18.
091 Introduction to Dynamical Systems
Spring 2005
Lecture 3: Bifurcation and the Quadratic Map
Prof. Emma Carberry
Lecture notes by: Alan Dunn
Note that all functions will be from R to R unless otherwise noted.
Types of cycles and their meaning
Before we proceed into discussing dierent types of bifurcations, we shall take the time to discuss
the concepts of attraction and repulsion, except this time, we will discuss how they relate to cycles
instead of points. These concepts will be important in the discussion of bifurcation, because certain
types of bifurcation involve the generation of cycles that originate from a xed point (though this
will be dened in more precise detail later).
Denition An attracting ncycle for a function F (x) is an ncycle for which |(F n ) | < 1 on the
ncycle, and a repelling ncycle is one for which |(F n ) | > 1 on the ncycle.
The reader will surely have noticed that the point at which the derivative is to be taken is not
specied, however, it is easy to see that (F n ) = F F n1 F F n2 . . . F , which means that
(F n ) has the same value for any x that is part of the ncycle since F a will range over the ncycle
for 0 a n 1.
There are two ways of interpreting the meaning of an attracting or repelling ncycle. One of
these ways is as an attracting or repelling point for the function F n , but another, perhaps more
graphically intuitive way of thinking about these cycles is as cycles in which every nth point in the
orbit is considered as a sequence that is attracted or repelled from a specic point of the ncycle.
Admittedly, this is not a rigorous mathematical formulation, but should help to provide intuition
for the concept.
Bifurcations
The notion of a bifurcation is that of a splitting, or division, of signicant points to a function by
the modication of a parameter that controls the function. More specically, we will be considering
a family of functions F (x) parameterized by and the behavior of their xed points and cycles
as we vary this parameter. We will nd that there are certain critical values of for which the
structure of the xed points and cycles of F changes; for critical values with specic properties we
will declare that a bifurcation has occurred.
There are two main types of bifurcations we will be discussing:
Saddlenode bifurcations
Perioddoubling bifurcations
1
In each of these types of bifurcations, a xed point splits into a xed point and some other special
point or points. The meaning of this will be cleared up momentarily.
2.1
The SaddleNo de bifurcation
Denition A family of functions F is said to undergo a saddlenode bifurcation if there exists
some value 0 and some open interval I that satises the following conditions:
1. such that 0 < < 0 F has no xed points on I.
2. F0 has exactly one xed point on I, and this xed point is neutral.
3. such that 0 < < 0 + F has exactly two xed points on I, one of which is repelling
and one of which is attracting.
Note that the way the denition is formulated it does not matter whether the point splits for
increasing or decreasing since it is always possible to formulate another family of functions F
which would have the appropriate behavior as 0 if the original family did not.
The denition is fairly clear in giving the intuition for the situation: a xed point essentially appears
in an interval at a specic value of and then for > 0 this xed point splits into two, one of
which is attracting and one of which is repelling.
The reason for the name SaddleNode bifurcation is that one typical situation in which this type
of bifurcation arises is in a function with a saddle, that is, a part in the function that has a
slope of exactly 1 at some point and a slope that continuously varies from something less than 1
to something greater than 1, call this function f . If we let F (x) = f (x) + , then as , we
can have a situation in which the graph of the saddle y = F (x) becomes tangent to the line y = x
at the point wherein the slope varies through 1. This eect can be seen in Figure 1.
Example:
Take Fc (x) =
1
1x
+ c. Solving F (x) = x we get:
1
p = (1 + c c2 2c 3)
2
At c = 1 these two are equal to 0, and then as c drops below 1 we get two distinct xed points.
1
Furthermore, we have Fc (x) = (1x)
2 , which is equal to 1 at x = 0, and thus the xed point for
c = 1 is neutral. Then when c drops below 1, we can observe that one of the ps becomes slightly
less and one becomes slightly more, which will cause Fc to be less than 1 at one and greater than 1
at the other, that is, we will have an attracting and a repelling xed point. Thus this is an example
of a SaddleNode bifurcation.
2.2
The PeriodDoubling bifurcation
Denition A family of functions F is said to undergo a perioddoubling bifurcation if there exists
some value 0 and some union of open intervals I that satises the following conditions:
2
y=x
y=x
y = f(x) + a
y = f(x) + b, b < a
y=x
Repelling fixed point
y = f(x) + c, c < b
Neutral ixed point
Attracting fixed point
Figure 1: A typical SaddleNode bifurcation.
1. such that 0 < < 0 F has exactly one attracting (respectively, repelling) ncycle
on I.
2. F0 has exactly one ncycle on I, and this ncycle is neutral.
3. such that 0 < < 0 + F has exactly one 2ncycle, which is attracting (respectively
repelling), and one ncycle, which is repelling (respectively attracting). Let the 2ncycle be
of the points pa for a = 1, 2, . . . , 2n.
4. lim+ pa = q(a) where each q(a) is a point in the original ncycle.
0
The intuition for this type of bifurcation is that we start with some sort of ncycle, and from this
ncycle we spawn a 2ncycle, that is, one whose period is doubled, and change the type of ncycle
that we have from attracting to repelling or repelling to attracting.
Example:
Take Fc (x) = x + x2 + c. Then we have:
Fc (x) = x + x2 + c = x
x = c
Fc2 (x) = x4 + 2x3 + (2 + 2c)x2 + (1 + 2c)x + c(2 + c) = x
x = c, 1 1 c
So for c innitesimally greater than 1 we have a xed point for Fc very close to 1, Fc (x) = 2x + 1,
and thus the xed point is attracting. As c gets to 1, this xed point becomes neutral, since the
xed point moves
c becomes innitesimally
to x = 1, which has F1 (1) = 2(1)+1 = 1. When
1 1 c, since these are each
less than 1, 1 c becomes real, and then we have a 2cycle
roots of Fc2 (x) = x and not roots of Fc (x) = x. (Fc2 ) (1 + 1 c) = 5 + 4c, and thus as c
becomes innitesimally less than 1, the two cycle becomes attracting. Furthermore, it is evident
that as c 1, these two xed points head toward x = 1. Thus, since we have a critical value
for which an attracting xed point splits into a repelling xed point and an attracting 2cycle,
we have an example of a PeriodDoubling bifurcation.
3
-3
-2.5
-2
-1.5
-1
-0.5
-1
-2
Figure 2: A bifurcation diagram for the previously considered example Fc (x) = x + x2 + c. In
this diagram we can see both the two xed points for Fc as well as the bifurcation that occurs at
x = 1, c = 1
2.3
Bifurcation Diagrams
An informative way of presenting some of the bifurcation data weve seen so far is through something
known as a bifurcation diagram. In such a diagram, we plot the paths of xed points and cycles
as we vary the parameter that controls the family of functions. An example is presented in Figure
2.
