IJPAM: Volume 66, No. 1 (2011)

A GEOMETRIC CONDITION FOR THE EXISTENCE OF
THE HOMOCLINIC ORBITS OF LIÉNARD SYSTEMS

Makoto Hayashi
Department of Mathematics
College of Science and Technology
Nihon University
7-24-1, Narashinodai, Funabashi-shi, Chiba, 274-8501, JAPAN
e-mail: [email protected]


Abstract.In this paper, a geometric condition in order to prove the existence of homoclinic orbits of Liénard systems is stated. As a result, the necessary and sufficient condition for the existence of the homoclinic orbit of the system will be given in the simple form. Our idea is to apply curves with some invariance to the existence of the homoclinic orbit. It will be shown that the positions of these curves on the phase space play an important role in the existence of homoclinic orbits of the system. The results will be applied to many examples.

Received: October 25, 2010

AMS Subject Classification: 34C07, 34C25, 34C26, 34D20

Key Words and Phrases: Liénard system, homoclinic orbits, invariant curves

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2011
Volume: 66
Issue: 1