title: 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 the 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 the 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 the 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.
authors.affiliation: Graduate School of Information Science and Engineering, Tokyo Institute of Technology, O-okayama 2-12-1, Meguro, Tokyo 152-8552, Japan.
authors.forename: Sho
authors.initials: S
authors.lastname: Shirasaka
authors.affiliation: Faculty of Software and Information Technology, Aomori University, Kobata 2-3-1, Aomori, Aomori 030-0943, Japan.
authors.forename: Wataru
authors.initials: W
authors.lastname: Kurebayashi
authors.affiliation: School of Engineering, Tokyo Institute of Technology, O-okayama 2-12-1, Meguro, Tokyo 152-8552, Japan.
authors.forename: Hiroya
authors.identifier: 0000000333940392
authors.initials: H
authors.lastname: Nakao