On the qualitative behavior of a class of generalized Liénard planar systems
This work is devoted to the problem of existence of limit cycles for a class of Liénard generalized differential systems \[ \frac{dx}{dt} = y - F(x,y), \quad \frac{dy}{dt} = -g(x), \] assuming that \(F : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) and \(g : \mathbb{R} \rightarrow \mathbb{R}\) are locally Lipschitz continuous functions, in order to guarantee the uniqueness of the solutions for the associated initial value problems. It is also assumed that \(g(0) = 0\), \(g(x)x > 0 \) for \(x = 0\) and that the origin is the only singular point of the system. At first, the authors discuss some basic facts of the mentioned system related to the use of the energy of the associated Duffing equations as a Lyapunov function. Then they study the case \(F(x, y) = \lambda B(y)A(x)\), where \(A(x)\) satisfies the standard assumptions on \(F(x)\) in the classical case.
qualitative theory of planar dynamical systems
scientific article; zbMATH DE number 7491607
scientific article