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School of Physics

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College of Sciences

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Center for Relativistic Astrophysics

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Mourigal Lab

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Now showing 1 - 9 of 9

Spatiotemporal Tiling of the Kuramoto-Sivashinsky equation

(Georgia Institute of Technology, 2020-12-06) Gudorf, Matthew

Motivated by space-time translational invariance and exponentially unstable dynamics, `spatiotemporally chaotic' or `turbulent' flows are recast as a (D+1)-dimensional spatiotemporal theory which treats space and time equally. Time evolution is replaced by a repertoire of spatiotemporal patterns taking the form of (D+1) dimensional invariant tori (periodic orbits). Our claim is that the entirety of space-time can be described as the shadowing of a finite collection of `fundamental orbits'. We demonstrate that not only can fundamental orbits be extracted from larger orbits, they can also be used as the `building blocks' of turbulence. In the future we aim to explain all of these results by constructing a (D+1)-dimensional symbolic dynamics whose alphabet is the set of fundamental orbits, however, in order to do so we must first find all fundamental orbits. These ideas are investigated in the context of the 1+1 dimensional space-time of the Kuramoto-Sivashinsky equation using the independently developed Python package 'orbithunter'.
Geometry of inertial manifolds in nonlinear dissipative dynamical systems

(Georgia Institute of Technology, 2017-04-05) Ding, Xiong

High- and infinite-dimensional nonlinear dynamical systems often exhibit complicated flow (spatiotemporal chaos or turbulence) in their state space (phase space). Sets invariant under time evolution, such as equilibria, periodic orbits, invariant tori and unstable manifolds, play a key role in shaping the geometry of such system’s longtime dynamics. These invariant solutions form the backbone of the global attractor, and their linear stability controls the nearby dynamics. In this thesis we study the geometrical structure of inertial manifolds of nonlinear dissipative systems. As an exponentially attracting subset of the state space, inertial manifold serves as a tool to reduce the study of an infinite-dimensional system to the study of a finite set of determining modes. We determine the dimension of the inertial manifold for the one-dimensional Kuramoto-Sivashinsky equation using the information about the linear stability of system’s unstable periodic orbits. In order to attain the numerical precision required to study the exponentially unstable periodic orbits, we formulate and implement “periodic eigendecomposition”, a new algorithm that enables us to calculate all Floquet multipliers and vectors of a given periodic orbit, for a given discretization of system’s partial differential equations (PDEs). It turns out that the O(2) symmetry of Kuramoto-Sivashinsky equation significantly complicates the geometrical structure of the global attractor, so a symmetry reduction is required in order that the geometry of the flow can be clearly visualized. We reduce the continuous symmetry using so-called slicing technique. The main result of the thesis is that for one-dimensional Kuramoto-Sivashinsky equation defined on a periodic domain of size L = 22, the dimension of the inertial manifold is 8, a number considerably smaller that the number of Fourier modes, 62, used in our simulations. Based on our results, we believe that inertial manifolds can, in general, be approximately constructed by using sufficiently dense sets of periodic orbits and their linearized neighborhoods. With the advances in numerical algorithms for finding periodic orbits in chaotic/turbulent flows, we hope that methods developed in this thesis for a one-dimensional nonlinear PDE, i.e., using periodic orbits to determine the dimension of an inertial manifold, can be ported to higher-dimensional physical nonlinear dissipative systems, such as Navier-Stokes equations.
Exact coherent structures in spatiotemporal chaos: From qualitative description to quantitative predictions

(Georgia Institute of Technology, 2015-11-16) Budanur, Nazmi Burak

The term spatiotemporal chaos refers to physical phenomena that exhibit irregular oscillations in both space and time. Examples of such phenomena range from cardiac dynamics to fluid turbulence, where the motion is described by nonlinear partial differential equations. It is well known from the studies of low dimensional chaotic systems that the state space, the space of solutions to the governing dynamical equations, is shaped by the invariant sets such as equilibria, periodic orbits, and invariant tori. State space of partial differential equations is infinite dimensional, nevertheless, recent computational advancements allow us to find their invariant solutions (exact coherent structures) numerically. In this thesis, we try to elucidate the chaotic dynamics of nonlinear partial differential equations by studying their exact coherent structures and invariant manifolds. Specifically, we investigate the Kuramoto-Sivashinsky equation, which describes the velocity of a flame front, and the Navier-Stokes equation for an incompressible fluid in a circular pipe. We argue with examples that this approach can lead to a theory of turbulence with predictive power.
State Space Partitions of Stochastic Chaotic Maps

(Georgia Institute of Technology, 2014-05-05) Heninger, Jeffrey M.

