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Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle

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Abstract

In this paper, we consider Liénard systems of the form

$$\frac{{dx}} {{dt}} = y, \frac{{dy}} {{dt}} = - \left( {x + bx^3 - x^5 } \right) + \varepsilon \left( {\alpha + \beta x^2 + \gamma x^4 } \right)y,$$

where b ∈ ℝ, 0 < |∈| ≪ 1, (α, β, γ) ∈ D ∈ ℝ3 and D is bounded. We prove that for |b| ≫ 1 (b < 0) the least upper bound of the number of isolated zeros of the related Abelian integrals is 2 (counting the multiplicity) and this upper bound is a sharp one.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, P. R. China

    Li Qin Zhao & De Ping Li

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  1. Li Qin Zhao
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  2. De Ping Li
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Correspondence to Li Qin Zhao.

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Supported by National Natural Science Foundation of China (Grant No. 11271046)

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Zhao, L.Q., Li, D.P. Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle. Acta. Math. Sin.-English Ser. 30, 411–422 (2014). https://doi.org/10.1007/s10114-014-2615-8

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  • DOI: https://doi.org/10.1007/s10114-014-2615-8

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