Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems

Abstract

Phase reduction framework for limit-cycling systems based on isochrons has been used as a powerful tool for analyzing rhythmic phenomena. Recently, the notion of isostables, which complements the isochrons by characterizing amplitudes of the system state, i.e., deviations from the limit-cycle attractor, has been introduced to describe transient dynamics around the limit cycle [Wilson and Moehlis, Phys. Rev. E 94, 052213 (2016)]. In this study, we introduce a framework for a reduced phase-amplitude description of transient dynamics of stable limit-cycling systems. In contrast to the preceding study, the isostables are treated in a fully consistent way with the Koopman operator analysis, which enables us to avoid discontinuities of the isostables and to apply the framework to system states far from the limit cycle. We also propose a new, convenient bi-orthogonalization method to obtain the response functions of the amplitudes, which can be interpreted as an extension of the adjoint covariant Lyapunov vector to transient dynamics in limit-cycling systems. We illustrate the utility of the proposed reduction framework by estimating optimal injection timing of external input that efficiently suppresses deviations of the system state from the limit cycle in a model of a biochemical oscillator.

19 pages, 2 figures

References in corpus (5)

• Applied Koopmanism
• Phase reduction of stochastic limit cycle oscillators
• Phase-amplitude descriptions of neural oscillator models
• Diffusion-induced instability and chaos in random oscillator networks
• Noise in oscillators: a review of state space decomposition approaches

Cited by in corpus (7)

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• Optimal Control of Oscillation Timing and Entrainment Using Large Magnitude Inputs: An Adaptive Phase-Amplitude-Coordinate-Based Approach
• Koopman spectral analysis of elementary cellular automata
• Asymptotic phase and amplitude for classical and semiclassical stochastic oscillators via Koopman operator theory