Sufficient conditions for oscillation of the Liénard equation
Sufficient conditions for the oscillation of all solutions to the generalized Liénard equation \[ \dot{x}=h(y)-F(x),\quad \dot{y}=-g(x), \tag{*} \] with \(F(0)=0\), \(yh(y)>0\), \(xg(x)>0\) for \(x,y\neq 0\), \(h(y)\) is strictly increasing, \(\lim_{y\to \pm\infty}h(y)=\pm \infty\) and \(\lim_{|x|\to \infty}G(x)=\infty\), with \(G(x)=\int_0^x g(s) ds\). Conditions on the functions \(h,F,g\) are given which guarantee that all solutions to (*) are oscillatory, i.e., the solution curves \(x=x(y)\) of any solution \(x,y\) to (*) crosses infinitely many times the \(y\)-axis in the phase plane. The principal results of the paper are established using the so-called generalized polar coordinates -- a special solution to the system \(\dot{x}=h(y)\), \(\dot{y}=-g(x)\). It is also shown that several recently established oscillation criteria for the (generalized) Liénard equation can be obtained as special cases of the results presented in the paper.
scientific article; zbMATH DE number 1445762