Smale – Williams Solenoids in a System of Coupled Bonhoeffer – van der Pol Oscillators

The principle of constructing a new class of systems demonstrating hyperbolic chaotic attractors is proposed. It is based on using subsystems, the transfer of oscillatory excitation between which is provided resonantly due to the difference in the frequencies of small and large (relaxation) oscillations by an integer number of times, accompanied by phase transformation according to an expanding circle map. As an example, we consider a system where a Smale – Williams attractor is realized, which is based on two coupled Bonhoeffer – van der Pol oscillators. Due to the applied modulation of parameter controlling the Andronov – Hopf bifurcation, the oscillators manifest activity and suppression turn by turn. With appropriate selection of the modulation form, relaxation oscillations occur at the end of each activity stage, the fundamental frequency of which is by an integer factor $M=2,3,4, \ldots$ smaller than that of small oscillations. When the partner oscillator enters the activity stage, the oscillations start being stimulated by the $M$-th harmonic of the relaxation oscillations, so that the transformation of the oscillation phase during the modulation period corresponds to the $M$-fold expanding circle map. In the state space of the Poincaré map this corresponds to an attractor of Smale – Williams type, constructed with $M$-fold increase in the number of turns of the winding at each step of the mapping. The results of numerical studies confirming the occurrence of the hyperbolic attractors in certain parameter domains are presented, including the waveforms of the oscillations, portraits of attractors, diagrams illustrating the phase transformation according to the expanding circle map, Lyapunov exponents, and charts of dynamic regimes in parameter planes. The hyperbolic nature of the attractors is verified by numerical calculations that confirm the absence of tangencies of stable and unstable manifolds for trajectories on the attractor (“criterion of angles”). An electronic circuit is proposed that implements this principle of obtaining the hyperbolic chaos and its functioning is demonstrated using the software package Multisim.