Analysis on the fractal structure of complexity solutions for nonlinear differential equations

在我们以前的研究中，根据有理数逼近的好坏程度对有理数进行分类，我们引入了参数化丢番图条件，我们称之为d_0-（D）条件，并提出了上下回归维数之间的差距作为衡量轨道不可预测性程度的指标参数。利用d_0-（D）条件的这些阶，我们估计了由圆映射或高斯映射给出的拟周期轨道的回归维数的间隙。这些结果已在COE会议（庆应义塾大学）、Yokohama Math.J.（[3]）和J. Nonlinear and Convex Analysis（[4]）上发表。为了分析各种类型的偏微分方程的拟周期解的复杂性，我们利用这些解的无理频率的d_0-（D）条件估计回归维数。首席研究员在台湾宣布了这些结果。J. Math.（[1]）和Proc.4th Conf.on NACA（[5]）以及共同研究者M. Misawa在[6]中对非线性偏微分方程给出了一些相关结果，这对于研究非线性动力系统的混沌行为是非常重要的。在[7]中，共同研究者Y.大岛还证明了随机性的一些相关结果，使用概率论。

In our previous research, classifying irrational numbers according to the orders of goodness or badness levels of approximation by rational numbers, we introduced the parameterized Diophantine condition, we say d_0- (D) condition, and we also proposed the gaps between the upper and the lower recurrent dimensions as the index parameters, which measure unpredictability levels of the orbits.In this research, using these orders of the d_0-(D) conditions, we estimate the gaps of recurrent dimensions of quasi-periodic orbits given by the circle mappings or the Gauss map. These results were announced by the head investigator in the COE conference (Keio University) and in Yokohama Math.J. ([3]) and in J. Nonlinear and Convex Analysis ([4]).To analyze complexity of quasi-periodic solutions given by various types of partial differential equations we estimate the recurrent dimensions by using the d_0- (D) conditions for their irrational frequencies of these solutions. The head investigator announced these results in Taiwan. J. Math. ([1]) and in Proc.4th Conf.on NACA ([5]) and the co-investigator M. Misawa in [6] has shown some related results on nonlin-ear P.D.E., which are important and essential to investigate chaotic behaviors of nonlinear dynamical systems. In [7] the co-investigator Y. Oshima also proved some related results for randomness, using probability theory.