Hyperbolic chaotic attractor in amplitude dynamics of coupled self-oscillators with periodic parameter modulation

The paper proposes an approach to constructing feasible examples of dynamical systems with hyperbolic chaotic attractors based on the successive transfer of excitation between two pairs of self-oscillators that are alternately active. An angular variable that measures the relations of the current amplitudes for the two oscillators of each pair undergoes a transformation in accordance with the expanding circle map during each cycle of the process. We start with equations describing the dynamics in terms of complex or real amplitudes and then examine two models based on van der Pol oscillators. One model corresponds to the situation of equality of natural frequencies of the partial oscillators, and another to a nonresonant ratio of the oscillation frequencies relating to each of the two pairs. Dynamics of all models are illustrated with diagrams indicating the transformation of the angular variables, portraits of attractors, Lyapunov exponents, etc. The uniformly hyperbolic nature of the attractor in the stroboscopic Poincaré map is confirmed for a real-amplitude version of the equations by computations of statistical distribution of angles between stable and unstable manifolds at a representative set of points on the attractor. In other versions of the equations the attractors relate presumably to the partially hyperbolic class.