Juan Patiño-Echeverría, Bernd Krauskopf, and Hinke M. Osinga
Abstract
Wild chaos is a higher-dimensional form of chaotic dynamics characterized by the robust presence of homoclinic tangencies. It can only arise in vector fields of dimension at least four, or in diffeomorphisms of dimension at least three. We study a four-dimensional Lorenz-like system known to exhibit a wild pseudohyperbolic attractor for a specific parameter point. Its global bifurcation structure in a two-parameter plane has previously been analysed numerically by computing kneading diagrams and Lyapunov spectra computations, revealing dense accumulations of Shilnikov-type homoclinic bifurcations and parameter regions satisfying necessary conditions for wild chaos.
Here, we build on this analysis by finding and continuing the codimension-one bifurcation curves associated with two fundamental symmetric periodic orbits and their asymmetric counterparts. We determine the criticality of symmetry-breaking, period-doubling, fold, and torus bifurcations, and construct the global two-parameter bifurcation diagram in a compactified parameter plane. These bifurcation curves are shown to form a nested structure that organizes both periodic windows and chaotic regions. In particular, they delimit transitions between periodic dynamics, chaotic attractors generated by period-doubling cascades, and wild pseudohyperbolic attractors. While no single bifurcation curve forms a sharp boundary of wild chaos, torus and fold bifurcations of symmetric periodic orbits are found to play a significant role in organizing the nature of attractors in the system; this finding is supported by numerical computation of Lyapunov spectra for said attractors.
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