Singular-Perturbation Homoclinic Hyperchaos

Abstract

The relaxation-oscillation or singular-perturbation principle, respectively, lends itself to the generation of chaotic flows out of two 2-D flows connected via two 1-D thresholds (“reinjection principle”). In the same vein, two 3-D flows can be overlaid to generate hyperchaos using two 2-D thresholds. A special case in spiral-flow-based singular-perturbation chaos is “homoclinic” reinjection, a codimension-one situation that is covered by Shil’nikov’s theorem after time inversion. An analogous reinjection in screw-flow-based hyperchaos is of codimension two. An infinitely often folded (in the limit noninvertible) hyperhorseshoe exists near the homoclinic trajectory. The latter can be part of a strictly self-similar basin boundary.