Attractivity criteria for intermittently damped second order nonlinear differential equations
The author studies second order nonlinear differential equations (1) \(x'' + g(t,x,x')x' + f(x) = 0\) \((t \geq 0)\). He gives sufficient conditions for asymptotic stability and, in some cases, global asymptotic stability of the zero solution of \((1)\). The hypotheses include the case of intermittent damping; i.e. the case that \(g\) is positive on a collection of intervals and zero elsewhere. He investigates the special case \(f(x) = | x | ^\alpha \text{sign}(x)\). In this case he shows the existence of a \(g(\cdot,\cdot,\cdot)=g(t)\) such that the zero solution is globally asymptotically stable if \(\alpha=1\) and asymptotically, but not globally, stable if \(\alpha > 1\).
second order nonlinear differential equation
scientific article; zbMATH DE number 1098670