Study on Methods of Analysis of Nonlinear Circuits with Symmetrical Structure

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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.