Bifurcation

Intorduction: 1610s, "the point at which something splits in two," noun of action
from bifurcate (v.). The meaning "a division into two forks" is from 1640s. "to
divide into two forks or branches," 1610s, from Medieval Latin bifurcatus, from
Latin bi- "two" (see bi-) + furca "two-pronged fork, fork-shaped instrument."
The word “bifurcation” was introduced by H. Poincare (1885) in his study of
equilibria of rotating liquid masses. But “bifurcation phenomena” have already
been understood by C. Jacobi (1834).
Bifurcation theory provides a strategy for investigating the bifurcations that occur
within a family of dynamic systems. Bifurcation theory has intensively investigated
varied topics that bear on chaotic and quasiperiodic dynamics. Much of this theory
has been developed in the context of discrete time dynamical systems defined by
iteration of mappings. From the dynamics of vector fields it can easily be interpret
that, the triviality of the dynamics dependence on the parameters. The qualitive
structure of the flow varies with the changes of parameter. So, bifurcation can be
defined as,” a sudden qualitative or topological change in the behavior of a
dynamic systems due to the changes in parameters”.
bifurcations occur in both continuous systems (described by ordinary, delay or
partial differential equations) and discrete systems (described by maps).
Classification of bifurcation: There are mainly two types of bifurcation:
1.Local bifurcation.
When the stability of equilibrium is changed due to the change of parameter is
known as “local bifurcation”. Some of the examples of local bifurcation are:
(i) Saddle-Node bifurcation;
(ii) Trans-critical bifurcation;
(iii) Pitch-fork bifurcation;
2.Global bifurcation.
When the changes in topology extend out to an arbitrarily large distance due to
the change or the collide with equilibria of larger variant of sets (such as periodic
orbit) is known as “Global bifurcation”.
Some of the examples of Global bifurcation are:
(i) Homoclinic bifurcation.
(ii) Heteroclinic bifurcation.
(iii) Infinite-period bifurcation.

Applications of Bifurcations:

Bifurcations theory is used in numerous field of science and engineering. It is also

used in “Biological Network”. Bifurcation theory has been applied to connect
quantum systems to the dynamics of their classical analogues in atomic systems,
molecular systems and resonant tunneling diodes. Bifurcation theory has also been
applied to the study of laser dynamics. Classical bifurcation theory is used in the
solution of nonlinear stability problems in many branches of engineering.
Bifurcations theory is also used in static ‘’Buckling” phenomena. Bifurcation
methods is also used in nonlinear flight dynamics problems. Many kinds of
bifurcations have been studied with regard to links between classical and quantum
dynamics including saddle node bifurcations, Hopf-bifurcations, umbilic
bifurcations, period doubling bifurcations, reconnection bifurcations, tangent
bifurcations, and cusp bifurcations.
Fixed Point
Before going into the topic of bifurcation ,we first need to discuss about fixed
point.
Let, 𝑥̇ = f(x) be a equation of one dimensional flow . if 𝑥̇ > 0 then the flow is right
and if 𝑥̇ < 0 then the flow is left. But if 𝑥̇ = 0 then there is no flow at all.
So, at those points 𝑥̇ = 0 such points are known as fixed points.
There two kinds of fixed points ---
1) Stable fixed point.
2) Unstable fixed point.
Stable fixed point: If the flows are coming towards a fixed point then that point
is known as a stable fixed point or simply a sink.
Unstable fixed point: If the flows are repealing from a fixed point then that point
is known as a unstable fixed point or simply a source.

Figure: Fixed points(where solid dots are stable whereas’s others are unstable).
Linear stability Analysis
Above we discuss the stability of fixed through graphical explanation. One can
easily state the stability of fixed point through numerical.
If x* is a fixed of 𝑥̇ = f(x) .
Then, x* is stable if f’(x*) < 0.
x* is unstable if f’(x*) > 0.
 Half stable fixed point:
There is a special kind of fixed point which is known as half-stable fixed point.
If a flow is a sink or attracting from the left or right of a fixed point and a source
or repelling from the right or left ,then that fixed point is known as half stable.

𝑭𝒊𝒈𝒖𝒓𝒆: 𝒂 𝒉𝒂𝒍𝒇 𝒔𝒕𝒂𝒃𝒍𝒆 𝒇𝒊𝒙𝒆𝒅 𝒑𝒐𝒊𝒏𝒕.

