Augmented Phase Reduction of (Not So) Weakly Perturbed Coupled Oscillators

Abstract

While phase reduction is a tremendously useful tool for understanding the dynamics of weakly perturbed limit cycle oscillators, its assumptions break down as perturbations become larger, limiting its practical utility in many applications. This fundamental limitation is often apparent when studying coupled populations of oscillators when the collective behavior approaches a limit cycle with transient behavior that decays slowly. Using the notion of isostables of periodic orbits, which define a coordinate system transverse to the limit cycle, we develop a strategy to augment the standard phase reduction in large populations of coupled oscillators to allow for the application of larger perturbations. The resulting augmented phase reduction yields a tremendous decrease in model complexity with a resulting dimensionality that does not depend on the number of oscillators in the population. Additionally, the augmented phase reduction replicates model behavior that standard phase reduction cannot. Finally, we propose and implement a direct method for the application of augmented phase reduction that would be applicable in systems of biological oscillators. Applying the augmented phase reduction to a model of coupled circadian oscillators yields novel insight about the possible mechanism behind the difference in reported jet-lag severity between eastward and westward travel.

Phase-amplitude dynamics in terms of extended response functions: Invariant curves and arnold tongues

“…In particular, we have described the saddle-node bifurcation of periodic orbits that the system undergoes when crossing the boundaries of these Arnold tongues. Compared to parallel results obtained with the 1D entrainment map (9), we see that the Arnold tongues are slightly different. However, for numerical reasons, we have kept at parameter values that were not favorable to show a clear distinction.…”

“…The asymptotic-state hypothesis is widely valid but it fails in transient states, generally because of strong, highly repetitive or noisy stimuli, or due to the slow convergence to the asymptotic oscillatory state, that is, the attracting limit cycle. In order to go beyond the limits of the phase reduction, recent literature has focused on the control of the phase response out of the limit cycle (see [4][5][6][7][8][9]), a problem that is intertwined with the progress about the computation of the so-called isochrons associated to the limit cycle (see [4,[10][11][12][13]). It is worth mentioning that under the asymptotic-state hypothesis, the PRC is a function that maps the phase θ of the oscillator onto the phase change elicited by a given stimulus I stim .…”

“…However, when the PRC concept is extended to a neighbourhood of the limit cycle (see [4]), one has to consider a (n − 1)-dimensional amplitude variable transversal to the limit cycle. Assuming that there is a privileged direction (corresponding to the eigenvalue of the Poincaré map closest to the unit circle, see [8,9,14]), that will control at first order the proximity to the limit cycle (technical details explained below), then the PRC naturally extends to a system of two variables, namely the phase and this privileged amplitude variable (see Figure 1), and two response functions, named phase and amplitude response functions (PRF and ARF, respectively), see [5] and Section II.…”

Isostable reduction of oscillators with piecewise smooth dynamics and complex Floquet multipliers

“…In this case, it is generally assumed that the isostable coordinates associated with these Floquet multipliers are always zero (c.f. [19], [20]), leading to a reduction in dimension as compared to Eq. (1).…”

“…(5) has been shown to be useful in contexts where the standard phase reduction from Eq. (4) is inadequate, i.e., when large magnitude perturbations are applied [19], [20]. However, the methodology described in both [19] and [20] requires all Floquet exponents to be real and requires that F(x) is sufficiently smooth in order to compute the reduced functions Z(θ), B k (θ), I(θ) and C k j (θ).…”