Analysis of chaotic saddles in low-dimensional dynamical systems: the derivative nonlinear Schrِdinger equation

In this paper, we present a computational study of nonattracting chaotic sets known as chaotic saddles in a low-dimensional dynamical system describing stationary solutions of the derivative nonlinear Schrödinger equation, a driven-dissipative model for Alfvén waves. These chaotic saddles have “gaps” which are filled at chaotic transitions, such as a saddle-node bifurcation and an interior crisis. We give a detailed explanation of how to numerically determine the chaotic saddles, and describe how a chaotic attractor after an interior crisis point can be “decomposed” into two chaotic saddles, dynamically connected by a set of coupling unstable periodic orbits created by a gap filling “explosion” after the crisis. This coupling between two chaotic saddles is responsible for the intermittent dynamics displayed by the chaotic system after the interior crisis.