The generalized Liénard systems (Q1848994)

This paper is concerned with the generalized Liénard system \[ x' = (h(y)-F(x))/a(x), \qquad y' = -a(x)g(x), \] where \(a(x), h(y), F(x), g(x)\) are continuous functions and \(a(x) > 0\). When \(h(y)=y\), \(a(x)= \exp\int_0^x k(s) ds\), \(F(x) = \int_0^x a(s) f(s) ds\), such a system reduces to a system equivalent to the second-order equation \(x''+ f(x)x'+ k(x)x'{^2} + g(x) = 0\). Under quite general assumptions, the authors give necessary and sufficient conditions for the global asymptotic stability of a critical point. Their techniques are an extension to such systems of techniques previously applied in the study of boundedness and stability properties of classical Liénard systems.