Analytical Homoclinic Solution of a Two-Dimensional Nonlinear System of Differential Equations

J O Maaita,E Meletlidou

doi:10.1155/2013/879040

Abstract

Analytical solution of the homoclinic orbit of a two-dimensional system of differential equations that describes the Hamiltonian part of the slow flow of a three-degree-of-freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment is described.

Highlights

A homoclinic orbit is the trajectory of a flow of dynamical system that joins a saddle equilibrium point to itself; that is, the homoclinic trajectory h(t) converges to the equilibrium point as t → ± ∞ [1]
The analytical solutions of homoclinic orbits are very important for many applications as in the use of the homoclinic Melnikov function, in order to prove the existence of transversal homoclinic orbits and chaotic behavior
We find the analytical solution of the homoclinic orbit of a one-degree-of-freedom system of differential equations that describes the Hamiltonian part of the slow flow of a three-degree-of-freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment [2]

Summary

Introduction

A homoclinic orbit is the trajectory of a flow of dynamical system that joins a saddle equilibrium point to itself; that is, the homoclinic trajectory h(t) converges to the equilibrium point as t → ± ∞ [1].The analytical solutions of homoclinic orbits are very important for many applications as in the use of the homoclinic Melnikov function, in order to prove the existence of transversal homoclinic orbits and chaotic behavior.In what follows, we find the analytical solution of the homoclinic orbit of a one-degree-of-freedom system of differential equations that describes the Hamiltonian part of the slow flow of a three-degree-of-freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment [2].The aim of the work in [2] was to study the asymptotic behavior of the system. The analytical solutions of homoclinic orbits are very important for many applications as in the use of the homoclinic Melnikov function, in order to prove the existence of transversal homoclinic orbits and chaotic behavior. We find the analytical solution of the homoclinic orbit of a one-degree-of-freedom system of differential equations that describes the Hamiltonian part of the slow flow of a three-degree-of-freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment [2].

Results

Conclusion