October 24, 2013 --- Class 17 --- Instability and Chaos Lyapunov Exponent,
Numerical Integration in One Dimension
Instability and Chaos
A key issue in chaotic systems is sensitivity to initial
conditions. This is discussed in CSM in Problem 6.3 on page
147. We looked at r=0.91 as described in part (a) of that
problem. It is easy to write a program that will follow two
trajectories starting at nearby values and see how they converge
or diverge. We found that for r=0.91, nearby starting points
diverge. The divergence is roughly exponential. We reproduced
Fig. 6.8 on page 158 using my code map_diff.c which is in my home
directory.
Lyapunov Exponent
The Lyapunov exponent describes the exponential growth or decay
of the difference between two nearby trajectories. Equations (6.15)
and (6.16) are the basis of a computer code to calculate the
Lyapunov exponent. We reproduced Fig. 6.9 on page 160.
Bifurcation points correspond to the Lyapunov exponent going to
zero. The big dips correspond to the superstable attractors.
The transtion to chaos occurs when the Lyapunov exponent first
becomes positive. Note some islands of stability above that
transition point where the Lyapunov exponent becomes negative.
Code for caluclating the lyapunov exponent can be found in
~sg/chap6/prob9.
Numerical Integration
We talked about some of the standard methods for numerical
integration in one dimension. This is discussed in Section 11.1
of CSM. Additional material is available in Numerical Recipes
or in Abramowitz and Stegun. We discussed the rectangle rule,
trapezoid rule, and Simpson's rule.
In the next class, we will discuss how to estimate the error in
each method and how to get Simpson's rule from the
trapeziod rule. We will then compare different methods in
in a simple example.