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Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions

机译:存在上下解的情况下Duffing方程周期解的稳定性

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We prove that a periodic solution of the Duffing equation x" + cx' + g (t, x) = h (t), is asymptotically stable if and only if it is bracketed by a lower solution 06 and an upper solution beta satisfying alpha(t) > beta(t) for every t, provided that the derivative of g with respect to x is not too large. We also produce a characterization of the asymptotic stability of the periodic solutions of the above equation in terms of certain stability properties of the corresponding fixed points of a related infinite dimensional order-preserving discrete-time dynamical system. These results have a local flavour and therefore they naturally apply to the study of the stability in cases where g is not defined everywhere as a function of x, or several periodic solutions exist. As an application, we briefly discuss the existence of stable and unstable periodic solutions for some classes of Duffing equations with singular or oscillating nonlinearities. (C) 2002 Published by Elsevier Science Inc. [References: 17]
机译:我们证明了Duffing方程x“ + cx'+ g(t,x)= h(t)的周期解是渐近稳定的,当且仅当它由一个较低解06和一个满足alpha的较高解beta包围(t)> beta(t),只要g相对于x的导数不太大,我们还可以根据某些稳定性,对上述方程的周期解的渐近稳定性进行刻画相关的无限维保序离散时间动力系统的对应不动点,这些结果具有局部风味,因此,当g到处都没有定义为x的函数时,它们自然适用于稳定性的研究,或存在几个周期解。作为应用,我们简要讨论某些具有奇异或振荡非线性的Duffing方程的稳定和不稳定周期解的存在。(C)2002由Elsevier Science Inc.发布[Ref等级:17]

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