Synthesis of Symmetrical Nonlinear-Circuits with Finite Group Representaion

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We considered possible combination of periodic solutions and their bifurcations of given nonlinear dynamical system for topologically classification. With symmetry point of view, corresponding finite group representations and some mathematical results are obtained. We would like to report and publish about these results after finishing preparation. Concretely we applied these mathematical results to neural networks and electric circuits. Firstly we reinvestigated symmetrically coupled BVP oscillators. The coupled oscillators can be chaotic under a reasonable assumption ; each oscillator has different rhythm. We clarified bifurcation diagrams in detail. All phenomena have been confirmed in laboratory experiments.Next, we investigated BVP oscillators synaptically coupled by an alpha-function to simulate delayed coupling system. Since generated periodic solutions can be differentiable, the Poincare mapping is well-defined. Thus bifurcation phenomena and chaos are numerically calculated. Even though the BVP osillator is introduced as a firing model of Hodgkin-Huxley dynamics, qualified analogy between this synaptically coupled system and HH equation have never treated before. We found out there exist similar bifurcation structures between them. Also possible periodic Solutions and non-periodic solutions are classified and enumerated by using finite group representations, then some synchronization phenomena and the state transition of bursting responses are clarified.We are also developing intensively some mathematical tools for interrupted dynamical systems since we found out that the switching point of the periodic orbit can be transferred into one of periodic point of the Poincare mapping.