The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems
The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems
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DOI：
10.1109/tcs.1979.1084636
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发表时间：
1979-04
影响因子：
14.7
通讯作者：
A. Mees;L. Chua
中科院分区：
文献类型：
--
作者：
A. Mees;L. Chua
研究自治非线性系统周期解的最有力的方法之一是由Hopf证明发展而来的理论。他表明，平衡点附近的振荡可以通过观察线性化方程的平衡扰动特征值以及方程的某些关键导数来理解。大量的工作已经做了最近在这个理论和本文总结了最近的结果，提出了一些新的，并显示它们如何可以用来研究几乎正弦振荡的非线性电路和系统。新的结果是证明的基本部分的霍普夫定理只使用基本的方法，和图形解释的非线性多回路反馈系统的定理。图形标准检查稳定或不稳定的周期振荡的存在的霍普夫条件。由于它让人想起线性系统的广义Nyquist准则，我们的图形程序可以被解释为频域版本的Hopf分岔定理。
One of the most powerful methods for studying periodic solutions In autonomous nonlinear systems is the theory which has developed from a proof by Hopf. He showed that oscillations near an equilibrium point can be understood by looking at the eigenvalues of the linearized equations for perturbations from equilibrium, and at certain crucial derivatives of the equations. A good deal of work has been done recently on this theory and the present paper summarizes recent results, presents some new ones, and shows how they can be used to study almost sinusoidal oscillations in nonlinear circuits and systems. The new results are a proof of the basic part of the Hopf theorem using only elementary methods, and a graphical interpretation of the theorem for nonlinear multiple-loop feedback systems. The graphical criterion checks the Hopf conditions for the existence of stable or unstable periodic oscillations. Since it is reminiscent of the generalized Nyquist criterion for linear systems, our graphical procedure can be interpreted as the frequencydomain version of the Hopf bifurcation theorem.