Chaotic diffusion of dissipative solitons: From anti-persistent random walks to Hidden Markov Models

Abstract

In previous publications, we showed that the incremental process of the chaotic diffusion of dissipative solitons in a prototypical complex Ginzburg-Landau equation, known, e.g., from nonlinear optics, is governed by a simple Markov process leading to an Anti-Persistent Random Walk of motion or by a more complex Hidden Markov Model with continuous output densities. In this article, we reveal the transition between these two models by examining the dependence of the soliton dynamics on the main bifurcation parameter of the cubic-quintic Ginzburg-Landau equation, and by identifying the underlying hidden Markov processes. These models capture the non-trivial decay of correlations in jump widths and sequences of symbols representing the symbolic dynamics of short and long jumps, the statistics of anti-persistent walk episodes, and the multimodal density of the jump widths. We demonstrate that there exists a physically meaningful reduction of the dynamics of an infinite-dimensional deterministic system to one of a probabilistic finite state machine and provide a deeper understanding of the soliton dynamics under parameter variation of the underlying nonlinear dynamics.