On the Center of the Generalized Liénard System

摘要

In this paper, we discuss the conditions for a center for the generalized Liénard system (E)1 $$frac{{dx}}{{dt}} = varphi left( y right)- Fleft( x right),frac{{dy}}{{dt}} = - gleft( x right),$$ or (E)1 $$frac{{dx}}{{dt}} = psi left( y right),frac{{dy}}{{dt}} = - fleft( x right)hleft( y right)- gleft( x right),$$ with f(x), g(x),ϕ(y), ψ(y), h(y)colon ℝ→ ℝ, F(x) = ∫0x f(x)dx, and xg(x) > 0 for x ≠ 0. By using a different technique, that is, by introducing auxiliary systems and using the differential inquality theorem, we are able to generalize and improve some results in [1], [2].