Straddle Trajectories

Abstract

The ideal behind the Dynamics program is that we must be able to visualize what is happening in a dynamical system. That means not just seeing chaotic attractors, but all important elements of dynamics. This chapter presents newly developed techniques for observing trajectories that lie in invariant sets which are not attractors. When trying to understand the global dynamics of a pendulum, a pendulum with friction, one must be familiar with the unstable solution that has the pendulum bob inverted, pointing straight up, as well as the attracting solution. Almost every trajectory starting near this unstable steady state will diverge from it, being pulled down by gravity. Most nonlinear systems have analogues of this unstable state. Sometimes the analogue is a single point, an unstable steady state, and sometimes an invariant chaotic set that is not attracting. We refer to such invariant chaotic sets that are not attracting as chaotic saddles. A chaotic saddle is an invariant set C that is neither attracting nor repelling and contains a chaotic trajectory which is dense in C. In other words, a chaotic saddle is an invariant set C that is neither attracting nor repelling, and there exists a point in C whose trajectory has a positive Lyapunov exponent and travels throughout C. An example of a chaotic saddle is the invariant Cantor set for the Henon map with parameter values RHO = 5, Cl = 0.3; see Example 2–19 which introduces Saddle Straddle Trajectories. Basin boundaries are examples of invariant sets that are not attractors; see Chapter 7 for the definition of basin boundary. They sometimes contain chaotic saddles.