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

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 ex-amining 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 dynam-ics 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 understand-ing of the soliton dynamics under parameter variation of the underlying nonlinear dynamics.(c) 2022 Elsevier Ltd. All rights reserved.