Q.Analyze the following equations graphically.In each case , sketch the vector field
on the real line , find all fixed points, classify their stability.
1. 𝑥̇ = 4x2 – 16.
2. 𝑥̇ = e-x Sinx.
Solution:
(1) . Given that, 𝑥̇ = 4x2 – 16.
For fixed points we should have 𝑥̇ = 0 .
So, 4x2 – 16 = 0 .
⟹ 4x2 = 16.
⟹ x2 = 4.
⟹ 𝑥 = ±2.
Figure: 4x2 – 16 . white dot represent stable fixed point and black dot represent
unstable fixed point.
Therefore, at x = -2 the fixed point is stable and x = 2 the fixed point is unstable.

3) Given that, 𝑥̇ = e-x Sinx.

For fixed points we should have 𝑥̇ = 0 .
So, e-x Sinx = 0
⟹ Sinx = 0 [ e-x can’t be zero ]
∴ x = (2n-1)𝜋 , 2n𝜋 ; ∀𝑛 ∈ ℤ.
Figure: e-x Sinx . Where x=(2n-1)𝜋 and white dot represents stable fixed points
whereas’s 2n𝜋 and black represents unstable fixed points.

Linear Stability Analysis

Use linear stability analysis to classify the fixed points of the following
systems.If linear stability analysis fails because f’(x*) = 0 , use graphical
argument to decide the stability.
1. 𝑥̇ = 𝑥 (1 − 𝑥 )(2 − 𝑥 )
2
2. 𝑥̇ = 1 − 𝑒 −𝑥
3. 𝑁̇ = −𝑎𝑁𝑙𝑛(𝑏𝑁);where N(t) is proportional to the number of cells in
tumor and a,b>0 are parameters.
 Solution: 1. Given that, 𝑥̇ = 𝑥 (1 − 𝑥 )(2 − 𝑥 )
= (𝑥 − 𝑥 2 )(2 − 𝑥 )
= (2𝑥 − 𝑥 2 − 2𝑥 2 + 𝑥 3 )
= 𝑥 3 − 3𝑥 2 + 2𝑥 .

Now, f’(x) = 3𝑥 2 − 6𝑥 + 2 .
For fixed point we need to find x at 𝑥̇ = 0 .
∴ 𝑥 (1 − 𝑥 )(2 − 𝑥 ) = 0 𝑆𝑜, 𝑥 = 0 ,1,2 .
∴ 𝑥1∗ = 0 , 𝑥2∗ = 1 , 𝑥3∗ = 3
From that, 𝑓 ′ (𝑥1∗ ) = 2 > 0 ; therefore 0 is a unstable fixed point.
𝑓 ′ (𝑥2∗ ) = −1 < 0 ; therefore 1 is a stable fixed point.
𝑓 ′ (𝑥3∗ ) = 2 > 0 ; therefore 3 is a unstable fixed point.
2
Solution: 2. Given that , 𝑥̇ = 1 − 𝑒 −𝑥
2 2
Let, 𝑓 (𝑥 ) = 1 − 𝑒 −𝑥 ∴ 𝑓 ′ (𝑥 ) = −2𝑥𝑒 −𝑥
For fixed point we need to find x at 𝑥̇ = 0
2
∴ 1 − 𝑒 −𝑥 = 0
1
⇒ 2 =1
𝑒𝑥
2
⇒ 𝑒𝑥 = 1
2
⇒ 𝑙𝑛𝑒 𝑥 = 𝑙𝑛1
⇒ 𝑥 2 = 0 ∴ 𝑥 = 0 . Therefore, x* = 0 . ∴ 𝑓 ′ (𝑥 ∗ ) = 0 .
So, we can’t claim whether it is a stable fixed point or unstable fixed point. We
have to take a graphical approach.
 2
Figure: 1 − 𝑒 −𝑥
From the graph we see that x=0 is neither a stable or unstable fixed point
it is a half-stable fixed point.

