Systematic template extraction from chaotic attractors: I. Genus-one attractors with an inversion symmetry

摘要

Describing the topological properties by a template is a powerful technique to classify chaotic attractors. Most of the time, reduced templates are used, but direct templates - in which all mechanisms (torsions, branch permutation, etc) identified in the attractor are explicitly described without any simplification - are of great interest for a better description of the subtleties of the dynamics. We introduce here two additional conventions for representing in a unique way the reduced template from a given linking matrix. We then propose an addition law for linking matrices which allows us to manipulate (combine) different mechanisms. A direct template can thus be described by using a series of linking matrices. On the other hand, we show how to analytically build the linking matrix associated with an attractor which is the image under an inversion symmetry of an attractor whose linking matrix is known.