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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 $$\ddot x+ f_1(x)\dot x+ f_2(x)\dot x^2+ g(x)= e(t).$${}.
34C11Qualitative theory of solutions of ODE: growth, boundedness
34D40Ultimate boundedness (MSC2000)
34D05Asymptotic stability of ODE