Solution: 3. Given that, 𝑁̇ = −𝑎𝑁𝑙𝑛(𝑏𝑁) . Let, 𝑓(𝑁) = −𝑎𝑁𝑙𝑛(𝑏𝑁)

𝑎𝑁
∴ 𝑓 ′ (𝑁) = − × 𝑏 − 𝑎𝑙𝑛(𝑏𝑁) = −𝑎 − 𝑎𝑙𝑛(𝑏𝑁)
𝑏𝑁
For fixed point we need to find N at 𝑁̇ = 0 .
∴ −𝑎𝑁𝑙𝑛(𝑏𝑁) = 0
1
⇒ 𝑁 = 0 𝑜𝑟 ln(𝑏𝑁) = 0 ⇒ 𝑏𝑁 = 1 ⇒ 𝑁 =
𝑏
1
∴ 𝑁∗ = 0 ,
𝑏
𝑓 ′ (0) = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑. 𝑎𝑠 ln(0) = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑.
1
𝑓 ′ ( ) = −𝑎 < 0 ; 𝑎𝑠 𝑎 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒.
𝑏
 1
Therefore, N = is a stable fixed point.
𝑏

Saddle-Node Bifurcation
The Saddle-Node Bifurcation is sometimes called a fold bifurcation or turning
point bifurcation. The name derives from vector fields on the plane where fixed
points known as saddles and nodes can collide and annihilate.
Definition: If two fixed points of a dynamical system move towards each other
then collide and mutually annihilate with the variation of parameters, then the
system is known as saddle-node bifurcation.
The general example of saddle-node bifurcation is : 𝑥̇ = 𝑟 + 𝑥 2 .

r<0 r=0 r>0

Above the graph of (r+x2) we see that at r<0 there are two fixed point ,when
we change the value of r=0 then two fixed point collide each other and become
half-stable finally when r>0 then there is no fixed point at all . We can say the
fixed point mutually annihilate when r>0. Here, r=0 is the node or nodal point.
N.B. white dot is stable and black dot is unstable fixed point.
For each of the following problem, sketch all the qualitatively different vector
fields that occur as r is varied. Show that a saddle-node bifurcation occurs at a
critical value of r, to be determined. Finally, sketch the bifurcation diagram of
fixed points x* versus r.
Questions:
1. 𝑥̇ = 1 + 𝑟𝑥 + 𝑥 2
2. 𝑥̇ = 𝑟 + 𝑥 − ln (1 + 𝑥)
Solution: Given that, 𝑥̇ = 1 + 𝑟𝑥 + 𝑥 2 . at 𝑥̇ = 0 we will have fixed point.
∴ 1 + 𝑟𝑥 + 𝑥 2 = 0
−𝑟 ± √𝑟 2 − 4
⇒𝑥=
2
From that we see that two fixed point exist at |𝑟| > 2 .
At, r=±2 there is only one fixed point. That’s mean two fixed point collide
when r=±2.Therefore, a saddle-node bifurcation occurs at a critical value of
r= -2 and r = 2.
There is no fixed-point if |𝑟| < 2.

r=3 r=-5
 r=-2 r=2

r = -1 r=1
This page is left for bifurcation diagram:
Solution: Given that, 𝑥̇ = 𝑟 + 𝑥 − ln(1 + 𝑥 ) . at 𝑥̇ = 0 we will have fixed point.
∴ 𝑟 + 𝑥 − ln(1 + 𝑥 ) = 0.
⇒ 𝑟 = ln(1 + 𝑥 ) − 𝑥;
To be defined ln(1+x) has to have values 𝑥 ≥ −1 . If 𝑥 → −1 𝑜𝑟 𝑟 →
−∞.So,there are no fixed for r > 0. For r < 0, the fixed-point approaching x = -
1 is stable, the other one is unstable. If r = 0 then the two fixed-point collide,
therefore 0 is the critical value of r where saddle node bifurcation occur.

r=2 r=0

r=1
This page is blank for bifurcation diagram:
 Trans-critical Bifurcation
A trans critical bifurcation occurs when there is an exchange of stabilities
between two fixed points. A trans critical bifurcation is a particular kind of
local bifurcation, meaning that it is characterized by an equilibrium having an
eigenvalue whose real part passes through zero.
Definition: A trans critical bifurcation is one in which a fixed point exists for all
values of a parameter and is never destroyed but it’s stability changes with the
variation of parameter.
General form of a trans critical bifurcation is: 𝑥̇ = 𝑟𝑥 − 𝑥 2 .
Now, let’s try to explain trans critical in graphical approach:

