Bounded and Unbounded Solutions of the SD Oscillator at Resonance
边静珂 1刘杰1
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作者信息
1. 河南理工大学数学与信息科学学院,焦作,454003
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摘要
本文研究如下的光滑-不连续振子(SD振子)解的有界性和无界性:x"+f(x)x'+x-x√x2+α2=p(t).由于f(x)≠ 0,该系统不是Hamilton系统,我们需要引入可逆性假设以便利用可逆系统的小扭转定理.此外,当非负参数α减小至0时,系统变得不连续.此时,我们需要引入适当的变换来克服正则性的缺失.我们证明:对于任意非负参数α和周期的奇函数p(t),当|∫2π0p(t)sin t d t|＜4时,方程所有解均有界;当|∫2π0p(t)sin t d t|＞4时,方程存在无界解;当|∫2π0p(t)sin t d t|≥4+|F|∞时,方程所有解均无界.
Abstract
In this paper we study the boundedness and unboundedness of the solutions of the smooth and discontinuous(SD)oscillator x"+f(x)x'+x-x/√x2+α2=p(t).Since f(x)≠0,the system is non-Hamiltonian,so we have to introduce some reversibility assumptions to apply a suitable twist theorem,for reversible maps with small twist.Moreover,when the nonnegative parameter α decreases to 0,the system becomes discontinuous.In this case,we need to introduce some suitable transformations to overcome the lack of regularity.We will prove that for any nonnegative parameter α,when p(t)is an odd periodic function satisfying|∫2π0p(t)sin t dt|＜4,all the solutions are bounded;when p(t)satisfies|∫2π0p(t)sin t d t|＞4,the SD oscillator has unbounded solutions,and when p(t)satisfies|∫2π0p(t)sin t d t|≥ 4+|F|∞,all the solutions are unbounded.
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