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

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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.