An example of a non-autonomous continuous-time system with attractor of Plykin type in the Poincaré map

S. P. Kuznetsov

Russian journal of nonlinear dynamics, Tome 5 (2009) no. 3, pp. 403-424

Résumé

A non-autonomous flow system is introduced, which may serve as a base for elaboration of real systems and devices demonstrating the structurally stable chaotic dynamics. The starting point is the map of the sphere composed of four stages of sequential continuous geometrically evident transformations. The computations indicate that in a certain parameter range the map posseses an attractor of Plykin type. Accounting the structural stability intrinsic to this attractor, a modification of the model is undertaken, which includes a variable change with passage to representation of instantaneous states on the plane. As a result, a set of two non-autonomous differential equations of the first order with smooth coefficients is obtained explicitly, which has the Plykin type attractor in the plane in the Poincaré cross-section. Results of computations are presented for the sphere map and for the flow system including portraits of attractors, Lyapunov exponents, dimension estimates. Substantiation of the hyperbolic nature of the attractors for the sphere map and for the flow system is based on a computer procedure of verification of the so-called cone criterion; in this context, some hints are applied, which may be useful in similar computations for some other systems.

Détail

Keywords: hyperbolic chaos, Plykin attractor, Lyapunov exponent, structural stability.