Relation of nonlinear oscillator design based on phase reduction method and fractional derivative

Abstract

Summary For wide application of the synchronization properties of nonlinear oscillators in control and signal processing engineering field, a common design method for nonlinear oscillator models is required. The output response of a nonlinear oscillator excited by a periodic signal implicitly includes information on both the instantaneous phase and the amplitude of the input signal. The design of a nonlinear oscillator model that can synchronize with an arbitrary cyclic phenomenon can be enabled by explicitly expressing the dynamics of both phase and amplitude with the phase reduction method and organizing them by the number of events in one period. Assuming the cyclic phenomenon to be a single periodic signal, the properties of the generalized nonlinear oscillator model depend on the phase resolution of the signal, which is associated with the phase‐shift of fractional calculus. Thus, this study verifies the validity of fractional calculus through the input‐output characteristic of the generalized nonlinear oscillator model. Numerical simulation using a fractional differentiator with backward‐difference demonstrated that increasing the phase resolution of a single periodic signal improves the estimation accuracy of the phase and amplitude of the signal by employing a generalized nonlinear oscillator model.