The finest resolution that can be achieved in any real chaotic system is limited by the presence of noise. This noise can be used to define neighborhoods of the deterministic periodic orbits using the local eigenfunctions of the Fokker-Planck operator and its adjoint. We extend the work of D. Lippolis to include hyperbolic periodic orbits. The dynamics along the stable and unstable directions are separated. The neighborhoods on the stable and unstable manifolds can be defined in the same way as the neighborhoods for entirely stable or entirely unstable orbits. The neighborhoods are then returned to the original coordinates. The Fokker-Planck evolution can be described as a finite Markov transition graph between these neighborhoods. Its spectral determinant is used to calculate the time averages of observables. We apply this technique to calculate long time observables of the Lozi map.
How well can one resolve the state space of a chaotic map?

(Georgia Institute of Technology, 2010-04-06) Lippolis, Domenico

All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching/contraction and the smearing due to noise. My goal is to determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additive white noise. That is achieved by computing the local eigenfunctions of the Fokker-Planck evolution operator in linearized neighborhoods of the periodic orbits of the corresponding deterministic system, and using overlaps of their widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables. The method applies in principle to both continuous- and discrete-time dynamical systems. Numerical tests of such optimal partitions on unimodal maps support my hypothesis.
Recurrent spatio-temporal structures in presence of continuous symmetries

(Georgia Institute of Technology, 2009-04-06) Siminos, Evangelos

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.
Charting the State Space of Plane Couette Flow: Equilibria, Relative Equilibria, and Heteroclinic Connections

(Georgia Institute of Technology, 2008-07-08) Halcrow, Jonathan

The study of turbulence has been dominated historically by a bottom-up approach, with a much stronger emphasis on the physical structure of ﬂows than on that of the dynam- ical state space. Turbulence has traditionally been described in terms of various visually recognizable physical features, such as waves and vortices. Thanks to recent theoretical as well as experimental advancements, it is now possible to take a more top-down approach to turbulence. Recent work has uncovered non-trivial equilibria as well as relative periodic orbits in several turbulent systems. Furthermore, it is now possible to verify theoretical results at a high degree of precision, thanks to an experimental technique known as Particle Image Velocimetry. These results squarely frame moderate Reynolds number Re turbulence in boundary shear ﬂows as a tractable dynamical systems problem. In this thesis, I intend to elucidate the ﬁner structure of the state space of moderate Re wall-bounded turbulent ﬂows in hope of providing a more accurate and precise description of this complex phenomenon. Computation of new undiscovered equilibria, relative equilibria, and their heteroclinic connections provide a skeleton upon which a numerically accurate description of turbulence can be framed. The behavior of the equilibria under variation of Reynolds number and cell aspect ratios is also examined. It is hoped that this description of the state space will provide new avenues for research into nonlinear control systems for shear ﬂows as well as quantitative predictions of transport properties of moderate Re ﬂuid ﬂows.
Chaotic Scattering in Rydberg Atoms, Trapping in Molecules

(Georgia Institute of Technology, 2007-11-20) Paskauskas, Rytis

We investigate chaotic ionization of highly excited hydrogen atom in crossed electric and magnetic fields (Rydberg atom) and intra-molecular relaxation in planar carbonyl sulfide (OCS) molecule. The underlying theoretical framework of our studies is dynamical systems theory and periodic orbit theory. These theories offer formulae to compute expectation values of observables in chaotic systems with best accuracy available in given circumstances, however they require to have a good control and reliable numerical tools to compute unstable periodic orbits. We have developed such methods of computation and partitioning of the phase space of hydrogen atom in crossed at right angles electric and magnetic fields, represented by a two degree of freedom (dof) Hamiltonian system. We discuss extensions to a 3-dof setting by developing the methodology to compute unstable invariant tori, and applying it to the planar OCS, represented by a 3-dof Hamiltonian. We find such tori important in explaining anomalous relaxation rates in chemical reactions. Their potential application in Transition State Theory is discussed.
Dynamical systems approach to one-dimensional spatiotemporal chaos -- A cyclist's view

(Georgia Institute of Technology, 2004-11-19) Lan, Yueheng

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.