Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems

TL;DR

This paper extends averaging theory to analyze limit cycles in higher-dimensional, piecewise polynomial, discontinuous differential systems, especially near linear systems with specific eigenvalue structures, revealing bifurcations of periodic solutions.

Contribution

It develops a generalized averaging method for non-autonomous, discontinuous, piecewise smooth systems with manifolds of periodic solutions, applied to higher-dimensional linear systems with complex eigenvalues.

Findings

Extended averaging theory for discontinuous systems.

Identified bifurcation of limit cycles from linear systems.

Analyzed systems with specific eigenvalue configurations.

Abstract

The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non-autonomous $n$-dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold $\mathcal{Z}\subset\mathbb{R}^n$ of periodic solutions satisfying $\dim(\mathcal{Z})<n.$ Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential systems, $x'=Mx$, when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the following differential system $x'=Mx+ \varepsilon F_1^n(x)+\varepsilon^2F_2^n(x),$ in $\mathbb{R}^{d+2}$ where $\varepsilon$ is a small parameter, $M$ is a $(d+2)\times(d+2)$ matrix having one pair of pure imaginary conjugate eigenvalues, $m$ zeros eigenvalues, and $d-m$ non-zero real eigenvalues.