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.

Recent advances in coupled oscillator theory

“…Understanding the dynamical behavior in directions transverse to the limit cycle (i.e., the amplitude coordinates) is critical to developing higher order approximations of the phase dynamics and there are many possible options for representing both the phase and amplitude coordinates. For instance, [21] and [41] use hyperplanes to denote surfaces of constant phase as part of a higher order asymptotic expansion, [23] and [38] use a moving orthonormal coordinate frame in the definition of phase-amplitude coordinates, and [5], [42], and [33] define amplitude coordinates based on Floquet theory. The coordinates based on Floquet theory have been shown to be particularly useful as they result in relatively simple second order accurate phase-amplitude reduced dynamics [40] [39].…”

Dynamic causal modelling of phase-amplitude interactions

“…The potential insights to be gained by generalizing phase-only models to include amplitude dynamics are being increasingly recognised in fields including neuroscience (Ashwin et al, 2016;Park and Ermentrout, 2016), physics (Kurebayashi et al, 2013;Pyragas and Novicenko, 2015), and chemistry (Shirasaka et al, 2017). We set out to derive a dynamic causal model in which oscillators operating close to a supercritical Poincaré-Andronov-Hopf bifurcation (Marsden and McCracken, 1976) -henceforth referred to simply as a Hopf bifurcation -can evolve beyond their limit cycles.…”

Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds

“…, as S σ is invertible near (η 0 , ξ 0 ). The equation governingξ follows analogously by left acting on equation (14) by the expression given in (12). (15) highlights the exchange of tangent vectors via the action of S σ that motivates our choice to name this the exchange operator.…”

“…Often the dynamics of an ordinary differential equation (ODE) defies rectangular coordinate schemes; that is, the geometry induced by a flow may be difficult to represent in Cartesian coordinates. In fact, a common early step in analysis is to exchange Cartesian coordinates for a geometry better suited for the problem [11,2,10,15,16,12,17]. In this work, we present a technique that allows us to use any embedded manifold (not necessarily invariant) to generate a natural coordinate frame for a dynamical system.…”