具有任意频率的拟周期驱动阻尼振子方程响应解的存在性

Existence of response solution to quasi-periodically forced damping oscillator equation with any frequency

摘要

致力于寻找一个具有任意频率的拟周期驱动阻尼振子方程x_tt+μx_t+x-βx²=εf(ωt)响应解的存在性(即与驱动频率相同的拟周期解)。当 μ≠0 且远离零时,系统是双曲的(特征值的实部非零),此时不会出现小除数问题。因此,在不对频率 ω 施加任何算术性条件,也不要求驱动项的平均是零的情况下,将原方程响应解的存在性转化为 Banach 空间中不动点问题,分别在解析、高阶可微的情形下用压缩映射原理证明方程响应解的存在性。

Abstract

This paper is devoted to finding the existence of the response solution (i.e., quasi-periodic solutions with the same frequency as the forcing) for a quasi-periodically forced damping oscillator equation x_tt+μx_t+x-βx²=εf(ωt) with arbitrary frequency. When μ≠0 and it is far away from zero, the system is hyperbolic (the real parts of eigenvalues are not zero), there is no small divisor problem at this time. Therefore, without imposing any arithmetic conditions on the frequency ω, nor requiring the average of the forcing to be 0, this paper formulates the existence of the response solution of the original equation into a fixed point problem in the Banach space, and proves the existence of the response solution for the equation by using the contraction mapping principle in the case of analytic and higher-order differentiability.

关键词

振子方程 / 响应解 / 压缩映射原理

Key words

oscillator equation / response solution / contraction mapping principle

参考文献

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基金资助

国家自然科学基金资助项目(12101434)

四川省自然科学基金资助项目(24NSFSC4934)