March 6, 2013 --- Class 18 --- Lyapunov Exponent, Numerical Integration
in One Dimension
Lyapunov Exponent
Near the end of class we briefly discussed the Lyapunov exponent
which 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.
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 compared the accuracy of
various methods like the rectangle rule, trapezoid rule and
Simpson's rule.
In the next class we will discuss how to estimate the error in
Simpson's rule and how to get Simpson's rule from the
trapeziod rule. We will then compare different methods in
in a simple example.