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

Recent years have witnessed increasing interest in phase-amplitude reduction of limit-cycle dynamics. Adding an amplitude coordinate to the phase coordinate allows us to take into account the dynamics transversal to the limit cycle and thereby overcome the main limitations of classic phase reduction (strong convergence to the limit cycle and weak inputs). While previous studies, mostly focus on local quantities such as infinitesimal responses, a major and limiting challenge of phase-amplitude reduction is to compute amplitude coordinates globally, in the basin of attraction of the limit cycle. In this paper, we propose a method to compute the full set of phase-amplitude coordinates in the large. Our method is based on the so-called Koopman (composition) operator and aims at computing the eigenfunctions of the operator through Laplace averages (in combination with the harmonic balance method). This yields a forward integration method that is not limited to two-dimensional systems. We illustrate the method by computing the so-called isostables of limit cycles in two-, three-, and four-dimensional state spaces, as well as their responses to strong external inputs.