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arXiv · 1804.08175

Bifurcations from families of periodic solutions in piecewise differential systems

Jaume Llibre·Douglas D. Novaes·Camila A. B. Rodrigues
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Abstract

Consider a differential system of the form $$ x'=F_0(t,x)+\sum_{i=1}^k \varepsilon^i F_i(t,x)+\varepsilon^{k+1} R(t,x,\varepsilon), $$ where $F_i:\mathbb{S}^1 \times D \to \mathbb{R}^m$ and $R:\mathbb{S}^1 \times D \times (-\varepsilon_0,\varepsilon_0) \to \mathbb{R}^m$ are piecewise $C^{k+1}$ functions and $T$-periodic in the variable $t$. Assuming that the unperturbed system $x'=F_0(t,x)$ has a $d$-dimensional submanifold of periodic solutions with $d<m$, we use the Lyapunov-Schmidt reduction and the averaging theory to study the existence of isolated $T$-periodic solutions of the above differential system.

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Jaume Llibre, Douglas D. Novaes, Camila A. B. Rodrigues. 2018-04-22. Bifurcations from families of periodic solutions in piecewise differential systems. https://doi.org/10.1016/j.physd.2020.132342

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