Unraveling the Chaos-Land and Its Organization in the Rabinovich System

Abstract

A suite of analytical and computational techniques based on symbolic representations of simple and complex dynamics, is further developed and employed to unravel the global organization of bi-parametric structures that underlie the emergence of chaos in a simplified resonantly coupled wave triplet system, known as the Rabinovich system. Bi-parametric scans reveal the stunning intricacy and intramural connections between homoclinic and heteroclinic connections, and codimension-2 Bykov T-points and saddle structures, which are the prime organizing centers of complexity of the bifurcation unfolding of the given system. This suite includes Deterministic Chaos Prospector (DCP) to sweep and effectively identify regions of simple (Morse-Smale) and chaotic structurally unstable dynamics in the system. Our analysis provides striking new insights into the complex behaviors exhibited by this and similar systems.

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

“…Topologically identical regions in the parametric plane can be identified by defining a formal convergent power series P [29,62] for a sequence {k i } q i=p defined as:…”

“…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]. These include homoclinic orbits and heteroclinic connections between saddle equilibria that are the key building blocks of deterministic chaos in most systems.…”

“…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 .…”