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

Summary: The author gives new sufficient conditions for the boundedness of solutions to the following second-order nonlinear differential system

x ˙=1 a(x)[h(y)-F(x)],y ˙=-a(x)[g(x)-e(t)],(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

x ¨+f 1 (x)x ˙+f 2 (x)x ˙ 2 +g(x)=e(t)·

.

MSC:
34C11Qualitative theory of solutions of ODE: growth, boundedness
34D40Ultimate boundedness (MSC2000)
34D05Asymptotic stability of ODE