Dynamics of a periodic differential equation with a singular nonlinearity of attractive type
This paper considers existence and stability of \(T\)-periodic solutions of the forced Liénard equation (1) \(x''+f(x)x'+g(x)=p(t)\), where the functions \(f\), \(g\) and \(p\) are continuous, \(f>0\), \(g\) is decreasing and \(\lim_{x\to0+} g(x)=+\infty\), \(\lim_{x\to+\infty} g(x)=0\). In case \(\overline p={1\over T}\int^T_0 p(s)ds>0\), the authors prove existence of a unique \(T\)-periodic solution of (1) which is unstable. Further, if the variation \(\sup f-\inf f\) is not too large, solutions of (1), defined on \([t_0,+\infty)\), go as \(t\to+\infty\) either to the periodic solution or to infinity. A third type of result concerns the existence of a global stable manifold described by a decreasing function \(h\): if \(x_0=h(v_0)\), solutions with initial conditions \((x_0,v_0)\) go to the periodic one, if \(x_0>h(v_0)\), these solutions go to infinity as \(t\) goes to infinity and, if \(x_0<h(v_0)\), these solutions go to the singularity in a finite time. Similar results are obtained for the global unstable manifold which is described by an increasing function \(x_0=\xi(v_0)\). As a consequence, there are no homoclinic points.
scientific article; zbMATH DE number 934221