Isostable reduction of periodic orbits

Abstract

The well-established method of phase reduction neglects information about a limit-cycle oscillator's approach towards its periodic orbit. Consequently, phase reduction suffers in practicality unless the magnitude of the Floquet multipliers of the underlying limit cycle are small in magnitude. By defining isostable coordinates of a periodic orbit, we present an augmentation to classical phase reduction which obviates this restriction on the Floquet multipliers. This framework allows for the study and understanding of periodic dynamics for which standard phase reduction alone is inadequate. Most notably, isostable reduction allows for a convenient and self-contained characterization of the dynamics near unstable periodic orbits.

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].…”

“…The following provides a summary of the work from [42] and [40]. Consider a general equation of the formẋ…”

“…. , n. For every non-unity eigenvalue λ k of J η with equal algebraic and geometric multiplicity, an isostable coordinates ψ k can be defined as in [42], [40] ψ k (x) = lim…”

“…This limiting behavior gives the isostable coordinate ψ k (x) which is defined for all locations in the basin of attraction of the limit cycle. Because (18) is defined according to the infinite time convergence of solutions to the limit cycle, one can show (as in [42]) that under the flow,ψ…”

Global computation of phase-amplitude reduction for limit-cycle dynamics

“…This not only allows to fully characterize the sensitivity to large perturbations, but also provides a global picture of the limit cycle dynamics. However, the global computation of amplitude coordinates is delicate and, to the authors knowledge, previous contributions mainly focused on local quantities such as the infinitesimal isostable (or phase) response (see [4,23] in the large and [28] along the limit cycle).…”