Estimating asymptotic phase and amplitude functions of limit-cycle oscillators from time series data

We propose a method for estimating the asymptotic phase and amplitude functions of limit-cycle oscillators using observed time series data without prior knowledge of their dynamical equations. The estimation is performed by polynomial regression and can be solved as a convex optimization problem. The validity of the proposed method is numerically illustrated by using two-dimensional limit-cycle oscillators as examples. As an application, we demonstrate data-driven fast entrainment with amplitude suppression using the optimal periodic input derived from the estimated phase and amplitude functions.