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The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems

Published online by Cambridge University Press:  02 April 2018

Y. PARK ,
K. M. SHAW ,
H. J. CHIEL  and
P. J. THOMAS
Y. PARK
Affiliation:
Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA email: yop6@pitt.edu, pjthomas@case.edu Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
K. M. SHAW
Affiliation:
Department of Biology, Case Western Reserve University, Cleveland, OH 44106, USA e-mail: hjc@case.edu Department of Anesthesia, Critical Care, Small and Pain Medicine, Massachusetts General Hospital, Boston, MA 02114, USA email: kmshaw@partners.org
H. J. CHIEL
Affiliation:
Department of Biology, Case Western Reserve University, Cleveland, OH 44106, USA e-mail: hjc@case.edu
P. J. THOMAS
Affiliation:
Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA email: yop6@pitt.edu, pjthomas@case.edu
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Abstract

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The asymptotic phase θ of an initial point x in the stable manifold of a limit cycle (LC) identifies the phase of the point on the LC to which the flow φt(x) converges as t → ∞. The infinitesimal phase response curve (iPRC) quantifies the change in timing due to a small perturbation of a LC trajectory. For a stable LC in a smooth dynamical system, the iPRC is the gradient ∇x(θ) of the phase function, which can be obtained via the adjoint of the variational equation. For systems with discontinuous dynamics, the standard approach to obtaining the iPRC fails. We derive a formula for the iPRCs of LCs occurring in piecewise smooth (Filippov) dynamical systems of arbitrary dimension, subject to a transverse flow condition. Discontinuous jumps in the iPRC can occur at the boundaries separating subdomains, and are captured by a linear matching condition. The matching matrix, M, can be derived from the saltation matrix arising in the associated variational problem. For the special case of linear dynamics away from switching boundaries, we obtain an explicit expression for the iPRC. We present examples from cell biology (Glass networks) and neuroscience (central pattern generator models). We apply the iPRCs obtained to study synchronization and phase-locking in piecewise smooth LC systems in which synchronization arises solely due to the crossing of switching manifolds.

Keywords

Mathematical biologyNonsmooth analysisPhase plane analysisNonlinear oscillators
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 5: Theory and applications of nonsmooth dynamical systems , October 2018 , pp. 905 - 940
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

†This work was supported in part by NSF grant DMS-1413770 and NSF grant DMS-1010434.

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