Title:
Recurrent spatio-temporal structures in presence of continuous symmetries

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dc.contributor.advisor Cvitanović, Predrag
dc.contributor.author Siminos, Evangelos en_US
dc.contributor.committeeMember Dieci, Luca
dc.contributor.committeeMember Grigoriev, Roman
dc.contributor.committeeMember Schatz, Michael
dc.contributor.committeeMember Wiesenfeld, Kurt
dc.contributor.department Physics en_US
dc.date.accessioned 2009-06-08T19:29:07Z
dc.date.available 2009-06-08T19:29:07Z
dc.date.issued 2009-04-06 en_US
dc.description.abstract 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 infinite-dimensional state space and (b) the associated measure, that is, the likelihood that asymptotic dynamics visits a given state 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 state 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 state 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 Poincare sections. en_US
dc.description.degree Ph.D. en_US
dc.identifier.uri http://hdl.handle.net/1853/28215
dc.publisher Georgia Institute of Technology en_US
dc.subject Spatio-temporal en_US
dc.subject Nonlinear en_US
dc.subject Recurrence en_US
dc.subject Moving frame en_US
dc.subject Symmetry reduction en_US
dc.subject Kuramoto-Sivashinsky en_US
dc.subject Turbulence en_US
dc.subject Dynamical system en_US
dc.subject Equivariance en_US
dc.subject Invariant en_US
dc.subject Chaos en_US
dc.subject Continuous symmetry en_US
dc.subject.lcsh Dynamics
dc.subject.lcsh Chaotic behavior in systems
dc.subject.lcsh Nonlinear theories
dc.title Recurrent spatio-temporal structures in presence of continuous symmetries en_US
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.advisor Cvitanović, Predrag
local.contributor.corporatename College of Sciences
local.contributor.corporatename School of Physics
relation.isAdvisorOfPublication 9e426c12-f8c3-45b7-b36c-aceab7799f3b
relation.isOrgUnitOfPublication 85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication 2ba39017-11f1-40f4-9bc5-66f17b8f1539
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