BIFURCATION OF LIMIT CYCLES IN SMALL PERTURBATIONS OF A HYPER-ELLIPTIC HAMILTONIAN SYSTEM WITH TWO NILPOTENT SADDLES

Rasool Kazemi; Hamid R. Z. Zangeneh

doi:10.11948/2012029

2012 Volume 2 Issue 4

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Rasool Kazemi, Hamid R. Z. Zangeneh. BIFURCATION OF LIMIT CYCLES IN SMALL PERTURBATIONS OF A HYPER-ELLIPTIC HAMILTONIAN SYSTEM WITH TWO NILPOTENT SADDLES[J]. Journal of Applied Analysis & Computation, 2012, 2(4): 395-413. doi: 10.11948/2012029

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Rasool Kazemi, Hamid R. Z. Zangeneh. BIFURCATION OF LIMIT CYCLES IN SMALL PERTURBATIONS OF A HYPER-ELLIPTIC HAMILTONIAN SYSTEM WITH TWO NILPOTENT SADDLES[J]. Journal of Applied Analysis & Computation, 2012, 2(4): 395-413. doi: 10.11948/2012029

BIFURCATION OF LIMIT CYCLES IN SMALL PERTURBATIONS OF A HYPER-ELLIPTIC HAMILTONIAN SYSTEM WITH TWO NILPOTENT SADDLES

Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran

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Abstract

Abstract

In this paper we study the first-order Melnikov function for a planar near-Hamiltonian system near a heteroclinic loop connecting two nilpotent saddles. The asymptotic expansion of this Melnikov function and formulas for the first seven coefficients are given. Next, we consider the bifurcation of limit cycles in a class of hyper-elliptic Hamiltonian systems which has a heteroclinic loop connecting two nilpotent saddles. It is shown that this system can undergo a degenerate Hopf bifurcation and Poincarè bifurcation, which emerges at most four limit cycles in the plane for sufficiently small positive ε. The number of limit cycles which appear near the heteroclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. Further more we give all possible distribution of limit cycles bifurcated from the period annulus.
- Melnikov function /
- Hilbert's 16th problem /
- limit cycles /
- heteroclinic loop /
- nilpotent saddle
MSC: 34C07;34C08;37G15;34M50

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References

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