Necessary and sufficient conditions for boundedness and convergence of solutions to a second-order nonlinear differential system. (Chinese. English summary) Zbl 1018.34038

Summary: The author gives new sufficient conditions for the boundedness of solutions to the following second-order nonlinear differential system \[ \dot x= {1\over a(x)} [h(y)- F(x)],\quad \dot y= -a(x)[g(x)- e(t)],\tag{\(S'\)} \] and obtains a necessary and sufficient condition for all solutions to \((S')\) to be bounded and to converge to zero. These results can be applied to the Liénard-type equation \[ \ddot x+ f_1(x)\dot x+ f_2(x)\dot x^2+ g(x)= e(t). \] {}.


34C11 Growth and boundedness of solutions to ordinary differential equations
34D40 Ultimate boundedness (MSC2000)
34D05 Asymptotic properties of solutions to ordinary differential equations