Abstract
In [J. Differential Equations 146 (2) (1998) 320–335], Françoise gives an algorithm for calculating the first nonvanishing Melnikov function M ℓ of a small polynomial perturbation of a Hamiltonian vector field and shows that M ℓ 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 ℓ belongs to the C[ logt,t,1/t] module above the Abelian integrals. We also establish the linear differential system verified by these Melnikov functions M ℓ( t).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.