Dynamical systems approach to one-dimensional spatiotemporal chaos-- A cyclist's view

Periodic orbit theory;Spatio-temporal chaos;Pattern formation

Lan, Yueheng ; Physics

University:Georgia Institute of Technology

Department:Physics

关键词: Periodic orbit theory;    Spatio-temporal chaos;    Pattern formation;   

Others  :  https://smartech.gatech.edu/bitstream/1853/4833/1/lan_yueheng_200412_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

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.