Geodesic
Some remarks on the Melnikov function
Electronic Journal of Differential Equations, Tome 2002 (2002).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We study the Melnikov function associated with a periodic perturbation of a differential equation having a homoclinic orbit. Our main interest is the characterization of perturbations that give rise to vanishing or non-vanishing of the Melnikov function. For this purpose we show that, in some cases, the Fourier coefficients of the Melkinov function can be evaluated by means of the calculus of residues. We apply this result, among other things, to the construction of a second-order equation whose Melnikov function vanishes identically for any $C^{1}, 2\pi$-periodic perturbation. Then we study the second order Melnikov function of the perturbed equation, and prove it is non-vanishing for a large class of perturbations.
Classification : 34C23, 34C37
Keywords: melnikov function, residues, Fourier coefficients 
@article{EJDE_2002__2002__a35,
     author = {Battelli, Flaviano and Fe\v{c}kan, Michal},
     title = {Some remarks on the {Melnikov} function},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2002},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a35/}
}
TY  - JOUR
AU  - Battelli, Flaviano
AU  - Fečkan, Michal
TI  - Some remarks on the Melnikov function
JO  - Electronic Journal of Differential Equations
PY  - 2002
VL  - 2002
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a35/
LA  - en
ID  - EJDE_2002__2002__a35
ER  - 
%0 Journal Article
%A Battelli, Flaviano
%A Fečkan, Michal
%T Some remarks on the Melnikov function
%J Electronic Journal of Differential Equations
%D 2002
%V 2002
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a35/
%G en
%F EJDE_2002__2002__a35
Battelli, Flaviano; Fečkan, Michal. Some remarks on the Melnikov function. Electronic Journal of Differential Equations, Tome 2002 (2002). http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a35/