Homoclinic chaos and its organization in a nonlinear optics model

Abstract

We developed a powerful computational approach to elaborate on onset mechanisms of deterministic chaos due to complex homoclinic bifurcations in diverse systems. Its core is the reduction of phase space dynamics to symbolic binary representations that lets one detect regions of simple and complex dynamics as well as fine organization structures of the latter in parameter space. Massively parallel simulations shorten the computational time to disclose highly detailed bifurcation diagrams to a few seconds.

Dynamics and bifurcations in multistable 3-cell neural networks

“…4 reveal the sequence of bifurcations that stable rhythms undergo near the borderlines. Such bifurcation diagrams have proven useful for studying the dynamics of small CPG-circuits and other nonlinear systems [44][45][46][47][48] . In addition to the fully symmetric motif in Fig.…”

(INVITED) Homoclinic puzzles and chaos in a nonlinear laser model

“…3,4 and detailed magnifications are presented in Fig. 8 [30]. Each characteristic spiral around a T-point in the parameter plane corresponds to a homoclinic loop of the saddle O in the phase space, such that with each turn of the spiral approaching the T-point, the outgoing separatrix of saddle makes increasing number of loops around a saddle-focus P ± , before finally forming a closed heteroclinic connection between the saddle and the saddle focus at the T-point.…”

“…We introduce and discuss the basic elements of homoclinic bifurcations in the text later, as well as how short-term symbolic dynamics can be introduced to disclose a stunning array of homoclinic structures and their organizing centers in all Lorenz-like systems, including the optically pumped laser (OPL) model under consideration in this paper. Some pilot results on the use of symbolic dynamics for the OPL model can be found in [17,30]. In addition to simple dynamics associated with stable equilibria and periodic orbits, this system reveals a broad range of bifurcation structures that are typical for many ODE models from nonlinear optics and ones with the Lorenz attractor [18,29,33,35,36].…”

“…Our motivation is to extend the existing theory of homoclinic bifurcations of low-codimensions [1,5,19,20,21,22,3,23,24,25,26,27,28] and to make it accessible and practical for the nonlinear dynamics community. Recently, we have advanced an approach called the "Deterministic Chaos Prospector" (DCP) that utilizes symbolic representations of simple and chaotic dynamics [29,30,31,32], based on our earlier works [18,33]. This allows for fast and effective identification of bifurcation structures underlying and governing deterministic chaos in various systems.…”

Homoclinic chaos in the Rössler model

“…This is the first step prior to applying more dedicated tools for examining a variety of homoclinic bifurcations. We previously developed a symbolic toolkit, code-named deterministic chaos prospector (DCP), running on graphics processing units (GPUs) to perform indepth, high-resolution sweeps of control parameters to disclose the fine organization of characteristic homoclinic and heteroclinic bifurcations and structures that have been universally observed in various Lorenz-like systems, see [13][14][15][16][17] and the reference therein. In addition to this approach capitalizing on sensitive dependence of chaos on parameter variations, the structural stability of regular dynamics can also be utilized to fast and accurately detect regions of simple and chaotic dynamics in a parameter space of the system in question 18 .…”