Analysis of dimensional and recursive properties for almost periodic solutions of nonlinear partial differential equations

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In recent years great efforts have been made to analyze complexity or chaotic behaviors in the study of dynamical systems. In this research we investigate fractal dimensions and recursive properties of orbits for quasi-periodic dynamical systems and then, we apply the abstract results to almost or quasi-periodic solutions for nonlinear partial differential equations. In [1] (of 11.REF.) we estimate correlation dimensions of discrete quasi-periodic orbits by using the parameters derived from some algebraic properties of the irrational frequencies. On the other hand, in [2], we study recursive properties of the quasi-periodic orbits by defining recurrent dimensions and show inequality relations between the correlation dimensions and the recurrent dimensions. To estimate these dimensions we introduce new class of irrational numbers, quasi Roth numbers, quasi or weak Liouville numbers, which are classified according to badly approximable properties or (extremely) good properties for the rational approximations, respectively.Furthermore, in [2] and [3] we investigate quasi-periodic solutions of nonlinear partial differential equations with quasi periodic perturbations and estimate these dimensions of the attractors.Fractal dimensions are most essential in the sense that they show the level of complexity, or selfsimilarity or randomness. On the other hand, it is well known that periodic or almost periodic states occupy the important positions as main gateways in various routes to chaos. In the following papers (of 11.REF.) by the head and co-investigators we have shown various fundamental results, which will play important and essential roles for investigating chaotic behaviors of nonlinear dynamical models.