Periodic forcing of a limit-cycle oscillator: Fixed points, Arnold tongues, and the global organization of bifurcations

The effects of periodic pulsatile stimuli on a nonlinear limit-cycle oscillation are analyzed for various relaxation rates to the limit-cycle oscillation. In the infinite relaxation limit, the effects of periodic stimuli are analyzed by consideration of the bifurcations of circle maps. In the case of a finite relaxation rate, it is necessary to analyze two-dimensional maps of the disk. However, the simple structure of the limit cycle allows us to carry out detailed theoretical and numerical analyses of the dynamics. Using the Brouwer fixed point theorem, we show that for any finite nonzero frequency and amplitude of the stimulus, there will be a period-1 fixed point associated with a period-1 phase locking. We use analytical and numerical methods to determine the stability boundary of the period-1 fixed point as a function of the stimulation frequency, amplitude, and relaxation rate to the limit cycle. These results, combined with numerical studies give insight into the changes in the global organization of the phase locking zones as a function of the relaxation rate to the limit cycle.