Phase reduction theory for hybrid nonlinear oscillators

Abstract

Hybrid dynamical systems characterized by discrete switching of smooth dynamics have been used to model various rhythmic phenomena. However, the phase reduction theory, a fundamental framework for analyzing the synchronization of limit-cycle oscillations in rhythmic systems, has mostly been restricted to smooth dynamical systems. Here we develop a general phase reduction theory for weakly perturbed limit cycles in hybrid dynamical systems that facilitates analysis, control, and optimization of nonlinear oscillators whose smooth models are unavailable or intractable.On the basis of the generalized theory, we analyze injection locking of hybrid limit-cycle oscillators by periodic forcing and reveal their characteristic synchronization properties, such as ultrafast and robust entrainment to the periodic forcing and logarithmic scaling at the synchronization transition. We also illustrate the theory by analyzing the synchronization dynamics of a simple physical model of biped locomotion.

On the concept of dynamical reduction: the case of coupled oscillators

“…In Refs. [47,48], phase equations for hybrid limit-cycle oscillators have been derived. A generalized adjoint equation with discontinuous jumps is derived and the sensitivity function for the phase is obtained.…”

Recent advances in coupled oscillator theory

“…This problem was handled in 2017 by Shirasaka et al, who rigorously defined isochrons for general hybrid systems, and introduced the phase reduction method for weakly perturbed hybrid systems using the saltation matrix [34].…”

Data-driven spectral analysis for coordinative structures in periodic systems with unknown and redundant dynamics

“…We adopted this approach with the assumption that there are underlying dynamical systems or limit cycles behind the segmental angles during locomotion. In limit cycles, phase description has been developed for locomotion such as using local joint angles [47,48,49,13] or right heel-contact cycle as global description [50]. Compared with these studies, we extracted global phases at the gait frequency and at additional harmonic frequencies in a data-driven manner.…”

“…In this study, we confirmed that the theoretical requirements are satisfied for locomotion data from the viewpoint of the convergence of estimation error. Although other locomotion studies obtained phase descriptions using local joint angles [47,48,49,13] or a global one [50], in this study, we obtain global phase descriptions with various frequencies from locomotion data at various speeds. Furthermore, we examine the difference and similarity between the conventional and our approaches by investigating the extracted dynamical information on the conventional coordinative structures.…”

“…One of the popular research subjects of the unknown and redundant dynamic phenomena is human locomotion, which can be performed at various speeds by solving the redundancy problem of many skeletal joints. This is because it seems to be based on specific rules (i.e., can be regarded as a cyclic movement) but the governing equation is not fully understood; thus, from a long time ago, it has attracted attention in many scientific and engineering fields such as neuroscience (e.g., [9,10]), physics (e.g., [11,12,13]), clinical medicine (e.g., [14,15,16]), behavioral science (e.g., [17,18]), robotics (e.g., [8,5]), and pattern recognition (e.g., [19,20]). For the redundant problem, some researchers have discovered synergistic, global, and coordinative structures among controlled components in joint motion (e.g., [10]) and muscle activities (e.g., [9]), which may contribute to solving the problem.…”