The Limit Cycles for a Class of Non-autonomous Piecewise Differential Equations

• Renhao Tian ^[1] ; Yulin Zhao ^[1]
  1. [1] Sun Yat-sen University

     Sun Yat-sen University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 4, 2024
• Idioma: inglés
• DOI: 10.1007/s12346-024-01050-8
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we study a class of non-autonomous piecewise differential equations defined as follows: dx/dt = a0(t) + n i=1 ai(t)|x| i , where n ∈ N+ and each ai(t) is real, 1-periodic, and smooth function. We deal with two basic problems related to their limit cycles isolated solutions satisfyingx(0) = x(1) . First, we prove that, for any given n ∈ N+, there is no upper bound on the number of limit cycles of such equations. Second, we demonstrate that if a1(t), . . . , an(t) do not change sign and have the same sign in the interval [0, 1], then the equation has at most two limit cycles. We provide a comprehensive analysis of all possible configurations of these limit cycles.

    In addition, we extend the result of at most two limit cycles to a broader class of general non-autonomous piecewise polynomial differential equations and offer a criterion for determining the uniqueness of the limit cycle within this class of equations

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