Spring 2000 Control in Chaos: Semester End Report

Author: Po-Jen Huang

Last Update: 5/26/2000

Introduction

This paper contains the idea of the Lyapunov exponent, for the study of the Quadratic map (1-D map), and the Ikeda map (2-D map). Appendix B contains all the programs that have been used to produce all the results shown in the paper. By the end of the paper, I hope the reader will become familiar with the concept of the Lyapunov exponent, and how to compute it.

Overview

What is the map?

We will be working with first order differential maps (in order to use the Lyapunov exponent to do analysis on the map), , so that , , and .

Where is any constant between 0 and 2. (Appendix A)

What is the Chaos?

While studying a system varying with time, one usually looks at the behavior of two nearby points. As time passes by, only one of the following three conditions might apply:

1. The two points get arbitrary close to each other, or
2. Distance between the two points remain the same, or
3. Two points diverge.

To avoid the problem of re-scaling we only consider system in a bounded domain. (For example in the quadratic map, we only consider a between 0 and 2, Appendix A) To be chaotic, the system must be sensitive to the initial condition. (Case 3)

What is the attractor?

The accractor is the subset of the domain, which attracts a set of initial condition in the limit. (I.e. get arbitrary close) The mathematical way of read it is:

iff such that such that

In the quadratic map for a between 0 and 0.6, the attractor is a single point, called the fixed or period one point. For example, when a = 0.2, the attractor = {0.3062}. The different value of the parameter will result in a different attractor.

Before defining the Lyapunov exponent, let's first introduce the idea of the Lyapunov number. The Lyapunov number is the measure (the ratio) of how fast two nearby points move apart after the first n iterations.

The Lyapunov exponent is defined by taking the natural log of the Lyapunov number, , and is given by the following equation

Where is the Jacobian matrix evaluates at , and is the Lyapunov exponent. Since it is not always easy to work with limits, it is convenient to define the k-th Lyapunov exponent as taking the first k product the Jacobian matrix. It is defined as:

Notice that for dimension greater than one, the above definition does not quite make sense. In fact, one can show that taking the log/k-th roof of a matrix is equivalent of taking the log/k-th root of the eigenvalue of the matrix. (Appendix C ) Surprising enough, from numerical simulation the Lyapunov exponent has no dependence on the initial point, but only depends on the value of the parameter (for example the value of a for the quadratic map). In the remaining of this report, the Lyapunov exponent is actually the k-th Lyapunov exponent. Below is the definition of the Lyapunov number.

Definition: Let f be a map of and let be a bounded orbit of f. the orbit is chaotic if

1. it is not asymptotically periodic,
2. Lyapunov number is not exactly zero, and
3. the Lyapunov number is greater than one

Note: The Lyapunov number is zero implies that the distance between two nearby points is zero, i.e. They must be the same point. We are only interested in how the distance changes for two different near by points.

Knowing how the k-th distance between the two points differs from the k-1st, by the third relation we can decide the orbit is either chaotic or periodic. Obviously, if the distance is increasing, then the ratio is larger than one, which indicates it have chaotic behavior. On the other hand if the distance is decreasing, then the two points will coalesce into a single point.

With all these ideas in hand, we can rewrite the definition for the chaos in terms of the Lyapunov exponent.

Definition: Let f be a map of and let be a bounded orbit of f. the orbit is chaotic if

1. it is not asymptotically periodic,
2. Lyapunov exponent is a real number, and
3. the Lyapunov exponent is greater than zero

We will use the above definition though the rest of this paper.

Let's discuss the behavior of the quadratic map. The plot of how the attractor changes with different parameter, the value of , is called the bifurcation diagram.

Notice there are white regions in between dense dotted strips, especially the one between 1.75 and 1.8. The white regions are the period regions, and the dense dotted regions are the chaotic regions, as we will show.

The Jacobin matrix for quadratic has a simple form:

So the Lyapunov exponent can be written out easily:

We are interested in how many iterations will be needed to see the Lyapunov exponent converge. Since the Lyapunov exponent should not depend on where we start, (Footnotes) we might just get rid of the first couple thousand points to start on the attractor. For the period one orbit the Lyapunov exponent is a constant, the fixed point, i.e. . The Lyapunov exponent is just:

The graph below is the Lyapunov exponent versus number of iterations for = 1.9 (). Notice that it is converges to a positive value, 0.5487. By the definition we made earlier, when the Lyapunov exponent is greater than zero, the map is chaotic.

The blue curve is when initial x = 0.8; green curve is when initial x = 0.9; magenta curve is when initial x = 1.0; and finally the red curve is when x = 1.1.

Notice that even though the shapes of the curves are different, all four curves converge to the same Lyapunov exponent, 0.5487.

The following is the plot of the Lyapunov exponent versus different value of .

Notice that the map bifurcates is where the the Lyapunov exponent reaches zero (first occur at = 0.7) is where the map first bifurcates. The bottom figure is the superposition of the Lyapunov exponent on top of the bifurcation diagram.

Notice that the above figure is consistent with our definition of Lyapunov exponent. Notice that the attractor is coplicated the Lyapunov exponent is positive, i.e. theere is chaos. Where the attractor is simple, the attractor with few points, the Lyapunov exponent is negative. Below is a better view for 1.5 < < 2.0. It is very easy to see that the Lyapunov exponent has a sharp drop in each of the windows, with period orbits.

Notice that how the Lyapunov exponent change when varying a. It can be seen really easily that the Lyapunov exponent drops whenever there is a window in the bifurcation diagram.

Ikeda Map

This is a slightly more complicated map. The map has the form: (z is a complex number)

Consider the following simple system,

The variable is the product of pump (the output intensity of the laser) and the transmission coefficient of the mirror, is the reflectivity of the mirror (the amplitude), , is 2 pi times the optical length of the cavity measured in wavelengths of the light, and is the strength of the nonlinear effect.

The system is builded by 4 mirrors, one of which is pastially reflective, are araged sa a righ cavity. In side the cavity is a nonlinear material, the gray object, and it has refractive index .

Since z is a complex number, we can rewrite the above equation as , which can be worked out using Euler's formula, and it has the form:

As one can see, the expression is getting more complicated. In fact, the situation gets worse, when trying to find the Jacobian matrix.

Where the four entries are:

Where

Notice that the determent is just . With the above expression for the Jacobian matrix, we can apply the definition of the Lyapunov exponent described before to compute it.

Recall

 Variable Value 1.0 0.9 0.4 6.0

Let’s start by looking at the attractor. (with the specify parameter above)

Below is the Lyapunov exponent for the Ikeda map. Notice that the value of the Lyapunov exponent converges to a positive value (chaotic), 0.588.

The numerical check is been done via evaluate the determinant does not change with number of iterations. (Footnote)

Below is the bifurcation diagram for the Ikeda map while varying the .

Below is the Lyapunov exponent while varying beta. Notice how the funny shape of the curve is.