Chaos and bifurcation of Mathieu-Duffing system with square damping term

XIE Jiaquan1,2, WANG Haijun1,2, SHI Wei2,3, ZHANG Jiale1,2, HUO Yiting1,2, CAO Jialin1,2, GAO Qiang1,2

Journal of Vibration and Shock ›› 2024, Vol. 43 ›› Issue (7) : 168-174.

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Journal of Vibration and Shock ›› 2024, Vol. 43 ›› Issue (7) : 168-174.

Chaos and bifurcation of Mathieu-Duffing system with square damping term

  • XIE Jiaquan1,2, WANG Haijun1,2, SHI Wei2,3, ZHANG Jiale1,2, HUO Yiting1,2, CAO Jialin1,2, GAO Qiang1,2
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Abstract

The purpose of this paper is to study the resonance and chaos of Mathieu Duffing system with square damping term. The amplitude frequency and phase frequency characteristics of the main resonance of the system under the combined excitation of parameters and force are explored by using the multi-scale method. Based on Lyapunov's first method, the stability conditions of the steady solution are given and the periodic solution branches of the system are determined. According to the Parametric equation of the heteroclinic orbits of the system, the necessary conditions for the heteroclinic orbits to cross and the chaos of the system are derived. According to the bifurcation diagram, phase trajectory diagram, and Poincare cross-section were used to study the effects of excitation amplitude and frequency on the chaotic motion behavior of the system. It was confirmed that changes in excitation frequency and amplitude can lead to the system entering a chaotic state through period doubling bifurcation.

Key words

Mathieu Duffing system / Square term damping / Multi scale method / Lyapunov first method / Period doubling bifurcation

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XIE Jiaquan1,2, WANG Haijun1,2, SHI Wei2,3, ZHANG Jiale1,2, HUO Yiting1,2, CAO Jialin1,2, GAO Qiang1,2. Chaos and bifurcation of Mathieu-Duffing system with square damping term[J]. Journal of Vibration and Shock, 2024, 43(7): 168-174

References

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