Dynamical systems approach to one-dimensional spatiotemporal chaos -- A cyclist's view
We propose a dynamical systems approach to the study of weak turbulence(spatiotemporal chaos) based on the periodic orbit theory, emphasizing
the role of recurrent patterns and coherent structures. After a brief review of the periodic orbit theory and its application to low-dimensional dynamics, we discuss its possible extension to study dynamics of spatially extended systems. The discussion is three-fold. First, we introduce a novel variational scheme for finding periodic orbits in high-dimensional systems.
Second, we prove rigorously the existence of periodic structures (modulated amplitude waves) near the first instability of the complex Ginzburg-Landau equation, and check their role
in pattern formation. Third, we present the extensive numerical exploration of the Kuramoto-Sivashinsky system in the chaotic regime: structure of the equilibrium solutions, our search for the shortest periodic orbits, description of the chaotic invariant set in terms of intrinsic coordinates and return maps on the Poincare section.
Advisor:Jean Bellissard; Turgay Uzer; Roman Grigoriev; Konstantin Mischaikow; Predrag Cvitanovic
School:Georgia Institute of Technology
School Location:USA - Georgia
Source Type:Master's Thesis
Date of Publication:11/19/2004