Advances in Differential Equations

Higher-order Melnikov functions for degenerate cubic Hamiltonians

I. D. Iliev

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It is shown that, in general, the first four Melnikov functions have to be taken into account in order to obtain definitive results concerning the limit cycles in quadratic perturbations of Hamiltonian systems in the plane with degenerate cubic Hamiltonians. An application is done in completing the proof that no more than two limit cycles can bifurcate out of homoclinic loops of quadratic Hamiltonian systems.

Article information

Adv. Differential Equations, Volume 1, Number 4 (1996), 689-708.

First available in Project Euclid: 25 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C05: Location of integral curves, singular points, limit cycles
Secondary: 34C23: Bifurcation [See also 37Gxx] 34C37: Homoclinic and heteroclinic solutions 58F21


Iliev, I. D. Higher-order Melnikov functions for degenerate cubic Hamiltonians. Adv. Differential Equations 1 (1996), no. 4, 689--708.

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