CHARACTERIZATION OF A SIMPLE CLASS OF MODULATED RELAXATION-OSCILLATORS

The phase-locking structure in modulated integrate-and-fire systems is explored. The existence of a smooth critical line where the Poincaré map has an infinite-slope inflection point is emphasized. At and below this line the system is related to circle map systems. This especially, allows realization of systems with higher-order scaling structures, qualitatively as well as quantitatively distinct from ordinary third-order circle map structures. Hourglass patterns develop in parameter space, and at small modulation amplitudes the behavior of the phase-locking regions (Arnold tongues) changes dramatically. Above the critical line the Arnold tongues complete the parameter space, leaving along any line a zero-dimensional Cantor set of points associated with irrational rotation numbers. The critical line is not associated with a transition to chaos. In particular, nonchaotic regions with complete phase locking exist. In the supercritical region a gap is present in the Poincaré map. The features at this gap are examined. Also, local hysteresis may occur. We discuss the applicability of the local approximation. Our picture provides a natural explanation for the missing hysteresis in the Belousov-Zhabotinsky reaction and for the bumps in the Arnold tongues observed in driven relaxation oscillators. © 1990 The American Physical Society.