**R. P. Agarwal, A. Aghajani, M. Mirafzal**

##
Exact Conditions for the Existence of Homoclinic Orbits in The Liénard
Systems

**abstract:**

We consider the Liénard system $\dot{x}=
y-F(x)$ and $\dot{y}= -g(x)$. Under the assumptions that the origin is a unique
equilibrium, we investigate the existence of homoclinic orbits of this system
which is closely related to the stability of the zero solution, center problem,
global attractively of the origin, and oscillation of solutions of the system.
We present the necessary and sufficient conditions for this system to have a
positive orbit which starts at a point on the vertical isocline $y=F(x)$ and
approaches the origin without intersecting the $x$-axis. Our results solve the
problem completely in some sense.

**Mathematics Subject Classification:**
Primary 37C29; Secondary 34A12

**Key words and phrases:** Homoclinic orbit, Liénard
system, oscillation