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

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Abstract

In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations \(\dot{x}=y, \ \dot{y}=-x-\varepsilon \cdot (f(x)\cdot y +\textrm{sgn}(y)\cdot g(x))\). Using the averaging method, we were able to generalize a previous result for Liénard systems. In our generalization, we consider g as a polynomial of degree m. We conclude that for sufficiently small values of \(|{\varepsilon }|\), the number \(h_{m,n}=\left[ \frac{n}{2}\right] +\left[ \frac{m}{2}\right] +1\) 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 \(\dot{x}=y\), \(\dot{y}=-x\). Furthermore, we demonstrate that it is indeed possible to obtain a system with \(h_{m,n}\) limit cycles.

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Acknowledgements

The São Paulo Research Foundation (FAPESP) partially supports R.M.M. Grants 2021/08031-9, 2018/03338-6 and supports T.M.P.A. Grant 2022/07654-5. The National Council for Scientific and Technological Development (CNPq) partially supports R. M. M. Grants 315925/2021-3 and 434599/2018-2 and partially supports T.M.P.A. Grant 132226/2020-0. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.

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  1. Departamento de Matemática, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz, Campinas, SP, 13083–859, Brazil

    Tiago M. P. de Abreu & Ricardo M. Martins

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  1. Tiago M. P. de Abreu
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  2. Ricardo M. Martins
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RMM proved the main results and typed the article. TMPA proved the main results and typed the article. All authors reviewed the manuscript.

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Correspondence to Ricardo M. Martins.

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de Abreu, T.M.P., Martins, R.M. Estimates for the Number of Limit Cycles in Discontinuous Generalized Liénard Equations. Qual. Theory Dyn. Syst. 23, 187 (2024). https://doi.org/10.1007/s12346-024-01048-2

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