New conditions for the intersection of orbits with the vertical isocline of the Liénard system

In this paper we investigate the problem of whether all orbits of the system dxdt=y−F(x) and dydt=−g(x) cross the vertical isocline y=F(x)y=F(x). We present some new necessary and sufficient conditions which guarantee the orbits of this system cross the vertical isocline. The conditions obtained are very sharp. Our results substantially extend and improve previous results presented by Aghajani and Moradifam [A. Aghajani, A. Moradifam, Some sufficient conditions for the intersection with the vertical isocline in the Liénard plane, Appl. Math. Lett. 19 (2006) 491–497; A. Aghajani, A. Moradifam, Oscillation of solutions of second-order nonlinear differential equations of Euler type, J. Math. Anal. Appl. 326 (2007) 1076–1089] which already include the results of Villari and Zanolin [G. Villari, F. Zanolin, On a dynamical system in the Liénard plane, Necessary and sufficient conditions for the intersection with the vertical isocline and applications, Funkcial. Ekvac. 33 (1990) 19–38] and Hara and Sugie [T. Hara, J. Sugie, When all trajectories in the Liénard plane cross the vertical isocline?, Nonlinear Differential Equations Appl. 2 (1995) 527–551] as special cases.