JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2026, Vol. 61 ›› Issue (2): 99-105.doi: 10.6040/j.issn.1671-9352.0.2024.290

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

SHU Xingkui, YANG Lian, WANG Fenfen*   

  1. School of Mathematical Science, Sichuan Normal University, Chengdu 610066, Sichuan, China
  • Published:2026-02-13

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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 equationxtt+μxt+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 theresponse solution for the equation by using the contraction mapping principle in the case of analytic and higher-order differentiability.

Key words: oscillator equation, response solution, contraction mapping principle

CLC Number: 

  • O175
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