Smale-Williams solenoids in autonomous system with saddle equilibrium.

Abstract

We construct an autonomous low-dimensional system of differential equations by replacement of real-valued variables with complex-valued variables in a self-oscillating system with homoclinic loops of a saddle. We provide analytical and numerical indications and argue that the emerging chaotic attractor is a uniformly hyperbolic chaotic attractor of Smale-Williams type. The four-dimensional phase space of the flow consists of two parts: a vicinity of a saddle equilibrium with two pairs of equal eigenvalues where the angular variable undergoes a Bernoulli map, and a region which ensures that the trajectories return to the origin without angular variable changing. The trajectories of the flow approach and leave the vicinity of the saddle equilibrium with the arguments of complex variables undergoing a Bernoulli map on each return. This makes possible the formation of the attractor of a Smale-Williams type in Poincaré cross section. In essence, our model resembles complex amplitude equations governing the dynamics of wave envelops or spatial Fourier modes. We discuss the roughness and generality of our scheme.