structures and Weak Tur-
Golubitsky, M., and D. Schaeffer, 1985, Singularities and bulence (Academic, Boston).
Groups in Bifurcation Theory, Vol. I (Springer, New York). Marsden, J., 1979, "On the geometry of the Liapunov-Schmidt
Golubitsky, M., and I. Stewart, 1987, "Generic bifurcation of procedure," in Global Analysis, edited by M. Grmela and J. E.
Hamiltonian systems with symmetry," Physica D 24, 391. Marsden, Springer Lecture Notes in Mathematics No. 755
Golubitsky, M., I. Stewart, and D. Schaeffer, 1988, Singularities (Springer, New York), p. 77.
and Groups in Bifurcation Theory, Vol. II (Springer, New Marsden, J., and M. McCracken, 1976, The Hopf Bifurcation
York). and its Applications (Springer, New York).
Graham, R., and J. A. Domaradzki, 1982, "Local amplitude Meunier, C , M. N. Bussac, and G. Laval, 1982, "Intermittency
equation of Taylor vortices and its boundary condition," Phys. at the onset of stochasticity in nonlinear resonant coupling
Rev. A 26, 1572. process," Physica D 4, 236.
Guckenheimer, J., and P. Holmes, 1986, Nonlinear Oscillations, Meyer, K., 1975, "Generic bifurcations in Hamiltonian sys-
Dynamical Systems, and Bifurcations of Vector Fields tems," in Dynamical Systems—Warwick, 1974, edited by A.
(Springer, New York). Manning, Springer Lecture Notes in Mathematics No. 468
Guillemin, V., and A. Pollack, 1974, Differential Topology (Springer, New York), p. 36.
(Prentice-Hall, Englewood Cliffs, NJ). Meyer, K., 1986, "Bibliographic notes on generic bifurcations

Rev. Mod. Phys., Vol. 63, No. 4, October 1991

 John David Crawford: Introduction to bifurcation theory 1037

in Hamiltonian systems/* in Multiparameter Bifurcation with Applications to Dynamical Systems and Statistical
Theory, edited by M. Golubitsky and J. Guckenheimer (Amer- Mechanics, Encyclopedia of Mathematical Sciences 2,
ican Mathematical Society, Providence), p. 373. (Springer, New York).
Newell, A., 1989, "The dynamics and analysis of patterns," in Sparrow, C , 1982, The Lorenz Equations (Springer, New York).
Lectures in the Sciences of Complexity 1, edited by D. L. Stein Spivak, M., 1965, Calculus on Manifolds (Benjamin, Menlo
(Addison-Wesley, Redwood City, CA), p. 107. Park, CA).
Newell, A., T. Passot, and M. Souli, 1990, "The phase diffusion Stakgold, I., 1979, Green's Functions and Boundary Value Prob-
and mean drift equations for convection at finite Rayleigh lems (Wiley, New York).
numbers in large containers," J. Fluid Mech. 220, 187. Stewart, I., 1988, "Bifurcations with Symmetry," in New Direc-
Olver, P., 1986, Applications of Lie Groups to Differential Equa- tions in Dynamical Systems, edited by T. Bedford and J. Swift
tions (Springer, New York). (Cambridge University, Cambridge, England), p. 235.
Pliss, V. A., 1964, "A reduction principle in stability theory of Taylor, G. I., 1923, "Stability of a viscous liquid contained be-
motions," Izv. Akad. Nauk SSSR, Ser. Matem. 28, 1297. tween two rotating cyclinders," Philos. Trans. R. Soc. London,
Pomeau, Y., and P. Manneville, 1979, "Stability and fluctua- Ser. A 233, 289.
tions of a spatially periodic convective flow," J. Phys. (Paris) Tsiveriotis, K., and R. A. Brown, 1989, "Bifurcation structure
Lett 40, 609. and the Eckhaus instability," Phys. Rev. Lett. 63, 2048.
Rasband, S. Neil, 1990, Chaotic Dynamics of Nonlinear Systems Tuckerman, L., and D. Barkley, 1990, "Bifurcation analysis of
(Wiley, New York). the Eckhaus instability," Physica D 46, 57.
Riecke, H., and H.-G. Paap, 1986, "Stability and wave-vector Vanderbauwhede, A., 1982, Local Bifurcation and Symmetry,
restriction of axisymmetric Taylor vortex flow," Phys. Rev. A Pitman Research Notes in Mathematics 75 (Pitman, Boston).
33, 547. Vanderbauwhede, A., 1989, "Centre manifolds, Normal forms,
Roberts, J. A. G., and G. R. W. Quispel, 1991, "Chaos and and Elementary bifurcations," Dynamics Reported 2, 89.
time-reversal symmetry," Phys. Rep., in press. Vanderbauwhede, A., and G. Iooss, 1991, "Centre manifold
Roberts, M., and I. Stewart, 1991, Eds., Singularity Theory and theory in infinite dimensions," Dynamics Reported, in press.
Its Applications: Warwick 1989, Vol. 2: Singularities, Bifurca- van der Meer, J. C , 1985, The Hamiltonian Hopf Bifurcation,
tions and Dynamics, Springer Lecture Notes in Mathematics Springer Lecture Notees in Mathematics No. 1160 (Springer,
No. 1463 (Springer, Berlin/New York). New York).
Ruelle, D., 1989, Elements of Differentiate Dyanmics of Bifur- Verhulst, F., 1990, Nonlinear Differential Equations and
cation Theory (Academic, New York). Dynamical Systems (Springer, New York).
Ruelle, D., and F. Takens, 1971, "On the nature of turbulence," Vul, E., Ya. Sinai, and K. Khanin, 1984, "Feigenbaum univer-
Commun. Math. Phys. 20, 167; Note concerning our paper sality and the thermodynamic formalism," Russ. Math. Sur-
"On the nature of turbulence," Commun. Math. Phys. 23, 343. veys 39, 41.
Sattinger, D., 1983, "Branching in the presence of symmetry," Vyshkind, S., and M. Rabinovich, 1976, "The phase stochasti-
CBMS-NSF Conference Notes 40 (SIAM, Philadelphia). zation mechanism and the structure of wave turbulence in dis-
Shoshitaishvili, A. N., 1972, "Bifurcations of topological type at sipative media," Sov. Phys. JETP 44, 292.
singular points of parametrized vector fields," Funct. Anal. Wersinger, J.-M., J. Finn, and E. Ott, 1980, "Bifurcation and
Appl. 6, 169. "strange" behavior in instability saturation by nonlinear
Shoshitaishvili, A. N., 1975, "The bifurcation of the topological three-wave mode coupling," Phys. Fluids 23, 1142.
type of the singular points of vector fields that depend on pa- Wiggins, S., 1988, Global Bifurcations and Chaos: Analytical
rameters," Trudy Seminara I. G. Petrovskogo 1, 279. Methods (Springer, New York).
Shub, M., 1987, Global Stability of Dynamical Systems Wiggins, S., 1990, Introduction to Applied Dynamical Systems
(Springer, New York). and Chaos (Springer, New York).
Sijbrand, J., 1985, "Properties of center manifolds," Trans. Am. Yorke, J., and K. Alligood, 1983, "Cascades of period-doubling
Math. Soc. 289,431. bifurcations: a prerequisite for horseshoes," Bull. Am. Math.
Sinai, Ya. G., 1989, Ed., Dynamical Systems II: Ergodic Theory Soc. 9, 319.

Rev. Mod. Phys., Vol. 63, No. 4, October 1991