Study on Methods of Analysis of Nonlinear Circuits with Symmetrical Structure

具有对称结构的非线性电路的状态有两个轨道：一个由其动力学系统决定，另一个由与对称性相关的群作用决定。在这项研究中，我们试图发展这类对称系统的数值分析方法。利用这些方法，我们分析了具有环的耦合振子的极限环的全局分支，具有指定群连接的耦合振子的对称破缺分支等。单向和/或双向耦合振子的分析。我们发现，不同的耦合会导致不同类型的多相振荡，如同相、反相或三相振荡。研究了平衡点的Hopf分叉和极限环的Neimark-Sacker分叉，并通过对称性破缺得到混沌状态。研究了与对称性相关的余维两个或三个分支，给出了寻找这些退化分岔点的数值分析方法。研究了线性和环形耦合神经网络的振动性。将一些混沌行程作为不变流形的全局分支进行了分析。研究了耦合振子的强迫同步问题。通过注入正弦输入，我们发现了同步态的同相和/或反相分叉特性。用同样的数值分析方法对一些中断的非线性电路进行了分析。我们发现了简单的三维系统，其混沌状态被周期性的开关作用所打断。这本小册子是出版的，其中收集了主要成果。

State of nonlinear circuits with symmetrical structure has two orbits : one is determined by its dynamical system and the other by the group action correlated with symmetry. In this research we tried to develop numerical methods of analysis for such systems with symmetry. By using these methods we analyzed the global bifurcation of limit cycles in coupled oscillators with a ring, symmetry breaking bifurcations of coupled oscillators with a prescribed group connection, etc.1. Analysis of a unidirectional and/or bidirectional coupled oscillators. We found that the different couplings cause different types of multi-phase oscillations, such as in-phase, anti-phase, or tri-phase oscillations. The Hopf bifurcation of equilibrium points and the Neimark-Sacker bifurcation of limit cycles are studied and chaotic states are obtained by symmetry breaking.2. Codimension two or three bifurcations correlated with symmetry are studied and some methods of numerical analysis for finding these degenerate bifurcation points are developed.3. Oscillations of neural networks coupled as linear and ring connections are investigated. Some chaotic itinerary are analyzed as a global bifurcation of invariant manifolds.4. Forced synchronization of coupled oscillators are studied. By injecting sinusoidal input we found the bifurcational properties of synchronized states with in phase and/or anti-phase.5. Some interrupted nonlinear circuits are analyzed by using the same method of numerical analysis. We found simple three dimensional systems with chaotic states interrupted by periodic switching action.6. The booklet is published in which main results are collected together.