HOMOCLINIC ORBITS IN GENERALIZED PLANAR SYSTEM OF LIENARD TYPE

HOMOCLINIC ORBITS IN GENERALIZED PLANAR SYSTEM OF LIENARD TYPE

T. K. Gharahasanlou, V. Roomi

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Abstract

In this manuscript a generalized Li´enard system will be considered. First, the existence and uniqueness of the solutions of the related initial value problem will be proven. Given some definitions, a necessary and sufficient condition for property (Z+ 1 ) will be presented. Some explicit conditions will also be given for the system to have or fail to have properties (Z+ 1 ). These results are very sharp and extend and improve the previous results in this subject. Finally, a necessary and sufficient condition will be presented about the existence and nonexistence of homoclinic orbits in the upper or lower half-plane. At the end, some examples will be provided to illustrate our results.

Keywords

Li´enard System, Homoclinic Orbit, Planar System, Stable Manifold.