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arXiv · 1801.01730

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

Jaume Llibre·Douglas Duarte Novaes·Iris de Oliveira Zeli
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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.

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Jaume Llibre, Douglas Duarte Novaes, Iris de Oliveira Zeli. 2019-12-06. Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems. https://doi.org/10.4171/rmi%2F1131

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