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

Henon在《具有奇异吸引子的二维映射》中提到了Cantor集的结构。数学课。图4至图6强烈地表明，“曲线”的乘法过程将无限期地持续下去，而且每一条明显的“曲线”实际上都是由无限的准平行曲线组成的。此外，这些数字表明存在层级序列，除比例因子外，每个级别的结构实际上是相同的。本研究的目的是揭示混沌与分形学之间的关系，并建立高阶动力系统中熟知映射产生的混沌存在的判据。Marotto在J.Math中给出了有限维动力系统中Jacobi矩阵在其不动点的特征值的绝对值大于1的混沌存在的判据。肛门。APPL63，第199-223页，1978年。不可能应用…关于参数为a=0.008，b=0.05，μ为-0.89至-0.995的Gumowski-Mira映射的更多判据，详见S、斋藤和石井：“论Warlas‘调整差分方程的混沌”，Kobeakuin的经济论文(待刊)。由于在某些参数情况下，特征值的绝对值小于1，所以在本研究中，我们考虑了特征值绝对值小于1的Gumowski-Mira映射，并且有大量的映射的仿真数据。在我们的方法中，我们研究了Cantor集的结构，通过两种方法：模糊分析和光子方法，我们建立了映射的Li-York ke混沌的存在准则。在模糊方法中，我们得到了关于边界模糊微分方程解的存在唯一性的定性准则的几个结果。我们得到了具有候选地点偏好的模糊设施选址问题的实用准则。在此基础上，从光子阵列计算的角度，给出了“光子DNA计算：概念与实现”的研究结果，其目的是得到Gumowski-Mira映射存在混沌的判据，并应用这些结果分析混沌特性，特别是初始数据的敏感相关性，提出一种用于加密的伪随机数发生器。在本文的参考文献中，我们对这项研究有一些合适和适用的结果。较少

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