Spatial phase sensitivity for oscillators close to the saddle-node homoclinic bifurcation

TL;DR

This paper investigates the spatial phase sensitivity function near saddle-node homoclinic bifurcations, revealing phase accumulation features and the influence of space-dependent noise on oscillator synchronization.

Contribution

It introduces the spatial phase sensitivity function for oscillators near SNH bifurcation and analyzes its implications for phase response and synchronization behavior.

Findings

Spatial PSF reveals phase accumulation on the limit cycle.

Type II phase response curve is the only possibility for 2D smooth systems.

Space-dependent noise coupling significantly affects synchronization.

Abstract

The traditional phase sensitivity function (PSF) has manifested its efficacy in investigating synchronization behaviors for limit-cycle oscillators. However, some subtle details may be ignored when the phase value is accumulated in space or the perturbation is space-dependent. In this paper, we compared spatial PSF with the traditional PSF for oscillators close to the saddle-node homoclinic (SNH) bifurcation, also known as saddle-node on invariant circle (SNIC) bifurcation. It is found that the spatial phase sensitivity function could reveal the phase accumulation feature on the limit cycle. Moreover, it is proved that for any two-dimensional smooth dynamical system, type II phase response curve is the only possible type. Finally, the synchronization distributions of uncoupled SNH oscillator driven by common and independent noises are studied, which shows the space-dependent coupling function of common noise could have significant influences on the synchronization behavior.