Fractal Parameter Space of Lorenz-like Attractors: A Hierarchical Approach

(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.…”

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 .…”

Lorenz attractors and the modular surface *

“…The equations have certain parameter values called Tparameters, at which there are two heteroclinic orbits, forming an interval connecting the three singular points in the equations [3,7,8,13,21]. Many T-points are numerically found in the Lorenz equations [5,10,28], and we consider the simplest one called the primary T-point at (ρ ≈ 30.87, σ ≈ 10.17, β = 8 3 ). At this point, the heteroclinic connections can be extended into an invariant trefoil knot passing through infinity [22], shown in figure 1.…”