Chaotic Diffusion of Dissipative Solitons: From Anti-Persistent Random Walk to Hidden Markov Models

TL;DR

This paper investigates the transition in chaotic diffusion behavior of dissipative solitons from simple Markov processes to complex Hidden Markov Models, revealing underlying stochastic mechanisms and reducing infinite-dimensional dynamics to finite-state models.

Contribution

It uncovers how soliton dynamics transition between Markov and Hidden Markov models and identifies the hidden processes governing their chaotic diffusion behavior.

Findings

Transition from simple Markov to Hidden Markov models with parameter change

Identification of hidden Markov processes underlying soliton dynamics

Reduction of infinite-dimensional dynamics to finite-state probabilistic 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 studying the soliton dynamics in dependence on the main bifurcation parameter of the Ginzburg-Landau equation and identify the underlying hidden Markov processes. These models capture the non-trivial decay of correlations in jump widths and symbol sequences representing the soliton motion, 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.