Furuta倒立摆的分岔与混沌动力学分析

丁玉梅;张琪昌

振动与冲击 ›› 2010, Vol. 29 ›› Issue (2) : 21-25.

PDF(762 KB)
PDF(762 KB)
振动与冲击 ›› 2010, Vol. 29 ›› Issue (2) : 21-25.
论文

Furuta倒立摆的分岔与混沌动力学分析

  • 丁玉梅1,2;张琪昌1
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BIFURCATION AND CHAOS ANALYSIS IN THE FURUTA PENDULUM

  • DING Yu-mei1,2;ZHANG Qi-chang1
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摘要

利用Hopf定理和规范形理论,讨论了Furuta旋转倒立摆非线性数学模型的Hopf分岔特性。给出系统存在Hopf分岔的条件,讨论了周期轨道的稳定性,利用数值模拟,得到系统的相轨迹图,进一步验证分析过程的正确性。利用Silnikov定理,讨论了旋转倒立摆的混沌动力学特征。利用卡尔达诺公式和微分方程级数解讨论了该系统的特征值和同宿轨道的存在性,比较严格地证明了系统存在Smale马蹄意义下的混沌现象,并给出发生Silnikov型Smale混沌的条件。

Abstract

Study Hopf bifurcations by normal form theory and the Hopf thory in the Furuta pendulum system. We calculate the normal forms of the Hopf bifurcation systems. The stability of the limit cycle is discussed.The space trajectory are investigated via numerical simulation, which are aslo verified the validity of our analysis. Based on the Silnikov criterion, the chaotic characters of the dynamical systems are discussed. Using Cardano formula and series solution of differential equation, eigenvalue problem and the existence of homoclinic orbit are studied respectively. Furthermore, a rigorous proof for the existence of Silnikov-sense Smale horseshoes chaos is presented and some conditions which lead to the chaos are obtained.

关键词

旋转倒立摆 / Hopf分岔 / 同宿轨道 / 混沌系统 / Silnikov定理

Key words

rotational inverted pendulum / Hopf Bifurcations / homoclinic orbit / chaotic system / Silnikov theorem

引用本文

导出引用
丁玉梅;张琪昌. Furuta倒立摆的分岔与混沌动力学分析[J]. 振动与冲击, 2010, 29(2): 21-25
DING Yu-mei;ZHANG Qi-chang. BIFURCATION AND CHAOS ANALYSIS IN THE FURUTA PENDULUM[J]. Journal of Vibration and Shock, 2010, 29(2): 21-25

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