Forced oscillation modes in a birhythmic system of two coupled relaxation oscillators near Andronoy - Hopf bifurcation

Abstract

A system of two identical relaxation oscillators of the FitzZHugh-Nagumo type with the parameters chosen in the vicinity of a bifurcation of limit cycle emergence was examined for the dynamic modes arising in the presence of a weak harmonic signal applied to both elements. Slow variable exchange between them gives rise to three stable limit cycles called in-phase, anti-phase, and extremely asymmetrical, in which only one of the oscillators generates spikes. In this study, we show that slow variable exchange also causes this system to respond to weak harmonic forcing in a manner quite different from what is known in the classical dynamics of forced oscillations. In addition to the expected synchronization tongues generated by interaction of the signal with the in-phase attractor, we observed at least three other consequences of the coexistence of different solutions: (i) there appeared broad bands of synchronization of the signal with the anti-phase solution at high frequencies multiple to the frequency of the anti-phase oscillations, whereas the base synchronization frequency band became much narrower; (ii) signal period ranges were found in which the limit cycles were such that several spikes were produced over the complete period and each oscillator was characterized with the same set of discrete