All inclusive synchronization phenomena-based studies on transmissions of energies and information in coupled systems of electric circuits

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In this study we have clarified synchronizing phenomena in the wide sense, which include quasi-periodic and chaotic synchronizations in various nonlinear systems.First, we returned to the context of nonlinear oscillation theory in which we began, in 1961, our experience with chaotic behavior. This context involves the systems of forced self-oscillators arising in electronic circuits. We combined earlier examples into a simple mixed system, and obtained information on the bifurcation diagram of this system. In particular, the parameter regime of the broken-egg chaotic attractor is mapped.Next, on behalf of the analysis on quasi-periodic oscillation and chaotic behavior in coupled electric system, a coupled magneto-elastic beam system, which has elastically continuous and magnetically discrete characteristics, was proposed. It made us possible to examine the physical phenomenon experimentally and to obtain the appropriate mathematical model. The results obtained by one of members showed the followings :1) The long wave length phenomena cannot be found in the finite dimensional system. Therefore, the partial differential equation is not physically appropriate to describe the short wave length phenomena.2) The dynamical behavior in the finite element nonlinear coupled systems, which imply between partial differential equations and ordinary differential equations, can be described by difference-differential equations qualitatively based on the experimental results on the synchronous phenomenon and the wave propagation.3) The unstable standing wave has an important role to realize the onset of wave propagation. The high dimensional heteroclinic structure of manifolds is substantial for the onset. These results give us the great knowledge on the relation between the phenomena in physical models and in the mathematical models.