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Limit cycles of generalized Liénard polynomial differential systems via averaging theory

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  • García, Belén
  • Llibre, Jaume
  • Pérez del Río, Jesús S.

Abstract

Using the averaging theory of first and second order we study the maximum number of limit cycles of the polynomial differential systemsẋ=y,ẏ=-x-ε(h1(x)+p1(x)y+q1(x)y2)-ε2(h2(x)+p2(x)y+q2(x)y2),which bifurcate from the periodic orbits of the linear center ẋ=y,ẏ=-x, where ε is a small parameter. If the degrees of the polynomials h1,h2,p1,p2,q1 and q2 are equal to n, then we prove that this maximum number is [n/2] using the averaging theory of first order, where [·] denotes the integer part function; and this maximum number is at most n using the averaging theory of second order.

Suggested Citation

  • García, Belén & Llibre, Jaume & Pérez del Río, Jesús S., 2014. "Limit cycles of generalized Liénard polynomial differential systems via averaging theory," Chaos, Solitons & Fractals, Elsevier, vol. 62, pages 1-9.
    • Handle: RePEc:eee:chsofr:v:62-63:y:2014:i::p:1-9
    DOI: 10.1016/j.chaos.2014.02.008
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    References listed on IDEAS

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    1. Chao Liu & Maoan Han, 2013. "The Number of Limit Cycles of a Polynomial System on the Plane," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-7, July.
    2. Llibre, Jaume & Valls, Clàudia, 2013. "Limit cycles for a generalization of polynomial Liénard differential systems," Chaos, Solitons & Fractals, Elsevier, vol. 46(C), pages 65-74.
    3. Yu, P. & Han, M., 2006. "Limit cycles in generalized Liénard systems," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1048-1068.
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