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

Abstract

We present a case study elaborating on the multiplicity and self-similarity of homoclinic and heteroclinic bifurcation structures in the 2D and 3D parameter spaces of a nonlinear laser model with a Lorenz-like chaotic attractor. In a symbiotic approach combining the traditional parameter continuation methods using MatCont and a newly developed technique called the Deterministic Chaos Prospector (DCP) utilizing symbolic dynamics on fast parallel computing hardware with graphics processing units (GPUs), we exhibit how specific codimension-two bifurcations originate and pattern regions of chaotic and simple dynamics in this classical model. We show detailed computational reconstructions of key bifurcation structures such as Bykov T-point spirals and inclination flips in 2D parameter space, as well as the spatial organization and 3D embedding of bifurcation surfaces, parametric saddles, and isolated closed curves (isolas).

Robust Stabilization and Synchronization of a Novel Chaotic System with Input Saturation Constraints

“…This study considers the discovery of a novel chaotic system, as well as the homoclinic and heteroclinic chaotic properties of this type of system. For example, in papers such as that in [ 58 ], it is demonstrated how some co-dimensions two bifurcations originate regions of chaotic and simple dynamics, and bifurcation structures such as Bykov T-points spirals are evinced. In papers such as that in [ 59 ], it is demonstrated how multiple chaos can arise from single parametric perturbations of a degenerated homoclinic orbit.…”

Homoclinic chaos in the Rössler model

“…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 . The Z 2 -symmetry is exploited to generate periodic or aperiodic bi- nary sequences that can be associated, respectively, with simple or chaotic flip-flop patterns of solutions of Lorenz-like systems.…”