Symbolic Toolkit for Chaos Explorations

Abstract

New computational technique based on the symbolic description utilizing kneading invariants is used for explorations of parametric chaos in a two exemplary systems with the Lorenz attractor: a normal model from mathematics, and a laser model from nonlinear optics. The technique allows for uncovering the stunning complexity and universality of the patterns discovered in the bi-parametric scans of the given models and detects their organizing centers -- codimension-two T-points and separating saddles.Comment: International Conference on Theory and Application in Nonlinear Dynamics (ICAND 2012

Symbolic Quest into Homoclinic Chaos

“…It sheds a light on the pivotal role of homoclinic and heteroclinic bifurcations as emergent centers for pattern formations in parameter spaces corresponding to complex dynamics. It also reveals universal principles of chaotic dynamics in deterministic systems with Lorenz-like attractors, which include the Lorenz equation itself and similar models [Barrio et al, 2012[Barrio et al, , 2013Xing et al, 2014a;Xing et al, 2014b]. All these systems feature various codimension-two heteroclinic and homoclinic bifurcations such as Bykov T-points, resonant saddles and inclination-switching.…”

“…(1): the saddle at the origin and two symmetric saddle-foci of the (1, 2)-type. Such points turn out to cause the occurrence of self-similar, fractal structures in the parameter region corresponding to chaotic dynamics in the known systems with the Lorenz attractor [ Barrio et al, 2012;Xing et al, 2014a;Xing et al, 2014b]. Figure 4 presents a Lyapunov exponent (LE) based sweep of the parameter space of the model with its attractors superimposed in the color-coded regions.…”

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

“…1), with all the quintessential organizing structures, pivotal to understand its complex dynamics. These include various homoclinic and heteroclinic bifurcation structures of co-dimensions one and two, the so-called Bykov T-points with the associated spirals, as well as parametric saddles for switching branches in the parameter plane of this model, that are also seen in the classical Lorenz and Shimizu-Morioka models [1,12,16,17,18,23,33,36,46]. Similar structures were also discovered in another non-linear optics model describing a laser with a saturable absorber, which can be locally reduced to the Shimizu-Morioka model near a steady-state solution with triple zero Lyapunov exponents [47,48].…”

“…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.…”