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

Autori
Jebrane, Ahmed ; Mardešić, Pavao ; Pelletier, Michèle

Izvornik
Bulletin des Sciences Mathématiques (0007-4497) 128 (2004), 9; 749-760

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Abelian integral; Melnikov function; Limit cycle; Fuchs system

Sažetak
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).

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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