On the homoclinic orbits of the generalized Liénard equations Original Research Article

In this work we study the existence of homoclinic orbits of the planar system of Liénard type View the MathML sourceẋ=1a(x)[h(y)−F(x)],ẏ=−a(x)g(x), Turn MathJax on where a(x)>0a(x)>0, for every x∈Rx∈R, and hh is strictly increasing, but it is not assumed that h(±∞)=±∞h(±∞)=±∞, h(y)≤myh(y)≤my, or h(y)≥myh(y)≥my. We present sufficient and necessary conditions for this system to have a positive orbit which starts at a point on the curve h(y)=F(x)h(y)=F(x) and approaches the origin without intersecting the xx-axis. The conditions obtained are very sharp. Our results extend the results presented by Hara and Yoneyama for this system with a(x)=1a(x)=1, and h(y)=yh(y)=y, and improve the results presented by Sugie.