Smale–Williams attractors at dissipative perturbations of an integrable Hamiltonian system

We study perturbations of an integrable Hamiltonian system with two degrees of freedom that possesses a saddle-focus equilibrium with a homoclinic skirt-a one-parameter family of homoclinic orbits of the saddle-focus whose stable and unstable invariant manifolds coincide. The perturbations are dissipative, and we consider those that split the skirt completely (i.e., no intersections remain). We are interested in the formation of attractors of the perturbed system, both simple (invariant tori) and chaotic ones. We construct three model three-dimensional diffeomorphisms that capture the properties of the system and study their invariant sets. In the first map, we find a stable invariant torus, whose destruction via the Afraimovich-Shilnikov scenario leads to the appearance of non-robust chaotic attractors. More interesting chaotic attractors are found for the second and third maps. Based on the results of our numerical experiments, we conclude that these attractors can be pseudo-hyperbolic and even hyperbolic of the Smale-Williams type. Their properties are verified by calculating both the Lyapunov exponents and the angles between the stable and unstable subbundles of the attractors. To the best of our knowledge, this is the first time such attractors have been observed in perturbed Hamiltonian systems.