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Chaos and Fractals: Complex Dynamic Systems

Bert G. Wachsmuth
Department of Mathematics and Computer Science
Seton Hall University

Setup: Suppose is a family of functions, and . Study the orbits of :

We will study the following two basic question:

fix the parameter c, what about the orbits of for various 's
fix a point , what about the orbits of for different values of c

Example: Take the quadratic family and fix c = 0: .

Clearly f(0) = 0 and f(1) = 1, so the orbit of 0 is {0, 0, 0 …} and the orbit of 1 is {1, 1, 1 …}. Also, f(-1) = 1, so the orbit of -1 is {-1, 1, 1, 1 …}. Moreover, if the orbit wil converge to 0, and if the orbit will converge to positive infinity.

Definition: If is differentiable, and is a fixed point, i.e. , then:

if then is called an attracting fixed point
if then is called a repelling fixed point
if then is called a non-hyperbolic fixed point

If is an attracting fixed point then the set of all points whose orbit converges to is called the basin of attraction of

Example: Take the quadratic family and fix c = -2: .

We can easily find the fixed points and periodic points of higher order using Maple …

Example: Take . Then so that the period points of period n are

infinitely many period points
period points are dense in
and periodic points are repelling.

Chaotic Dynamical Systems: is called chaotic if:

f has sensitive dependence on initial conditions
period points are dense in D
f is topologically transitive (technical)

Example: is chaotic (call the function f for short):




so that f also has a dense number of repelling periodic points in [-2, 2]

Question: How does the quadratic family go from completely simply (c = 0) to chaotic (c = -2) ? In other words, we now want to look at the second question: fix and vary the parameter. The best point to be fixed is the critical point where

Bifurcation Diagram

Example: , fix critical point , vary c from 0 to -2:

Bifurcation Applet - requires Java-enabled browser !

Bifurcation Applet

Click Controls to set parameters
Drag mouse to "zoom in"
Source Code:

 Algorithm for Picture:

Start with c = 0
Compute sequence without plotting
Compute sequence and plot orbits
Replace c by c - (something) and repeat

Sarkovskii's Theorem

Reorder the natural number as follows (Sarkovskii Ordering):

Then if is continuous, and f has a period point of period k, then f also has periodic points of period l for all in the Sarkovskii ordering.


If f has period 3, then f has periodic points of any period

If f has only finitely many periodic points, then these points are powers of 2

If f "wants" to have infinitely many periodic points, it must happen via "period-doubling" bifurcation

Iterations of Complex Functions

Same setup as before, but . Talking about families of (analytic) function in C:

A family of analytic functions is normal:

is equicontinuous

is locally uniformly bounded

omits more than 2 points (Montel's Theorem)

Example: with c = 0 and z in the complex plane. Inside of unit circle and outside of unit circle is where the family is normal. Interior of unit circle is the basin of attraction for fixed point z = 0, outside of unit circle is basin of attraction for fixed point infinity.

In particular, equicontinuity implies that there is no sensitive dependence on initial conditions, therefore: normal families are not chaotic.


Fatou Set:
Julia Set: = complement of Fatou Set

Originally by Fatou and Gaston Julia, 1920's.

Facts about the Fatou Set:

Number of components of F is either infinite or less than or equal to 2
F can be decomposed into attracting domains, superattracting domains, parabolic domains, Siegel disks, and Herman rings. The dynamics on each of these pieces of F is understood.

Facts about the Julia Set:

where (basin of attraction)
J is the closure of all repelling periodic points
If the orbits of finite critical points are unbounded, then J is totally disconnected. If these orbits are bounded, J is connected
If then
J is the set where "chaos" takes place

Algorithm to generate J:

Choose a point in C
Iterate some 50 times, check if orbit is larger than, say, 10
If orbit is larger than 10, color black. Otherwise color according to when it is larger than 10
Choose another and continue

The Mandelbrot Set

Consider the function . Then infinity is an attracting fixed point, regardless of c, and 0 is the only critical point. Julia set is the boundary of the basin of attraction of infinity, and it is either connected or totally disconnected. For this particular family of functions, define:

Then the boundary of is the Julia set for this particular c, and the set M in parameter space is called the Mandelbrot set.

Algorithm to generate M:

Fix (the critical point). Choose a parameter Iterate 50 times. If orbit is bounded, color black. Otherwise, color depending on when it's bigger than, say, 10.

Mandelbrot Applet for Java-enabled browsers !

Mandelbrot Applet:

Click Controls to set parameters
Drag mouse to "zoom in"
Source Code:

It is interesting to put the Mandelbrot set and the Bifucation diagram right on top of eachother, using exactly the same scale. There you can see that the big cardiot region of the Mandelbrot set corresponds to Julia sets with one attracting fixed point, the second circular bulb corresponds to areas with an attracting orbit of period 2, and so on.

The Henon Map

Setup: The family of Henon Maps is defined as

Simple Facts:

Almost affine map
Has an inverse map
If b = 0, then is topologically conjugate to
There is no "distinguished" point (such as the critical point).
When varying the parameters, we get:
If then H has no fixed points
If then H has one real fixed point
If then H has two fixed points (a sink and a saddle)
If then H has two fixed points (saddles) and a periodic two orbit


Simple Definitions:

Compute Eigenvalues of the Jacobian matrix

Note that

If is a fixed point, then:

If , the fixed point is attracting (and there is a basin of attraction), or a sink
If , the fixed point is repelling (and there is a basin of attraction for the inverse function), or a source
If , the fixed point is a saddle point (and there is a stable and unstable manifold in the direction of the Eigenvectors corresponding to the Eigenvalues)
If or , the fixed point is non-hyperbolic (and difficult)


There is a combination of (a, b) near b = 0.3 and a = -1.4 where the map is chaotic and the iterates has a "strange attractor".

A strange attractor is a set that attracts an open neighborhood, but is not the union of sinks.

Hanon Applet:

Simple applet to draw iterates of seed point after 100 "pre-iterates" for fixed, specified parameters.
Hit Start to plot
Change parameters or starting points, hit Start again
Source Code:

Bert G. Wachsmuth (bgw)
March 27, 1999


Bert G. Wachsmut
Last modified: 05/03/00