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MODULE: Dynamical Systems.

Part 2: Parametrized Families and Periodic Points

Parametrized Families

So far, the discussion has been about functions and the behavior of orbits under iteration. In fact, in this lab, we do not study one function, but a whole parametrized family of functions. That is, we look at a whole series of related functions at once, and try to understand how the eventual behavior under iteration changes as the function changes. For example, in the introduction, you considered the function f(x)=x^2. This is just one function within the parametrized family f_c(x)=x^2+c. This family of maps is called the quadratic family.

This picture shows the setup of Chaos Lab #2, along with maps from the quadratic family for four different values of the parameter c.

Fixed Points

Remember that p is a fixed point for g if g(p)=p. Here is an observation that makes it easy to find fixed points visually: every point of the identity function i(x)=x is a fixed point. Therefore if g(p)=i(p), then p is a fixed point. Fixed points are located where the graph of g intersects the graph of i. Notice that the dotted diagonal line in the graph window is actually the graph of the identity function.

Periodic Points

Denote the nth iterate of g by g^n. For example, g^2(x)=g(g(x))= the second iterate of x.

Definition: For a function g(x), a point p is periodic with least period n if g^k(p)=p for k=n, and this is not true for any smaller value of k. The orbit of p consists of only n points, since g^n(p)=p. Since saying "least period" all the time is cumbersome, if it is clear from the context, whenever we refer to a period n point, it is assumed to be least period n.

The method of finding fixed points graphically also works to find periodic points. For example, to find the points of least period two for function g, look for points of intersection of the graphs of i and g^2. The only catch is that fixed points are also period two points, but not of least period two. Thus you must first find the fixed points for g and discard them when looking at intersections of the graphs of i and g^2.

Just as with fixed points, periodic orbits can be attracting, repelling, or neutral. For a given periodic orbit, if orbits of nearby points converge to the periodic orbit, it is attracting. If orbits of nearby points move away from the periodic orbit, it is repelling. Otherwise it is neutral.

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Author: Evelyn Sander
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Created: Jun 09 1996 --- Last modified: Jul 31 1996