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

在我们之前的研究中，根据有理数逼近的好坏程度对无理数进行分类，我们引入了参数化的丢番图条件，我们称d_0-(D)条件，并且我们还提出了上下循环维之间的间隙作为指标参数，衡量轨道的不可预测性水平。在本研究中，利用d_0-(D)条件的这些阶数，我们估计了循环维的间隙由圆映射或高斯映射给出的准周期轨道。这些结果由首席研究员在 COE 会议（庆应义塾大学）和横滨 Math.J 上公布。 ([3]) 和 J. 非线性与凸分析 ([4]) 中。为了分析各种类型的偏微分方程给出的准周期解的复杂性，我们通过使用这些解的无理频率的 d_0- (D) 条件来估计循环维数。首席研究员在台湾宣布了这些结果。 J.马斯。 ([1]) 和 Proc.4th Conf.on NACA ([5]) 以及共同研究者 M. Misawa 在 [6] 中展示了非线性 P.D.E. 的一些相关结果，这对于研究非线性动力系统的混沌行为非常重要且必不可少。在[7]中，共同研究者 Y. Oshima 也利用概率论证明了随机性的一些相关结果。

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.