Study of Global Structures and Bifurcations of Dynamical Systems including Systems with Large Degrees of Freedom

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A great improvement of the Conley-Morse graph method, a computer-assisted method that analyzes global dynamics and bifurcations by combining a topological method and validated numerical computation has been done by developing a computer-algorithm for generating so-called clutching graphs which describe relation between Conley-Morse graphs on adjacent parameter domains. Another improvement is a method of non-uniform grid decomposition of the phase space in order to substantially decrease computational cost of the Conley-Morse graph method.A new approach to the bifurcation theory of dynamical systems from topological-computation theory viewpoint has been proposed which captures bifurcation of dynamics from changes of the corresponding Conley-Morse graphs. From this viewpoint, we have obtained a new mathematical framework for the crisis bifurcation.Moreover, we have studied the global dynamics of a Coupled Map Lattice system and a Coupled Oscillator system by using the Conley-Morse graph method, and have obtained the creation and bifurcations of some global dynamics such as unstable invariant tori and their connecting orbits.