# Browsing by Subject "Chaos"

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Item Bringing order out of chaos : an examination of continuity and discontinuity in young children's experiences of household and classroom chaos during early childhood(2013-05) Bobbitt, Kaeley Celeste; Gershoff, Elizabeth T.Show more Early childhood—a period of development that research has established as a critical period for establishing a foundation to support later development and well-being—is increasingly likely to take place in multiple contexts. Continuity and discontinuity in children’s exposure to environmental chaos across two important contexts for their early development: (1) the home and (2) the early learning and care (ELC) setting were examined using data from a large representative sample of low-income preschool children attending Head Start in order to determine how children’s exposure to chaos in each context combine to either promote or interfere with their social-emotional and cognitive development over a year of preschool. A series of multi-level models tested whether children’s experiences of chaos, operationalized in three ways: (1) as individual indicators of crowding, lack of routines, and instability in each setting; (2) as a cumulative index of chaos in each setting; and (3) as a profile that incorporated children’s experiences across setting, influenced children’s social-emotional and cognitive development. Both household and classroom chaos predicted children’s development, but children’s experiences in their home environments were the predominant influence, indicating that children who had non-chaotic home environments gained more over the preschool year than did children who had chaotic homes. These findings provide additional support that effective and high-quality early education and care settings must incorporate children’s home and family experiences.Show more Item Chaos, quasibound states, and classical periodic orbits in HOCI(2011-05) Barr, Alexander Michael; Reichl, L. E.; Bengtson, Roger D.; Kopp, Sacha; Sitz, Greg O.; Wyatt, Robert E.Show more We study the classical nonlinear dynamics and the quantum vibrational energy eigenstates of the molecule HOCl. The classical vibrational dynamics, at energies below the HO+Cl dissociation energy, contains several saddle-center and period doubling bifurcations. The saddle-center bifurcations are shown to be due to a 2:1, and at higher energies a 3:1, nonlinear resonance between bend and stretch motions in various periodic orbits. The sequence of bifurcations takes the system from nearly integrable at low energies to almost completely chaotic at energies near the HO+Cl dissociation energy. At energies above dissociation we study the chaotic scattering of the Cl atom off the HO dimer. This scattering is governed by a homoclinic tangle formed by the stable and unstable manifolds of a parabolic periodic orbit at infinity. We construct the first three segments of the homoclinic tangle in phase space and use scattering functions to investigate its higher-order structure. For the quantum system we use a discrete variable representation to efficiently calculate the Hamiltonian matrix. We find 365 even and 357 odd parity eigenstates with energies below the dissociation energy. By plotting the eigenstates in configuration space we show that almost every quantum eigenstate can be associated with one or more of the classical periodic orbits. The classical bifurcations that give rise to new periodic orbits are manifest quantum mechanically through the sudden appearance of new classes of eigenstates. Despite the high degree of chaos in the classical dynamics at energies near the dissociation energy most quantum eigenstates remain highly ordered with recognizable nodal patterns. We use R-matrix theory together with a discrete variable representation to calculate quasibound states with energies above the dissociation energy. We find quasibound states with lifetimes ranging over 5 orders of magnitude. Using configuration space plots and Husimi distributions we show that the long-lived quasibound states are supported by unstable periodic orbits in the classical dynamics and medium-lived quasibound states are spread throughout the chaotic region of the classical phase space. Short-lived quasibound states show some similarity to unstable periodic orbits that stretch along the dissociation channel.Show more Item Electostatic plasma edge turbulence and anomalous transport in SOL plasmas(2014-08) Meyerson, Dmitry; Gentle, Kenneth W.; Waelbroeck, F.Show more Controlling the scrape-off layer (SOL) properties in order to limit divertor erosion and extend component lifetime will be crucial to successful operation of ITER and devices that follow, where intermittent thermal loads on the order of GW/m² are expected. Steady state transport in the edge region is generally turbulent with large, order unity, fluctuations and is convection dominated. Owing to the success of the past fifty years of progress in magnetically confining hot plasmas, in this work we examine convective transport phenomena in the SOL that occur in the relatively "slow", drift-ordered fluid limit, most applicable to plasmas near MHD equilibrium. Diamagnetic charge separation in an inhomogeneous magnetic field is the principal energy transfer mechanism powering turbulence and convective transport examined in this work. Two possibilities are explored for controlling SOL conditions. In chapter 2 we review basic physics underlying the equations used to model interchange turbulence in the SOL and use a subset of equations that includes electron temperature and externally applied potential bias to examine the possibility of suppressing interchange driven turbulence in the Texas