Complex dynamics in systems with many degrees of freedom.

摘要

Complex dynamics in systems with many degrees of freedom are investigated with two classes of computational models. The models in the first class are motivated by the experimental observation of spatiotemporal chaos in strongly driven convection cells, and are designed to display chaotic evolution in discrete space and time. A local conservation law is incorporated into the equations of motion, and its importance is discussed. The central limit theorem is applied to characterize fluctuations over large uncorrelated regions, and a simple theory predicting the long wavelength properties of the models is developed and verified numerically. The applicability of the fluctuation-dissipation theorem and the maximum entropy principle to nonequilibrium systems is tested extensively. A possible application to an experimental situation is outlined.; The models in the second class are motivated by the concept of self-organized criticality, which predicts that driven dynamical systems naturally evolve to a statistically stationary state displaying scale invariance. Several scenarios of how scaling behavior can occur in dynamical systems are discussed, using ideas from dimensional analysis. A simple mean field theory for a large class of cellular automata models is developed. Extensive numerical simulations are described which test the validity of scaling forms and demonstrate possible errors resulting from finite size effects.