Construction of homoclinic and heteroclinic trajectories in mechanical systems with several equilibrium positions

A new approach for a construction of homo and heteroclinic trajectories of some principal non-linear dynamical systems is utilized here, namely the non-linear Schrodinger equation, non-autonomous Duffing equation and the equation of a parametrically excited damped pendulum are considered. Pade’ and quasi-Pade’ approximants and a convergence condition used earlier in the theory of non-linear normal vibration modes made possible to solve a boundary-value problems formulated for the orbits and to determine initial amplitude values of the trajectories with admissible precision. The approach proposed here is more exact than the generally accepted one because it is not necessary to use here separatrix trajectories of the corresponding autonomous equations.