Routes to chaos and complete phase locking in modulated relaxation oscillators

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Abstract
Relaxation oscillations are very commonly found in nature. When modulated by an external field, such systems show phase-locked, quasiperiodic, or chaotic behavior, dependent on the specific parameters. We present an exact analysis for a triangular modulated relaxation oscillator and determine the parameter-space phase diagram. We identify critical lines associated with qualitatively different transitions to chaos and complete phase locking (CPL). One transition is related to overlapping of phase-locked regions and is also identified as the transition from quasiperiodicity to chaos described by a noninvertibility of the Poincaré map. Another is a nonchaotic transition to a CPL regime, where a gap appears in the Poincaré map. Also, we find a sudden transition between this CPL regime and a regime where all attractors are chaotic. The critical lines separating the different regimes are found and attributed to either a horizontal or vertical line segment in the Poincaré map. Moreover, we find analytically a number of scaling relations for the phase-locked stability intervals on and nearby the critical lines. We also comment on the situation when the modulation is sinusoidal and when damping is present.