Analysis of Chaos and Fractals via Photonic Computing and Approaches of Fuzzy Mathematical Method

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Henon mentions that there exists the structure of a Cantor set in "A Two-dimensional Mapping with a Strange Attractor," Commun. Math. Phys., 30, pp69-77, 1976 as follows : Figure 4 to 6 strongly suggest that the process of multiplication of "curves" will continue indefinitely, and that each apparent" curve" is in fact made of an infinitely of quasi-parallel curves. Moreover these figures indicate the existence of a hierarchical sequence of "levels," the structure being practically identical at each level save for a scale factor. Our goal of this research is to show the relationship between chaos and fractals and to establish a criterion for the existence of chaos arising from well-known mappings in higher order dynamical systems.Marotto gave a criterion for the existence of chaos in finite dimensional dynamical systems where the absolute values of eigenvalues to the Jacobi matrix at its fixed point are larger than 1 in J. Math. Anal. Appl. 63, pp199-223, 1978. It is impossible to apply … More the criterion to the Gumowski-Mira's mapping with parameters a=0.008, b=0.05 and μ being through -0.89 to -0.995, in detail see S, Saito and H.Ishii : "On Chaos of the Warlas' Adjustment Difference Equation", Economic papers of Kobegakuin (to appear). Because in some case of parameters the absolute values of the eigenvalues are smaller than 1.In this research we considered the Gumowski-Mira's mapping with absolute values of eigenvalues being smaller than 1 and also had a lot of simulation data of the mapping. In our approaches we study the structure of Cantor sets and we are establishing the existence criterion of Li-Yorke chaos of the mapping via two methods : fuzzy analysis and photonic method.In fuzzy approach we had several results on qualitative criteria for the existence and uniqueness of solutions to boundary fuzzy differential equations. We got practical criteria for fuzzy facility location problems with preference of candidate sites. Moreover we gave results on "Photonic DNA computing : concept and implementation" in the viewpoint of photonic array computing.Our aims of this research are to get criteria for the existence of chaos of the Gumowski-Mira's mapping and to propose a generator of pseudo-random number for encryption by applying the results on analyzing chaos properties, specifically the sensitive dependence of initial data. We had some suitable and applicable results for this research in our papers of the references of this abstract. Less