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Approximate limit cycles of coupled nonlinear oscillators with fractional derivatives

机译：具有分数衍生物的耦合非线性振荡器的近似极限循环

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摘要

In this paper, the homotopy analysis method (HAM) is presented to establish the accurate approximate analytical solutions for multi-degree-of-freedom (MDOF) coupled nonlinear oscillators with fractional derivatives. Approximate limit cycles (LCs) of two systems of the coupled fractional van der Pol (VDP) oscillators and the fractional damped Duffing resonator driven by a fractional VDP oscillator are exampled for illustrating the validity and great potential of the HAM. The presented approach can provide approximate LCs very accurately and efficiently compared with some direct simulation results. This method can keep high accuracy and efficiency for both weakly and strongly nonlinear problems with any given fractional order. Furthermore, it is capable of tracking unstable LCs which cannot be generated by some time-marching numerical algorithm. Based on the obtained results, we analyze effect of different fractional orders, coupling coefficient, and nonlinear coefficient of the coupled equations on amplitudes and frequencies of the LCs.
机译：本文提出了同型分析方法（火腿）以确定具有分数衍生物的多程度自由度（MDOF）耦合非线性振荡器的准确近似分析解。耦合分数van der POL（VDP）振荡器的两个系统的近似极限循环（LCS）和由分数VDP振荡器驱动的分数阻尼Dumbing谐振器被检查用于说明火腿的有效性和巨大潜力。随着一些直接模拟结果，所提出的方法可以非常准确和高效地提供近似LCS。这种方法可以对任何给定的分数顺序保持弱且强烈的非线性问题的高精度和效率。此外，它能够跟踪不稳定的LCS，该LC不能通过一些行进的数值算法产生。基于所得的结果，我们分析耦合方程的不同分数令，耦合系数和非线性系数对LCS的幅度和频率的影响。

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  • 来源
    《Applied Mathematical Modelling》 |2020年第1期|1294-1309|共16页
  • 作者

    Guoqi Zhang; Zhiqiang Wu;

  • 作者单位

    Department of Mechanics School of Mechanical Engineering Tianjin University Tianjin 300350 China Tianjin Key Laboratory of Nonlinear Dynamics and Control Tianjin University Tianjin 300372 China;

    Department of Mechanics School of Mechanical Engineering Tianjin University Tianjin 300350 China Tianjin Key Laboratory of Nonlinear Dynamics and Control Tianjin University Tianjin 300372 China;

  • 收录信息 美国《科学引文索引》(SCI);美国《工程索引》(EI);
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

    Coupled nonlinear oscillators; Fractional derivative; Approximate limit cycle; Homotopy analysis method;

    机译：耦合非线性振荡器;分数衍生物;近似极限循环;同型分析方法;

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