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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
nonhyperbolic 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
 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.
Consequences:
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
"perioddoubling" 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.
Definition:
 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:
 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 
 WOLOG: 
 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 nonhyperbolic (and difficult) 
Theorem:
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 "preiterates" 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
