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Estimates for the Number of Limit Cycles in Discontinuous Generalized Liénard Equations

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Autor(es):
de Abreu, Tiago M. P. ; Martins, Ricardo M.
Número total de Autores: 2
Tipo de documento: Artigo Científico
Fonte: Qualitative Theory of Dynamical Systems; v. 23, n. 4, p. 14-pg., 2024-09-01.
Resumo

In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations x(center dot)=y,y(center dot)=-x-epsilon<middle dot>(f(x)<middle dot>y+sgn(y)<middle dot>g(x))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=y, \ \dot{y}=-x-\varepsilon \cdot (f(x)\cdot y +\textrm{sgn}(y)\cdot g(x))$$\end{document}. Using the averaging method, we were able to generalize a previous result for Li & eacute;nard systems. In our generalization, we consider g as a polynomial of degree m. We conclude that for sufficiently small values of |epsilon|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\varepsilon }|$$\end{document}, the number hm,n=n2+m2+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{m,n}=\left[ \frac{n}{2}\right] +\left[ \frac{m}{2}\right] +1$$\end{document} serves as a lower bound for the maximum number of limit cycles in this system, which bifurcates from the periodic orbits of the linear center x(center dot)=y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=y$$\end{document}, y(center dot)=-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{y}=-x$$\end{document}. Furthermore, we demonstrate that it is indeed possible to obtain a system with hm,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{m,n}$$\end{document} limit cycles. (AU)

Processo FAPESP: 18/03338-6 - Dinâmica global de sistemas dinâmicos suaves por partes
Beneficiário:Ricardo Miranda Martins
Modalidade de apoio: Auxílio à Pesquisa - Regular
Processo FAPESP: 21/08031-9 - Campos vetoriais suaves por partes definidos em variedades compactas
Beneficiário:Ricardo Miranda Martins
Modalidade de apoio: Auxílio à Pesquisa - Regular
Processo FAPESP: 22/07654-5 - Conjuntos invariantes e comportamento caótico em sistemas dinâmicos suaves por partes definidos em variedades compactas de baixa dimensão
Beneficiário:Tiago Miguel Pires de Abreu
Modalidade de apoio: Bolsas no Brasil - Doutorado