Homoclinic orbits in generalized planar system of Liénard type

Abstract

In this manuscript a generalized Liénard 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.