Math 145, Winter 99: Chaos Theory


Lab #5, Instructions from Ely Kerman

Introduction

It is difficult to give a precise definition of chaotic behaviour in a dynamical system. However, all proposed definitions include the notion of ``sensitive dependence on initial conditions.'' This phenomenon can be described quite simply: two trajectories that start very close together (i.e. have nearly equal initial conditions) run away from each other very quickly.

Such behaviour is very undesirable in physical models because, in the real world, we can not measure initial conditions to infinite accuracy. If the system we are interested in displays sensitive depenpence on initial conditions then the unavoidable inaccuracy in our initial measurements will make any longterm predictions we might base on our model totally unreliable. In fact, because Lorentz observed this behaviour even in his simplistic model of the atmosphere he came to the dreary conclusion that long term weather prediction would never be possible.

Exercise 1. No Sensitive Dependence

In order to understand sensitive dependence on initial conditions we first look back at a dynamical system in which it does not occur...the simple pendulum. Enter the portion of the worksheet in Fig.1 that appears above the graph. Try a variety of initial conditions, always choosing ini1 and ini2 to be very close, and note that the trajectories always remain close together (most times you can't even distinguish between them). (Can you find any exceptions?). To obtain numerical evidence of this behaviour enter the lines of code below the graph in Fig.1. The first two commands tell the computer to solve the dynamical system numerically starting at ini and ini2 respectively. The next two commands display the values of x and y, for the two different solutions, at time t=2. Again enter a variety of nearly equal initial conditions and verify that these numbers remain close for long periods of time.

Exercise 2. Sensitive Dependence and the Lorentz Attractor

In contrast to the pendulum, the Lorentz attractor does display sensitive dependence on initial conditions for many values of the parameters r, s, and b. Enter the worksheet in Fig. 2. Leave ini1 fixed and see how small you have to make the difference in the initial conditions in order to have the positions at t=40 be similar. Next, change the value of r and see if such behaviour still occurs, i.e. try r=1.

Exercise 3. Sensitive Dependence and the Logistic Map

Fortunately, we can observe sensitive dependence on initial conditions in dynamical systems much simpler than that of Lorentz. Indeed, we can see such behaviour in 1-dimensional maps. A 1-dimensional map is a rule like
x_{n+1}=f(x_n)
where f(x) is some function. Starting at some initial condition x_0, the rule tells us how to calculate x_1, then x_2,...etc. We end up with a sequence of numbers
{x_0,x_1,x_2,.... }
These sequences play the same role as trajectories do in continuous dynamical systems.

The most famous 1-dimensional map is the ``logistic equation''' which is given by the rule

x_{n+1}=rx_n(1-x_n)
Enter the portion of the worksheet in Fig.3 that appears before the graph. This allows you to plot the sequence that the logistic equation determines from any initial condition x_0. The values x_1, x_2,... correspond to the height of the graph over n=1,2, ,,,. For different values for the parameter r these graphs change drastically. Start with r=1 and note that the {x_i} quickly converge to 0. Then try r=2.8 and note that they quickly converge to some constant number near 0.6. Next let r=3.3. Here the values quickly begin to jump back and forth between two fixed values. Hence every second x_i is equal and we call this a period-2 cycle. Let r=3.5, print out your graph and write down, for your portfolio, a description of how the sequence is behaving.

Now try the following values for r: 3.449, 3.54409, 3.5644, 3.9. For r=3. we stop seeing any kind of pattern and we guess that the behaviour is chaotic. To test this guess we will see if the system displays sensitive dependence on initial conditions. Enter the two lines in the worksheet that appear below the graph. These lines calculate the twentieth points, x_{20}, in the sequences starting at x_0=0.1 and x_0=0.10001 respectively. Note that these values are very different and it appears that we have sensitive dependence on initial conditions. See how close you need to make the initial conditions in order for their values at n=20 to be similar. Try different values of r and see if this behaviour still occurs.

Posted 11 Feb 1999 by Ralph