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RESEARCH PRODUCT

A note on a generalization of Françoise's algorithm for calculating higher order Melnikov functions

Pavao MardešićAhmed JebraneMichèle Pelletier

subject

Abelian integralMathematics(all)GeneralizationGeneral MathematicsHomotopyMathematical analysisApplied mathematicsOrder (group theory)Abelian integral; Melnikov function; Limit cycle; Fuchs systemMelnikov methodMathematics

description

In [J. Differential Equations 146 (2) (1998) 320–335], Françoise gives an algorithm for calculating the first nonvanishing Melnikov function M of a small polynomial perturbation of a Hamiltonian vector field and shows that M is given by an Abelian integral. This is done under the condition that vanishing of an Abelian integral of any polynomial form ω on the family of cycles implies that the form is algebraically relatively exact. We study here a simple example where Françoise’s condition is not verified. We generalize Françoise’s algorithm to this case and we show that M belongs to the C[log t, t, 1/t] module above the Abelian integrals. We also establish the linear differential system verified by these Melnikov functions M(t).

yearjournalcountryeditionlanguage
2004-10-01
10.1016/j.bulsci.2004.03.012https://doi.org/10.1016/j.bulsci.2004.03.012