Hindawi Publishing Corporation
Journal of Nonlinear Dynamics
Volume 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
J. O. Maaita and E. Meletlidou
Physics Department, Aristotle University of Tessaloniki, 54124 Tessaloniki, Greece
Correspondence should be addressed to J. O. Maaita; jmaay@physics.auth.gr
Received 12 August 2013; Accepted 3 November 2013
Academic Editor: Mitsuhiro Ohta
Copyright © 2013 J. O. Maaita and E. Meletlidou. Tis 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.
Analytical solution of the homoclinic orbit of a two-dimensional system of diferential equations that describes the Hamiltonian
part of the slow fow 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 fow of dynamical
system that joins a saddle equilibrium point to itself; that is,
the homoclinic trajectory ℎ() converges to the equilibrium
point as →±∞ [1].
Te analytical solutions of homoclinic orbits are very
important for many applications as in the use of the homo-
clinic Melnikov function, in order to prove the existence of
transversal homoclinic orbits and chaotic behavior.
In what follows, we fnd the analytical solution of the
homoclinic orbit of a one-degree-of-freedom system of dif-
ferential equations that describes the Hamiltonian part of
the slow fow of a three-degree-of-freedom dissipative system
of linear coupled oscillators with an essentially nonlinear
attachment [2].
Te aim of the work in [2] was to study the asymptotic
behavior of the system. Specifcally, the initial dissipative sys-
tem composed of two linear and one nonlinear oscillators
was reduced to a nonautonomous damped strongly nonlinear
second-order diferential equation. With the use of the com-
plexifcation-averaging technique (CX-A), we obtained the
slow fow of the system, that is, a system of two, frst-order,
diferential equations governed by slow time
=−
2
+
2
−
3
8
(
2
+
2
)−
2
+
2
sin (
20
),
=−
2
−
2
+
3
8
(
2
+
2
)−
2
−
2
cos (
20
),
(1)
where , are the variables and , , , , ,
20
are
parameters.
From the study of the dynamics of the slow fow [3], we
concluded that the slow fow may do regular or chaotic oscil-
lations. Te computation of the analytical solution of the
homoclinic orbit of the unperturbed problem is the frst step
in order to investigate the chaotic behavior, of the above
system, with the use of the homoclinic Melnikov function.
2. Main Results
Te unperturbed part of the above system is
=
2
−
3
8
(
2
+
2
)−
2
+
2
sin ,
=−
2
+
3
8
(
2
+
2
)−
2
−
2
cos ,
(2)
and is a parameter. Te equilibrium points are found con-
sidering
=0,
=0. Afer some simple algebraic manipu-
lations, we have
=
+ cos
−+ sin
, (3)