Stability of Periodic Solutions for Hysteresis-Delay Differential Equations

TL;DR

This paper investigates the stability of periodic solutions in systems with delay and hysteresis, introducing a novel functional framework and reducing the stability analysis to a finite-dimensional spectral problem.

Contribution

It develops a new analytical approach using fractional Sobolev spaces to linearize and analyze the stability of periodic solutions in delay hysteresis systems.

Findings

Established existence and uniqueness of solutions

Reduced spectral stability analysis to finite-dimensional problem

Provided explicit linearization of the Poincaré map

Abstract

We study an interplay between delay and discontinuous hysteresis in dynamical systems. After having established existence and uniqueness of solutions, we focus on the analysis of stability of periodic solutions. The main object we study is a Poincar\'e map that is infinite-dimensional due to delay and non-differentiable due to hysteresis. We propose a general functional framework based on the fractional order Sobolev--Slobodeckij spaces and explicitly obtain a formal linearization of the Poincar\'e map in these spaces. Furthermore, we prove that the spectrum of this formal linearization determines the stability of the periodic solution and then reduce the spectral analysis to an equivalent finite-dimensional problem.