Abstract
We consider the Li?nard system in the plane and present general assumptions to obtain some new explicit conditions under which this system has or fails to have a positive orbit which starts at a point on the vertical isocline and approaches the origin without intersecting the x-axis. This arises naturally in the existence of homoclinic orbits and oscillatory solutions. Our investigation is based on the notion of orthogonal trajectories of orbits of the system.
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