r=-4 r=0 r=5

Above the graph we see that at r = -4 there are two fixed point and at x = 0 we
have stable fixed point. But ,if we change the value of r = 0 then x = 0 becomes
a half-stable point. Finally, at r = 5 , x = 0 becomes unstable.
So, we can say that there is always a value of x* for any kind of real values of r .
Moreover these fixed don’t mutually annihilate after they collide , they just
changes their stability .
Problem: For each of the following problem, sketch all the qualitatively
different vector fields that occur as r is varied. Show that a trans critical
bifurcation occurs at a critical value of r, to be determined. Finally, sketch the
bifurcation diagram of fixed points x* vs. r .
1. 𝑥̇ = 𝑟𝑥 − ln(1 + 𝑥 )
2. 𝑥̇ = 𝑥 (𝑟 − 𝑒 𝑥 )
Solution: 1. Given that, 𝑥̇ = 𝑟𝑥 − ln(1 + 𝑥 ) .
Here, x = 0 is a fixed point for all values of r. Now,
𝑥̇ = 𝑟𝑥 − ln(1 + 𝑥 ).
1
= 𝑟𝑥 − (𝑥 − 𝑥 2 + 𝑂(𝑥 3 )).
2

1
= 𝑟𝑥 − 𝑥 + 𝑥 2 − 𝑂 (𝑥 3 )
2
1
= 𝑥 (𝑟 − 1) + 𝑥 2 + 𝑂(𝑥 3 )
2
Hence a trans critical bifurcation occurs at a critical value of rc = 1 .

r = - 0.5 r = 0.5
 r = 1. r = 2.5

from the graph we see that if r < 1 then 0 is a stable fixed point but when we
increase the value of r > 1 then 0 become a unstable fixed point.
This page is left for bifurcation diagram
Solution: Given that, 𝑥̇ = 𝑥 (𝑟 − 𝑒 𝑥 )
Here, x = 0 is a fixed point for all values of r. Now,
𝑥̇ = 0
⇒ 𝑥 (𝑟 − 𝑒 𝑥 ) = 0
⇒ 𝑟 = 𝑒𝑥
⇒ 𝑙𝑛𝑟 = 𝑥
Hence a trans critical bifurcation occurs at a critical value of rc = 1.

r = -4 r=0
 r = 1. r = 1.5
from the graph we see that if r < 1 then 0 is a stable fixed point but when we
increase the value of r > 1 then 0 become a unstable fixed point.
Bifuraction Diagram
 Pitch-Fork Bifurcation
Pitchfork bifurcations occur in systems with symmetry. The pitchfork
bifurcations occur in physical models where fixed points appear and disappear
in pairs due to some intrinsic symmetry of the problem. More conveniently
pitchfork bifurcation is a particular type of local bifurcation where the system
transitions from one fixed point to three fixed points.
There are two kind of Pitch-Fork bifurcation.
1.Super-Critical Pitch-Fork Bifurcation.
2.Sub-Critical Pitch-Fork Bifurcation.
Super-Critical Pitch-Fork Bifurcation: The general form of super-critical pitch-
fork bifurcation is 𝑥̇ = 𝑟𝑥 − 𝑥 3 .

r<0 r=0

r>0
Sub-Critical Pitch-Fork Bifurcation: The general form of sub-critical pitch-fork
bifurcation is 𝑥̇ = 𝑟𝑥 + 𝑥 3 .

r>0 r=0

r>0
Problem: For each of the following problem, sketch all the qualitatively
different vector fields that occur as r is varied. Show that a trans critical
bifurcation occurs at a critical value of r, to be determined. Finally, sketch the
bifurcation diagram of fixed points x* vs. r .
1. 𝑥̇ = 𝑟𝑥 − 𝑠𝑖𝑛ℎ𝑥
𝑟𝑥
2. 𝑥̇ = 𝑥 +
1+𝑥 2

1. Solution: Given that, 𝑥̇ = 𝑟𝑥 − 𝑠𝑖𝑛ℎ𝑥

r = -2 r=0

r=1 r = 1.5
Here x* = 0 is a fixed point for all values of r. A super-critical pitch-fork
bifurcation occurs at rc = 1.
Bifurcation Diagram
 𝑟𝑥
2. Solution: Given that, 𝑥̇ = 𝑥 +
1+𝑥 2

r=3 r=0

r = -1 r = -2.5
Here x* = 0 is a fixed point for all values of r. A sub-critical pitch-fork
bifurcation occurs at rc = -1.
Bifurcation Diagram