The Maximum Number of Small-Amplitude Limit Cycles in Liénard-Type Systems with Cubic Restoring Terms

• Hongwei Shi ^[1]
  1. [1] Sun Yat-sen University

     Sun Yat-sen University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 5, 2024
• Idioma: inglés
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• Resumen

  • In this paper, we investigate the small-amplitude limit cycles of two classes of Liénard systems of the form x˙ = y−F(x, μ), y˙ = −g(x), where the damping term F(x, μ)is either a polynomial or a rational function qn (x) pm(x), (qn(x) and pm(x) are polynomials in x with degrees n and m, respectively), μ represents coefficients and the cubic restoring term g(x) = x − x2 − 1 k x3 with a non-zero constant k. By utilizing Picard-Fuchs equation, we gain the upper bounds of the number of small-amplitude limit cycles of these two systems for any k = 0. Moreover, the smaller upper bounds are obtained for k < −4, k = −9 2 , and the upper bound is sharp if the damping term is polynomial.