Search results for "Melnikov method"
showing 3 items of 3 documents
A note on a generalization of Françoise's algorithm for calculating higher order Melnikov functions
2004
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 ver…
Melnikov functions and Bautin ideal
2001
The computation of the number of limit cycles which appear in an analytic unfolding of planar vector fields is related to the decomposition of the displacement function of this unfolding in an ideal of functions in the parameter space, called the Ideal of Bautin. On the other hand, the asymptotic of the displacement function, for 1-parameter unfoldings of hamiltonian vector fields is given by Melnikov functions which are defined as the coefficients of Taylor expansion in the parameter. It is interesting to compare these two notions and to study if the general estimations of the number of limit cycles in terms of the Bautin ideal could be reduced to the computations of Melnikov functions for…
Chaotic behaviour in deformable models: the asymmetric doubly periodic oscillators
2002
Abstract The motion of a particle in a one-dimensional perturbed asymmetric doubly periodic (ASDP) potential is investigated analytically and numerically. A simple physical model for calculating analytically the Melnikov function is proposed. The onset of chaos is studied through an analysis of the phase space, a construction of the bifurcation diagram and a computation of the Lyapunov exponent. Theory predicts the regions of chaotic behaviour of orbits in a good agreement with computer calculations.