Analysis on the fractal structure of quasi periodic orbits for nonlinear evolution equations

在我们之前的研究中，我们引入了离散动力系统的循环维数，并估计了离散准周期轨道的上下循环维数，以分析各种类型的偏微分方程给出的准周期解的复杂性。为了研究各种混沌轨道，我们还提出了上循环维度和下循环维度之间的间隙作为指标参数，用于测量轨道的不可预测性水平。在这项研究中，根据有理数逼近的好坏程度对无理数进行分类，并参数化丢番图条件，我们称d_0-(D)条件，我们利用这些参数化丢番图条件的阶数来估计准周期轨道的循环维数间隙。这些结果由首席研究员在国际会议NACA2005（[1]）和Discr.Conti.Dyn.Systems ([4]) 和其他期刊。计算准周期动力系统的维数差距是为了衡量其复杂性和随机性水平。在[5]、[6]中，共同研究者 Y.Oshima 使用概率论证明了随机性的一些相关结果。另一方面，在[7]-[9]中，共同研究员M.Misawa展示了P.D.E.的各种基本结果，这对于研究非线性动力学模型的混沌行为将发挥重要且必不可少的作用。在[10]、[11]中，共同研究员T.Sadahiro对各种准周期动力系统的分形平铺结构进行了数值估计。

In our previous research we introduced recurrent dimensions of discrete dynamical systems and we have estimated the upper and lower recurrent dimensions of discrete quasi-periodic orbits to analyze complexity of quasi-periodic solutions given by various types of partial differential equaitions. To investigate various chaotic orbits 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, classifying the irrational numbers according to the orders of goodness or badness levels of approximation by rational numbers and parametrizing the Diophantine conditions, we say d_0-(D) condition, we estimate the gaps of recurrent dimensions of quasi-periodic orbits by using these orders of the parametrizing Diophantine conditions.These results were announced by the head investigator in the international conference NACA2005 ([1]) and in Discr.Conti.Dyn.Systems ([4]) and in the other journals.Calculating the dimensions gaps of quasi-periodic dynamical systems is to measure their level of complexity and randomness. In [5], [6] the co-investigator Y.Oshima proved some related results for randomness, using probability theory. On the other hand, in [7]-[9] the co-investigator M.Misawa has shown various fundamental results on P.D.E., which will play important and essential roles for investigating chaotic behaviors of nonlinear dynamical models. In [10],[11] the co-investigator T.Sadahiro numerically estimated the fractal tiling structures of various quasi-periodic dynamical systems.