物理学报2016，Vol.65Issue(9)：094502-0-094502-10,11.DOI:10.7498/aps.65.094502
分数阶时滞反馈对Duffng振子动力学特性的影响∗
Dynamical analysis of Duffng oscillator with fractional-order feedback with time delay
摘要
Abstract
With increasingly strict requirements for control speed and system performance, the unavoidable time delay becomes a serious problem. Fractional-order feedback is constantly adopted in control engineering due to its advantages, such as robustness, strong de-noising ability and better control performance. In this paper, the dynamical characteristics of an autonomous Duffng oscillator under fractional-order feedback coupling with time delay are investigated. At first, the first-order approximate analytical solution is obtained by the averaging method. The equivalent stiffness and equivalent damping coeffcients are defined by the feedback coeffcient, fractional order and time delay. It is found that the fractional-order feedback coupling with time delay has the functions of both delayed velocity feedback and delayed displacement feedback simultaneously. Then, the comparison between the analytical solution and the numerical one verifies the correctness and satisfactory precision of the approximately analytical solution under three parameter conditions respectively. The effects of the feedback coeffcient, fractional order and nonlinear stiffness coeffcient on the complex dynamical behaviors are analyzed, including the locations of bifurcation points, the stabilities of the periodic solutions, the existence ranges of the periodic solutions, the stability of zero solution and the stability switch times. It is found that the increase of fractional order could make the delay-amplitude curves of periodic solutions shift rightwards, but the stabilities of the periodic solutions and the stability switch times of zero solution cannot be changed. The decrease of the feedback coeffcient makes the amplitudes and ranges of the periodic solutions become larger, and induces the stability switch times of zero solution to decrease, but the stabilities of the periodic solutions keep unchanged. The sign of the nonlinear stiffness coeffcient determines the stabilities and the bending directions of delay-amplitude curves of periodic solutions, but the bifurcation points, the stability of zero solution and the stability switch times are not changed. It could be concluded that the primary system parameters have important influences on the dynamical behavior of Duffng oscillator, and the results are very helpful to design, analyze or control this kind of system. The analysis procedure and conclusions could provide a reference for the study on the similar fractional-order dynamic systems with time delays.关键词
分数阶微分/Duffng振子/时滞/平均法Key words
fractional-order derivative/Duffng oscillator/time delay/averaging method引用本文复制引用
温少芳,申永军,杨绍普..分数阶时滞反馈对Duffng振子动力学特性的影响∗[J].物理学报,2016,65(9):094502-0-094502-10,11.基金项目
国家自然科学基金(批准号:11372198)、河北省高等学校创新团队领军人才计划(批准号：LJRC018)、河北省高等学校高层次人才科学研究项目(批准号：GCC2014053)和河北省高层次人才资助项目(批准号：A201401001)资助的课题 （批准号:11372198）
