  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.
