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

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, $x^{\prime }=Mx$, 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 $x^{\prime }=Mx+\epsilon F_1^n(x)+{\epsilon }^2{F}_2^n(x),$ in ${\mathbb{R}}^{d+2}$ where $\epsilon$ 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.

References in corpus (3)

• Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones
• Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction
• Birth of limit cycles for a class of continuous and discontinuous differential systems in (d+2)-dimension

Cited by in corpus (1)

• Bifurcations from families of periodic solutions in piecewise differential systems