Toward a General Procedure for Extracting Templates from Chaotic Attractors Bounded by High Genus Torus

Abstract

The topological analysis of chaotic attractor by means of template is rather well established for simple attractors as solution to the Rössler system. Lorenz-like attractors are already slightly more complicated because they are bounded by a genus-3 bounding torus, implying the necessity to use a two-component Poincaré section. In this paper, we enriched the concept of linking matrix to correctly describe an algebraic template for an attractor with (g - 1) components of Poincaré section and whose bounding torus has g interior holes aligned. An example with g = 5 — a multispiral attractor — is explicitly treated.

Templates of Two Foliated Attractors — Lorenz and Chen Systems

“…The procedure of topological characterisation of attractors is presented for a solution of the Sprott system [9]. Recently [8], it is also presented and applied on an attractor solution of the Rössler system [10] and of the Lorenz system [1]. To remind the major steps of this procedure, firstly we summarize the topological characterization of a Rössler attractor, already studied by Letellier et al [11].…”

“…Also from this unimodal map, a symbolic dynamic is constructed in order to obtain a partition of the map where trajectories have the same behaviour over a period. Finally, the linking numbers between pair of orbits are computed and a template is proposed to summarized the relative organization of these orbits by providing the theoretical linking number (see [11,8] for details).…”

“…The theory of toroidal boundary introduced by Tsankov & Gilmore [6] indicates that a two components Poincaré section is required to get a discretization of trajectories. As previously done for this system [8], a Poincaré section is defined using two symmetric components…”

“…where ρ A,n and ρ B,n are normalised in ]0; 1[ and represent a component from the inside to the outside of the flow for each focus [8]; the indicator function of a component is a value, it is 1 if the trajectory is in the component and 0 else. Fig.…”

“…In their second paper [7] they explained how the Lorenz template can be rewritten in order to fit its genus-3 toroidal boundary. Recently, the general method has been extended in order to obtain the template of chaotic attractors bounded by higher genus torus [8]. Using this method, the topological characterization of two foliated attractors is performed, one is a solution of the Lorenz system, and the other attractor is solution of the Chen system [2] with more foliations.…”