Revista Matemática Iberoamericana


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Volume 36, Issue 1, 2020, pp. 291–318
DOI: 10.4171/rmi/1131

Published online: 2019-11-25

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

Jaume Llibre[1], Douglas Duarte Novaes[2] and Iris de Oliveira Zeli[3]

(1) Universitat Autònoma de Barcelona, Bellaterra, Spain
(2) Universidade Estadual de Campinas, Brazil
(3) Instituto Tecnológico de Aeronáutica (ITA), São José dos Campos, Brazil

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: \[ x'=Mx+ \varepsilon F_1^n(x)+\varepsilon^2 F_2^n(x), \] 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.

Keywords: Limit cycle, averaging method, periodic orbit, polynomial differential system, nonsmooth polynomial differential systems, nonsmooth dynamical system, Filippov system

Llibre Jaume, Novaes Douglas Duarte, Zeli Iris de Oliveira: Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems. Rev. Mat. Iberoam. 36 (2020), 291-318. doi: 10.4171/rmi/1131