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

Abstract

Phase response curves (PRCs) have been extensively used to control the phase of oscillators under perturbations. Their main advantage is the reduction of the whole model dynamics to a single variable (phase) dynamics. However, in some adverse situations (strong inputs, high-frequency stimuli, weak convergence,...), the phase reduction does not provide enough information and, therefore, PRC lose predictive power. To overcome this shortcoming, in the last decade, new contributions have appeared that allow to reduce the system dynamics to the phase plus some transversal variable that controls the deviations from the asymptotic behaviour. We call this setting extended response functions. In particular, we single out the phase response function (PRF, a generalization of the PRC) and the amplitude response function (ARF) that account for the above-mentioned deviations from the oscillating attractor. It has been shown that in adverse situations, the PRC misestimate the phase dynamics whereas the PRF-ARF system provides accurate enough predictions. In this paper, we address the problem of studying the dynamics of the PRF-ARF systems under periodic pulsatile stimuli. This paradigm leads to a two-dimensional discrete dynamical system that we call 2D entrainment map. By using advanced methods to study invariant manifolds and the dynamics inside them, we construct an analytico-numerical method to track the invariant curves induced by the stimulus as two crucial parameters of the system increase (the strength of the input and its frequency). Our methodology also incorporates the computation of Arnold tongues associated to the 2D entrainment map. We apply the method developed to study inner dynamics of the invariant curves of a canonical type II oscillator model. We further compare the Arnold tongues of the 2D map with those obtained with the map induced only by the PRC, which give already noticeable differences. We also observe (via simulations) how high-frequency or strong enough stimuli break up the oscillatory dynamics and lead to phase-locking, which is well captured by the 2D entrainment map.