An example of physical system with hyperbolic attractor of Smale – Williams type S . P . Kuznetsov

A simple and transparent example of a non-autonomous flow system, with hyper-bolic strange attractor is suggested. The system is constructed on a basis of two coupled van der Pol oscillators, the characteristic frequencies differ twice, and the parameters controlling generation in both oscillators undergo a slow periodic counter-phase variation in time. In terms of stroboscopic Poincaré section, the respective four-dimensional mapping has a hyperbolic strange attractor of Smale – Williams type. Qualitative reasoning and quantitative data of numerical computations are presented and discussed, e.g. Lyapunov exponents and their parameter dependencies. A special test for hy-perbolicity based on statistical analysis of distributions of angles between stable and unstable subspaces of a chaotic trajectory has been performed. Perspectives of further comparative studies of hyperbolic and non-hyperbolic chaotic dynamics in physical aspect are outlined. Mathematical theory of chaotic dynamics based on rigorous axiomatic foundation exploits a notion of hyperbolicity, which implies that all relevant trajectories in phase space of a dynamical system are of saddle type, with well defined stable and unstable directions [1, 2, 3, 4]. Hyperbolic systems of dissipative type, contracting the phase space volume, manifest robust strange attractors with strong chaotic properties. The robustness (structural stability) implies insensitivity of the motions in respect to variations of equations governing the dynamics. In particular, positive Lyapunov exponent depends on parameters in smooth manner, without flops into negative region characteristic to non-hyperbolic attractors. A Cantor-like structure of the strange attractor persists without qualitative changes (bifurca-tions), at least while the variations are not too large. Textbook examples of these robust strange attractors are represented only by artificial mathematical constructions associated with discrete-time models, e.g. Plykin attractor and Smale – Williams attractor (solenoid). It seems that the mathematical theory of hyperbolic chaos has been never applied conclusively to any physical object, although concepts of this theory are widely used for interpretation of chaotic behavior of realistic nonlinear systems. On the other hand, feasible nonlinear systems with complex dynamics, such as Lorenz and Rössler equations, chaotic self-oscillators, driven nonlinear oscillators etc. do not relate to the true hyperbolic class [4, 5, 6]. As a rule, observable chaos in these systems is linked with a so-called quasiat-tractor, a set in phase space, on which chaotic trajectories coexist with stable orbits of