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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.

• In the Chaos Lab #2, under the Function menu, choose the option Quadratic. Push the Add Graph button, marked with a "+". You should see a function in the quadratic family: y^2+c.

• Change the parameter: Notice the number under the column marked Parameter. This is the value of c. Edit this number. How does the graph of the function change as you vary the parameter? (You could have predicted this from your knowledge of graphing parabolas.)

• Try simultaneously graphing two maps from the quadratic family. To do this, hit the Add Graph (+) button twice and change the values in the Parameter column.

• Choose another family off the Function menu. Notice the equation for the graph you see. Try changing the parameter. Could you have guessed how the graph of the function changed as you varied the parameter?

### 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.
• Algebraically verify that 0 and 1 are fixed points of the square root function and the squaring function. Verify that +1 and -1 are fixed points of the function f(x)=1/x. Verify that 0 is a fixed point of the "+/-" function.

• Verify that the quadratic family f(x)=x^2+c has fixed points of the form
.5 (1+ sqrt(1-4c)) and .5(1-sqrt(1-4c)).

• For a variety of values of c, compare the graphs of functions in the quadratic family to the identity map. Use this to determine where these functions have fixed points. For each c, how many fixed points are there? Are there the same number of fixed points for every c? Relate your findings to the previous problem.

• Look at another family of functions by making another choice from the Function menu. Determine where functions in this family have fixed points. How many fixed points are there?

### 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.

• What is the period of a point under application of the "1/x" button? What about the "+/-" button?

• Check that for g(x)=x^2-1, points (1+ square root(5))/2 and (1-square root(5))/2 are fixed points. Check that 0 and -1 are period two points.

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.

• Use Chaos Lab #2 to graph the second iterate of the quadratic family. To do this, change the value in the Iterate column from 1 to 2. For parameter c=-1, check graphically the number of points of least period two. You may wish to simultaneously graph the first and second iterates to see which points are least period two. Does this agree with what you found above analytically?

• Graphically determine the number of period two points for functions in the quadratic family at c=-0.4, -1.1, and -1.3? Remember to exclude fixed points when you count intersections.

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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