Partial Exponential Stability Analysis of Slow-fast Systems via Periodic Averaging

TL;DR

This paper introduces new criteria for partial exponential stability in slow-fast systems using a novel periodic averaging approach, and applies it to analyze synchronization in oscillator networks.

Contribution

It develops a new averaging-based method for stability analysis of slow-fast systems and establishes related Lyapunov and perturbation theorems.

Findings

Averaged system stability implies original system stability.

Detuning oscillator frequency enhances synchronization robustness.

New converse Lyapunov theorem for partial exponential stability.

Abstract

This paper presents some new criteria for partial exponential stability of a slow-fast nonlinear system with a fast scalar variable using periodic averaging methods. Unlike classical averaging techniques, we construct an averaged system by averaging over this fast scalar variable instead of the time variable. We then show that partial exponential stability of the averaged system implies partial exponential stability of the original one. As some intermediate results, we also obtain a new converse Lyapunov theorem and some perturbation theorems for partially exponentially stable systems. We then apply our established criteria to study remote synchronization of Kuramoto-Sakaguchi oscillators coupled by a star network with two peripheral nodes. We analytically show that detuning the natural frequency of the central mediating oscillator can increase the robustness of the remote synchronization against phase shifts.