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

Yu.V Mikhlin,G.V Manucharyan

doi:10.1016/s0960-0779(02)00404-6

Abstract

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.

Highlights

Homo and heteroclinic trajectories (HT) have been extensively studied in the literature [1,2]
A formation of HT is a criterion of a chaotic behavior beginning in dynamical systems
It is important that the HT criterion of the chaos beginning proposed here is more exact than the generally accepted Melnikov criterion, because it is not necessary to use here separatrix trajectories of the corresponding autonomous equations

Summary

Introduction

Homo and heteroclinic trajectories (HT) have been extensively studied in the literature [1,2]. A new approach for the HT construction in the non-linear dynamical systems with phase space of dimensions equal to two is proposed. It is important that the HT criterion of the chaos beginning proposed here is more exact than the generally accepted Melnikov criterion, because it is not necessary to use here separatrix trajectories of the corresponding autonomous equations. Comparisons between analytical estimates and numerical calculations are presented with a discussion of the main features of the proposed approach

Convergence condition

Non-linear Schrodinger equation

Non-autonomous Duffing equation

Equation of a parametrically excited damped pendulum

Concluding remarks