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.

37 pages, 5 figures

References in corpus (5)

• Phase reduction of stochastic limit cycle oscillators
• Collective Phase Sensitivity
• Phase Reduction Method for Strongly Perturbed Limit Cycle Oscillators
• Phase reduction approach to synchronization of spatiotemporal rhythms in reaction-diffusion systems
• Phase description of oscillatory convection with a spatially translational mode

Cited by in corpus (14)

• Phase reduction approach to synchronization of nonlinear oscillators
• On the Concept of Dynamical Reduction : The Case of Coupled Oscillators
• Recent Advances in Coupled Oscillator Theory
• Methods of Continuation and their Implementation in the COCO Software Platform with Application to Delay Differential Equations
• Semiclassical Phase Reduction Theory for Quantum Synchronization
• Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction and Floquet theory
• The Infinitesimal Phase Response Curves of Oscillators in Piecewise Smooth Dynamical Systems
• Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling
• Optimization of periodic input waveforms for global entrainment of weakly forced limit-cycle oscillators
• Optimization of linear and nonlinear interaction schemes for stable synchronization of weakly coupled limit-cycle oscillators
• Phase dynamics of noise-induced coherent oscillations in excitable systems
• Koopman Analysis of the Singularly-Perturbed van der Pol Oscillator
• Sensitivity Analysis for Periodic Orbits and Quasiperiodic Invariant Tori Using the Adjoint Method
• Guaranteed phase synchronization of hybrid oscillators using symbolic Euler's method: The Brusselator and biped examples