2013] “Homoclinic spirals: Theory and numerics

Abstract In this paper we examine spiral structures in bi-parametric diagrams of dissipative systems with strange attractors. First, we show that the organizing center for spiral structures in a model with the Shilnikov saddle-focus is related to the change of the structure of the attractor transitioning between the spiral and screw-like types located at the turning point of a homoclinic bifurcation curve. Then, a new computational technique based on the symbolic description utilizing kneading invariants is proposed for explorations of parametric chaos in Lorenz like attractors. 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.