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

Abstract

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.

On the concept of dynamical reduction: the case of coupled oscillators

“…In Refs. [47,48], phase equations for hybrid limit-cycle oscillators have been derived. A generalized adjoint equation with discontinuous jumps is derived and the sensitivity function for the phase is obtained.…”

Phase reduction theory for hybrid nonlinear oscillators

“…This model exhibits a stable limit-cycle oscillation for appropriate parameter values that corresponds to periodic movements of the legs. This is a four dimensional hybrid dynamical system with impacts, hence it cannot be dealt with by the conventional methods [10,33,34] mentioned above.…”

Recent advances in coupled oscillator theory

“…Existence and mechanisms of chaos as a result of a discontinuous voltage reset has been shown in a Fitzhugh-Nagumo model by calculating the Lyapunov exponents: a result enabled by the saltation matrix [28]. Assuming continuous solutions in a discontinuous vector field Park et al (2018) derived the size of discontinuities in the iPRC from first principles [29] and showed that the calculation is closely related to the saltation matrix [3]. For hybrid systems with discontinuous solutions, Coombes et al (2012) derived the iPRC for a piecewise linear IF model [8] by normalizing on each segment away from discontinuities.…”