Abstract :
We consider the generalized Liénard system
dx
dt =
1
a(x) h(y) − F(x) ,
dy
dt =−a(x)g(x),
where a, F, g, and h are continuous functions on R and a(x) > 0, for x ∈ R. Under the assumptions that
the origin is a unique equilibrium, we study the problem whether all trajectories of this system intersect
the vertical isocline h(y) = F(x), which is very important in the global asymptotic stability of the origin,
oscillation theory, and existence of periodic solutions. Under quite general assumptions we obtain sufficient
and necessary conditions which are very sharp. Our results extend the results of Villari and Zanolin, and
Hara and Sugie for this system with h(y) = y, and a(x) = 1 and improve the results presented by Sugie
et al. and Gyllenberg and Ping.
© 2007 Elsevier Inc. All rights reserved.
