Slow-fast dynamics of strongly coupled adaptive frequency oscillators

TL;DR

This paper analyzes the slow-fast dynamics of strongly coupled adaptive frequency oscillators, revealing how layered invariant manifolds facilitate robust frequency learning and synchronization in complex oscillator networks.

Contribution

It provides a formal analysis of the adaptation mechanism in strongly coupled oscillators, uncovering the layered slow-fast dynamics and invariant manifolds responsible for frequency adaptation.

Findings

Existence of layered stable and unstable invariant slow manifolds.

Input signals induce jumps between manifolds, leading to exponential convergence.

The invariant manifold structure persists in networks with amplitude adaptation.

Abstract

Oscillators have two main limitations: their synchronization properties are limited (i.e they have a finite synchronization region) and they have no memory of past interactions (i.e. they return to their intrinsic frequency whenever the entraining signal disappears). We previously proposed a general mechanism to transform an oscillator into an adaptive frequency oscillator which adapts its parameters to learn the frequency of any input signal. The synchronization region then becomes infinite and the oscillator retains the entrainment frequency when the driving signal disappears. While this mechanism has been successfully used in various applications, such as robot control or observer design for active prosthesis, a formal understanding of its properties is still missing. In this paper, we study the adaptation mechanism in the case of strongly coupled phase oscillators and show that non-trivial slow-fast dynamics is at the origin of the adaptation. We show the existence of a layered structure of stable and unstable invariant slow manifolds and demonstrate how the input signal forces the dynamics to jump between these manifolds at regular intervals, leading to exponential convergence of the frequency adaptation. We extend the idea to a network of oscillators with amplitude adaptation and show that the slow invariant manifolds structure persists. Numerical simulations validate our analysis and extend the discussion to more complex cases.