Abstract
Abstract The existence of multiple stable states of higher order m:n locking in coupled limit cycle oscillators has been studied by prior authors in the context of injection-locking in systems driven by an external periodic force. The current work builds on this concept to study the higher order locking characteristics of pairs of limit cycle oscillators self-synchronizing under coupling forces. To this end we analyze three oscillator systems: Van der Pol oscillators using numerical analysis, a simplified model for MEMS oscillators using numerical analysis as well as perturbation theory, and a full model of thermo-optically driven MEMS oscillators using numerical analysis. For the Van der Pol system, higher order locking is observed for the strongly nonlinear case corresponding to relaxation oscillations and the transition from weak to strong nonlinearity is studied using a parameter sweep. Additionally, coupling of a different nature such as quadratic coupling is also capable of inducing higher order coupling in Van der Pol oscillators. For the MEMS systems with linear coupling, higher order locking is observed when a strong cubic stiffness nonlinearity exists. Devil’s staircase-like structures are obtained for the coupling strength-frequency ratio parameter space which suggest overlapping Arnold locking regions for m:n locks corresponding to different integers.
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