Posted: September 13th, 2017

Nonlinear Daynamics

Paper, Order, or Assignment Requirements

HW1: The objective of this homework is to introduce the concepts of fixed points, stability, bifurcations, and chaos via experimental (i.e., numerical) computation. We choose a very simple mathematical model; although it is highly abstract, we will argue later in the course that the model is also to a significant degree representative of the behavior of many real systems.

Let us parameterize a system by a single time­dependent variable, x(t). The variable x could represent, say, the globally averaged temperature on the Earth’s surface, the size of a particular population of animals on some secluded island, or perhaps even a particular stock market average. Whatever x(t) represents, let us suppose that we are interested only in values x_j at discrete times t = jτ , where τ is some suitably small interval of time. The evolution of x_j in time may then be written as

x_j+1 = R(x_j)

where R is some function which describes the dynamics.

Case1. Consider first perhaps the most simple R, such that

x_j+1 = µx_j

and −∞ < µ < ∞ is a parameter. Without using the computer, what can you predict about the value of x_j given an initial condition x₀? Qualitatively, what is different about the cases |µ| < 1, | | µ > 1, and |µ| = 1? What is the qualitative difference between the evolution with µ > 0 and µ < 0? Justify your answers using the concepts learned so far.

The Matlab function iterate.m has been set up to solve equation (2). If you have any doubts concerning how to run the computer, or any doubts concerning your answers above, use the computer to verify your predictions for above.

Case 2: Consider the nonlinear evolution

x_j+1 = 4µx_j(1 − x_j)

Show that if x_j = x*, where

Then x* is a fixed point of the dynamics.

Case3. The parameter µ in case 2 determines whether the fixed points are stable. Let us investigate the question of stability experimentally. Modify your computer program so that it simulates the evolution of case 2. Verify with a few tests that if 0.25 < µ < 0.75, then for any initial x₀ such that 0 < x₀ < 1, the iterates x_j eventually converge to the value predicted by x*. These fixed points are called stable because, even when x₀ is chosen to be different from x*, the evolution eventually returns to x_j = x*.

Case 4: Try the same experiment by choosing µ = 0.76. What happens now? Is the fixed point predicted by x* still stable? What is the long­term, asymptotic behavior?

Case 5: The value µ = 0.75 is called a critical value of µ because the long­term evolution of x_j qualitatively changes as µ increases to µ = 0.75 from below. We call changes to the dynamics “qualitative” if, say, a fixed point changes from stable to unstable, or the period of oscillation changes from, say, 2 iterations to 4 iterations. These qualitative changes are called bifurcations. Can you find another critical value of µ between µ = 0.75 and µ = 1.0 by numerical experimentation? (Restrict the initial condition to 0 < x0 < 1.) What qualitative change occurs now?

Case 6: Graph the evolution of case 2 for the µ = 0.89, for some initial condition x₀. Then do the same for an initial condition x₀ +ε , where ε is very small (say, less than 0.001). Is there any difference in the long­term behavior of x_j ?

Case 7: Repeat this for µ = 0.91. What happens now?