Phase dynamics of coupled oscillators reconstructed from data

Abstract

We systematically develop a technique for reconstructing the phase dynamics equations for coupled oscillators from data. For autonomous oscillators and for two interacting oscillators we demonstrate how phase estimates obtained from general scalar observables can be transformed to genuine phases. This allows us to obtain an invariant description of the phase dynamics in terms of the genuine, observable-independent phases. We discuss the importance of this transformation for characterization of strength and directionality of interaction from bivariate data. Moreover, we demonstrate that natural (autonomous) frequencies of oscillators can be recovered if several observations of coupled systems at different, yet unknown coupling strengths are available. We illustrate our method by several numerical examples and apply it to a human electrocardiogram and to a physical experiment with coupled metronomes.

Oscillatory patterns in the light curves of five long-term monitored type 1 active galactic nuclei

“…The first step is to transform given time series y(t) = (y k (t)), k = 1, ..N of each object into a cyclic observable. This is completed via construction of a two-dimensional embed-ding (y, y H ) (see Kralemann et al 2008, and references therein), where following equation…”

Improving Reduced Variable Models for Complex Systems via Experiment

“…We compare results of the two modeling methods and discuss limitations of models based on phase di↵erence. We calculate from experimental data a two-phase model using a previously introduced technique [55]. Our experimental results verify phase reconstruction in a system with noise and connect the two-dimensional and one-dimensional models [8,56,57].…”

“…For example, phase defined from percentage of trajectory length between consecutive Poincaré surface of section crossings is useful for more complicated oscillations (e.g. electrocardiograms) [55].…”

“…Note that definitions of phase based on the Hilbert transform have inherent From the transformed, genuine phases j we find the coupling functions Q, following the methods of [55]. First, the phase of each oscillator is cleansed of nonuniform phase advance.…”

“…(18) in [55] (when the phase space is not well-covered, phase velocities are instead fit with a two-dimensional kerneling function). The coupling functions are then further cleansed using the method in Section IV, part B in [55]. Numerics demonstrate that satisfactory results can be obtained already after the first cleansing, since the second cleansing is small compared to the first.…”