Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

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• 作者：Má ; rcio R.A. Gouveia^a ; ^{maralves@ibilce.unesp.br" class="auth_mail" title="E-mail the corresponding author} ; Jaume Llibre^b ; ^{jllibre@mat.uab.cat" class="auth_mail" title="E-mail the corresponding author} ; Douglas D. Novaes^c ; ^{ddnovaes@ime.unicamp.br" class="auth_mail" title="E-mail the corresponding author} ; Claudio Pessoa^a ; ^{pessoa@ibilce.unesp.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：34A36 ; 34C29 ; 37G15
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 April 2016
• 年：2016
• 卷：260
• 期：7
• 页码：6108-6129
• 全文大小：495 K

文摘

We consider an n  -dimensional piecewise smooth vector field with two zones separated by a hyperplane Σ which admits an invariant hyperplane Ω transversal to Σ containing a period annulus c378a" title="Click to view the MathML source">A$\mathcal{A}$ fulfilled by crossing periodic solutions. For small discontinuous perturbations of these systems we develop a Melnikov-like function to control the persistence of periodic solutions contained in c378a" title="Click to view the MathML source">A$\mathcal{A}$. When n=3$n=3$ we provide normal forms for the piecewise linear case. Finally we apply the Melnikov-like function to study discontinuous perturbations of the given normal forms.