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<p>This study discusses the resonance, bifurcation, and chaotic motion of a class of fractional order high power damped dissipative systems. By combining multi-scale methods with averaging methods, the amplitude frequency relationship expressions of the steady-state response of the system under linear viscous damping, square damping, cubic damping, and fourth power damping were derived. The existence and stability of fixed points in the mean equation, which correspond to non-trivial periodic solutions of the original system, are studied. For the studied system, only additive combination parameter resonance is possible. The Melnikov theory for high power damping systems of the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11360_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-th power has been extended to decouple complex systems, and the conditions for the existence of homoclinic/heteroclinic bifurcations have been obtained through approximating the time variable and Fourier transform. We have studied the period doubling bifurcation caused by external excitation amplitude changes to understand the chaotic high-power system potential behavior exhibited by the extended Duffing oscillator under parameter excitation. In addition, the safety basin exhibiting fractal patterns was analyzed by us, and the chaotic behavior of the system was verified through the analysis.</p>

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Research on system resonance and Melnikov chaos of fractional order systems with power damping dissipation

  • 期刊论文

  • 发表日期：

    2025年6月4日

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  • Jialin Si,
  • Jiaquan Xie,
  • Jianhua Yang,
  • Di Liu,
  • Huidong Xu,
  • Wei Shi,
  • Zhanlong Li,
  • Yuanming Liu

摘要

This study discusses the resonance, bifurcation, and chaotic motion of a class of fractional order high power damped dissipative systems. By combining multi-scale methods with averaging methods, the amplitude frequency relationship expressions of the steady-state response of the system under linear viscous damping, square damping, cubic damping, and fourth power damping were derived. The existence and stability of fixed points in the mean equation, which correspond to non-trivial periodic solutions of the original system, are studied. For the studied system, only additive combination parameter resonance is possible. The Melnikov theory for high power damping systems of the \(n\) n -th power has been extended to decouple complex systems, and the conditions for the existence of homoclinic/heteroclinic bifurcations have been obtained through approximating the time variable and Fourier transform. We have studied the period doubling bifurcation caused by external excitation amplitude changes to understand the chaotic high-power system potential behavior exhibited by the extended Duffing oscillator under parameter excitation. In addition, the safety basin exhibiting fractal patterns was analyzed by us, and the chaotic behavior of the system was verified through the analysis.