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Journal of Nonlinear Dynamics
Volume 2013 (2013), Article ID 879040, 4 pages
http://dx.doi.org/10.1155/2013/879040
Research Article

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

Physics Department, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Received 12 August 2013; Accepted 3 November 2013

Academic Editor: Mitsuhiro Ohta

Copyright © 2013 J. O. Maaita and E. Meletlidou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. 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 converges to the equilibrium point as [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. Specifically, the initial dissipative system composed of two linear and one nonlinear oscillators was reduced to a nonautonomous damped strongly nonlinear second-order differential equation. With the use of the complexification-averaging technique (CX-A), we obtained the slow flow of the system, that is, a system of two, first-order, differential equations governed by slow time where , are the variables and , , , , , are parameters.

From the study of the dynamics of the slow flow [3], we concluded that the slow flow may do regular or chaotic oscillations. The computation of the analytical solution of the homoclinic orbit of the unperturbed problem is the first step in order to investigate the chaotic behavior, of the above system, with the use of the homoclinic Melnikov function.

2. Main Results

The unperturbed part of the above system is and is a parameter. The equilibrium points are found considering , . After some simple algebraic manipulations, we have and the third-order equation When (4) has three real roots, we have three equilibrium points. It is well known that, in order (4) to have three real roots, it must hold that the determinant , where , , , , , and . This holds for where .

The equilibrium points are where .

The Hamiltonian of system (2) is given by We perform the canonical transformation , , and the hamiltonian becomes From the square of (9) by adding in both sides of the equation the quantity , we have From the hamiltonian (8), (10) becomes We denote by , the unstable equilibrium point , and (10) becomes By substituting (for the case ) in the hamiltonian, we have and (11) becomes For the right hand side of the previous equation after some simple algebra manipulations, we have We calculate and derive and for the equilibrium point we have Using (12) and the above equality, (15) becomes Then from (14) and (18), we have which is our main differential equation and is easily solved [4].

As it is seen in Figure 1 when our system has three equilibrium points, then depending on the parameters, we may have two homoclinic orbits. In our analysis, this result is given by the absolute value in (19).

879040.fig.001
Figure 1: Phase space of the hamiltonian for , , , , and .

For the case , the homoclinic solution is where . After substituting the solution (20) in (16) and integrate we derive where For the case , the homoclinic solution is

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

  1. A. J. Homburg and B. Sandstede, “Homoclinic and heteroclinic bifurcations in vector fields,” in Handbook of Dynamical Systems, vol. 3, Chapter 8, pp. 379–524, 2010.
  2. J. O. Maaita, E. Meletlidou, A. F. Vakakis, and V. Rothos, “The effect of slow flow dynamics on the oscillations of a singular damped system with an essentially nonlinear attachment,” Journal of Applied Nonlinear Dynamics, vol. 2, no. 4, pp. 315–328, 2013. View at Publisher · View at Google Scholar
  3. J. O. Maaita, E. Meletlidou, A. F. Vakakis, and V. Rothos, “The dynamics of the slow flow of a singular damped nonlinear system and its Parametric Study,” Journal of Applied Nonlinear Dynamics. In press.
  4. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, NY, USA, 7th edition, 2007.