Type of Document	Dissertation
Author	Lan, Yueheng
Author's Email Address	yueheng_lan@yahoo.com
URN	etd-10282004-154606
Title	Dynamical systems approach to one-dimensional spatiotemporal chaos -- A cyclist's view
Degree	Doctor of Philosophy
Department	Physics
Advisory Committee	Advisor Name Title Predrag Cvitanovic Committee Chair Jean Bellissard Committee Member Konstantin Mischaikow Committee Member Roman Grigoriev Committee Member Turgay Uzer Committee Member
Keywords	periodic orbit theory spatiotemporal chaos pattern formatiion
Date of Defense	2004-05-27
Availability	unrestricted
Abstract 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.
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