Recurrent spatio-temporal structures in presence of continuous symmetries

2009 2009

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Abstract (summary)

When statistical assumptions do not hold and coherent structures are present in spatially extended systems such as fluid flows, flame fronts and field theories, a dynamical description of turbulent phenomena becomes necessary. In the dynamical systems approach, theory of turbulence for a given system, with given boundary conditions, is given by (a) the geometry of its 8-dimensional phase space and (b) the associated measure, i.e., the likelihood that asymptotic dynamics visits a given phase space region.

In this thesis this vision is pursued in the context of Kuramoto-Sivashinsky system, one of the simplest physically interesting spatially extended nonlinear systems. With periodic boundary conditions, continuous translational symmetry endows phase space with additional structure that often dictates the type of observed solutions. At the same time, the notion of recurrence becomes relative: asymptotic dynamics visits the neighborhood of any equivalent, translated point, infinitely often. Identification of points related by the symmetry group action, termed symmetry reduction, although conceptually simple as the group action is linear, is hard to implement in practice, yet it leads to dramatic simplification of dynamics.

Here we propose a scheme, based on the method of moving frames of Cartan, to efficiently project solutions of high-dimensional truncations of partial differential equations computed in the original space to a reduced phase space. The procedure simplifies the visualization of high-dimensional flows and provides new insight into the role the unstable manifolds of equilibria and traveling waves play in organizing Kuramoto-Sivashinsky flow. This in turn elucidates the mechanism that creates unstable modulated traveling waves (periodic orbits in reduced space) that provide a skeleton of the dynamics. The compact description of dynamics thus achieved sets the stage for reduction of the dynamics to mappings between a set of Poincaré sections.

Indexing (details)

Theoretical physics;
Plasma physics
0753: Theoretical physics
0759: Plasma physics
Identifier / keyword
Pure sciences; Continuous symmetries; Moving frames; Symmetry reduction
Recurrent spatio-temporal structures in presence of continuous symmetries
Siminos, Evangelos
Number of pages
Publication year
Degree date
School code
DAI-B 70/06, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
Cvitanovic, Predrag
Georgia Institute of Technology
University location
United States -- Georgia
Source type
Dissertations & Theses
Document type
Dissertation/thesis number
ProQuest document ID
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
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