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

• Autores: Tiago M. P. de Abreu, Ricardo M. Martins
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 4, 2024
• Idioma: inglés
• DOI: 10.1007/s12346-024-01048-2
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• Resumen

  • In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations x˙ = y, y˙ = −x −ε ·( f (x)· y +sgn(y)· 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 |ε|, the number hm,n = n 2 + m 2 +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 x˙ = y, y˙ = −x. Furthermore, we demonstrate that it is indeed possible to obtain a system with hm,n limit cycles.

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