This dissertation treats two related problems in the area of chaos synchronism. In the first part, we address one question that is fundamental to the understanding of chaotic systems: how does a low-dimensional chaotic invariant set arise in high- or infinite-dimensional phase space? The question is motivated by the fact that the phase space of many dynamical systems in nature are infinite-dimensional, yet it often occurs that the dynamical invariant sets responsible for many observable phenomena of physical interest lie in some low-dimensional manifold. From an applied point of view, the occurrence of low-dimensional chaotic invariant sets is highly desirable, as these sets can be understood, controlled, and even predicted to certain degree. The main contribution of this part of this dissertation is an understanding of a possible scenario for dynamical systems to exhibit a low-dimensional asymptotic chaotic invariant set. The key is synchronization. We provide arguments and strong numerical evidence for our conjecture that generalized chaotic synchronism is a sufficient condition for the occurrence of low-dimensional chaotic invariant sets in high-dimensional phase space.; The second part of the dissertation treats the problem of phase synchronization in systems of coupled chaotic oscillators. In particular, the phenomenon of phase synchronization in weakly coupled nonidentical chaotic oscillators has received a tremendous amount of recent attention. Consider the situation where each individual oscillator exhibits a chaotic attractor in phase space. Due to the recurrence of chaotic trajectories, the motion resembles that of a complicated rotation and, as such, a proper angle of rotation, or phase, can be defined. When two such chaotic oscillators are coupled, their phases tend to follow each other in the sense that the phase difference remains bounded even when the coupling is weak, in contrast to the uncoupled case where the phase difference increases linearly with time. The amplitudes of the chaotic rotations, however, might remain uncorrelated despite synchronization in their phases. Chaotic phase synchronization appears to be a general phenomenon in systems of coupled nonlinear oscillators.; The issue of high performance scientific computing and the process of constructing the Beowulf supercomputer are then described at the end of the dissertation. (Abstract shortened by UMI.)
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