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# Bifurcations of limit cycles in a class of hyper-elliptic Lie´nard systems

In this work we consider the number of limit cycles that can bifurcate from

periodic annulus of the quintic Hamiltonian vector fields XH ¼ 2y @

@x 2xP0ðx2Þ @

@y

with P as an arbitrary polynomial of degree three under small perturbations of

the form "ð þ x2 þ x4Þy @

@y, where 05j"j 1 and ; ; are real constants.

We show that the least upper bound for the number of limit cycles bifurcated

from the periodic annulus of XH surrounding the origin is two, when origin is

a global or local centre.

April, 2009

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Month/Season:

April

Year:

2009