Poincaré-Melnikov method for parabolic points (Q2741137)

The authors established sufficient conditions for the applicability of the Poincaré-Melnikov method to finding transversal homoclinic trajectories to parabolic periodic orbits. It is assumed that the unperturbed system, an autonomous Hamiltonian system on the plane, has an orbit homoclinic to a parabolic equilibrium; the perturbation is time-periodic, not necessary Hamiltonian and vanishes at the equilibrium. Under these assumptions the parabolic equilibrium is obviously preserved, but it is necessary to add one more condition in order to validate the applicability of the Melnikov method. The authors also provided several examples, where the conditions are not satisfied and the Poincaré-Melnikov methods fails.NEWLINENEWLINEFor the entire collection see [Zbl 0963.00030].