含平方阻尼项的Mathieu-Duffing系统混沌与分岔研究

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振动与冲击 ›› 2024, Vol. 43 ›› Issue (7) : 168-174.

论文

解加全1,2, 王海军1,2, 师玮2,3, 张佳乐1,2, 霍逸婷1,2, 曹佳琳1,2, 高蔷1,2

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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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摘要

本文旨在研究含平方阻尼项Mathieu-Duffing系统的共振与混沌。利用多尺度法探究系统在参数和受迫联合激励作用下主共振的幅频与相频特性。基于Lyapunov 第一方法给出定常解的稳定性条件并判定系统存在的周期解支。依据系统异宿轨道参数方程推导系统出现异宿轨道横截相交及系统发生混沌的必要条件。根据分岔图、相轨迹图以及Poincare截面研究激励幅值与激励频率对系统进入混沌运动性态的影响，证实激励频率与激励幅值的变化均可导致系统经倍周期分岔进入混沌状态。

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.

导出引用

解加全1,2, 王海军1,2, 师玮2,3, 张佳乐1,2, 霍逸婷1,2, 曹佳琳1,2, 高蔷1,2. 含平方阻尼项的Mathieu-Duffing系统混沌与分岔研究[J]. 振动与冲击, 2024, 43(7): 168-174

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

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