Computer Algebra Recipes

Chapter 8
Nonlinear Diagnostic Tools
In the Appetizers, the reader was introduced to the concept of phase-plane
analysis of nonlinear ODE models. This involved the creation of phase-plane
portraits and the location and identi¯cation of the relevant stationary points of
the ODE system. This graphical approach was extended in the last chapter to
¯nding solitary wave solutions of physically important nonlinear PDEs.
Physicists and mathematicians have developed a wide variety of other graphical
tools for exploring the frontiers of nonlinear dynamics and understanding
what is observed. In this chapter, a few of the simpler diagnostic tools for
nonlinear ODEs and nonlinear di®erence equations are presented for the reader
who craves a ¯nal light but scrumptious intellectual dessert.
8.1 The Poincare Section
An important approach to studying the forced motion of nonlinear oscillator
systems is to create a Poincare section. If the driving frequency is !, onetakesa
\snapshot" of the phase plane after each period T0 =2¼=! of the driving force.
After an initial transient time, the ODE system will settle down in steady state
to either a periodic or a chaotic motion. For the periodic case, the system is said
to display a period-n response if its period T equals nT0, wheren =1; 2; 3; ::::
In other words, the frequency response of a period-n solution is !n = !=n.
If the system evolves to a period-1 solution, with T = T0, the Poincare
section will consist of a single plot point in the phase plane that is reproduced
at each multiple of the driving period. On the other hand, if the system evolves
to a period-2 solution, with T =2T0, thePoincaresectionwillhavetwopoints
between which the system oscillates as multiples of T0 elapse. And so on.
In contrast to the periodic situation, for chaotic motion a point is produced
at a di®erent location at each multiple of T0, and the \sum" of the individual
snapshots can produce strange, localized patterns (\strange attractors") of plot
points with complex boundaries in the phase plane.
In the following recipe, the \period-doubling route to chaos" is explored once
again for the forced Du±ng oscillator, now from the Poincare section viewpoint.
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