Helimak. Simulated scans in E₀×B₀ flow shear, driven by changes in the potential bias on the endplates appears to alter turbulence levels as measured by the mean amplitude of fluctuations. In broad agreement with experiment negative biasing generally decreases the fluctuation amplitude. Interaction between flow shear and interchange instability appears to be important, with the interchange rate forming a natural pivot point for observed shear rates. In chapter 3 we examine the possibility of resonant magnetic perturbations (RMPs) or more generally magnetic field-line chaos to decrease the maximum particle flux incident on the divertor. Naturally occurring error fields as well as RMPs applied for stability control are known to cause magnetic field-line chaos in the SOL region of tokamaks. In chapter 3 2D simulations are used to investigate the effect of the field-line chaos on the SOL and in particular on its width and peak particle flux. The chaos enters the SOL dynamics through the connection length, which is evaluated using a Poincaré map. The variation of experimentally relevant quantities, such as the SOL gradient length scale and the intermittency of the particle flux in the SOL, is described as a function of the strength of the magnetic perturbation. It is found that the effect of the chaos is to broaden the profile of the sheath-loss coefficient, which is proportional to the inverse connection length. That is, the SOL transport in a chaotic field is equivalent to that in a model where the sheathloss coefficient is replaced by its average over the unperturbed flux surfaces. Both fully chaotic and the flux-surface averaged approximation of RMP application significantly lower maximum parallel particle flux incident on the divertor.Show more Item Experimental study of turbulent jet and lifted jet flame unsteadiness from a non-linear dynamics perspective(2021-07-24) Rafati, Sina; Clemens, Noel T.; Moser, Robert; Varghese, Philip L; Raja, Laxminarayan L; Bisetti, FabrizioShow more This research aims to investigate the nonlinear dynamics of the non-reacting jets and non-premixed lifted jet flames. The goal is to understand better how the flow system dynamics change over time and identify the path toward unwanted conditions such as flashback, extinction, or blowout to limit combustors' dynamical failure. The existence of these undesirable conditions is bound to the fluid's history, meaning that initiated perturbation may persist in the system for time scales comparable to large-scale flow timescales. Hence, the notion is to utilize jet and jet flames as a study test case to work out how the flow evolves dynamically with the hope of understanding how to limit occurrences of the chaotic unwanted condition. Initially, planar particle image velocimetry has been used for the development of the methodologies. I have used planar data to investigate the nonlinear dynamics of non-reacting turbulent jets, with a low-to-moderate Reynolds number using the single-trajectory framework and ensemble framework. I have used Lyapunov exponents to calculate the spectra of scaling indices of the attractor. Then, I used Lagrangian Coherent Structures (LCSs), which are defined as manifolds that are locally Euclidean and invariant, to study the relationship between Lyapunov exponent changes with flow topological features. These LCSs behave as hypersurfaces with maximally repelling or attracting properties. These various methodologies were used to investigate flame-turbulence interaction in lifted jet flames. The Lagrangian framework is shown to be effective at revealing the kinematics associated with flame-turbulence interaction. The LCSs' time history represents how eddy structures interact with the flame and highlight their role in the dynamics of the lifted jet flames. Finally, I have investigated the flame and turbulence interaction using high-speed luminosity imaging and simultaneous three-dimensional particle image velocimetry. The three-dimensional Lagrangian structures provide us a more detailed flow-flame interaction. It is shown that the flow features associated with attracting LCSs can create a barrier attracting the flame that makes the flame move upstream. In contrast, the presence of repelling LCSs near stationary flames breaks the balance between the gas velocity and flame propagation speed, causing the flame to become non-stationary and move downstream. It was also found that the repelling LCSs induce negative curvature on the flame surface whereas pushing the flame toward the products. However, the attracting LCSs induce positive curvature on the flame surface and draws the flame toward the reactantsShow more Item Fractals : an exploration into the dimensions of curves and sufaces(2011-08) Wheeler, Jodi Lynette; Armendáriz, Efraim P.; Daniels, MarkShow more When many people think of fractals, they think of the beautiful images created by Mandelbrot’s set or the intricate dragons of Julia’s set. However, these are just the artistic stars of the fractal community. The theory behind the fractals is not necessarily pretty, but is very important to many areas outside the world of mathematics. This paper takes a closer look at various types of fractals, the fractal dimensionality of surfaces and chaotic dynamical systems. Some of the history and introduction of creating fractals is discussed. The tools used to prevent a modified Koch’s curve from overlapping itself, finding the limit of a curves length and solving for a surfaces dimensional measurement are explored. Lastly, an investigation of the theories of chaos and how they bring order into what initially appears to be random and unpredictable is presented. The practical purposes and uses of fractals throughout are also discussed.Show more Item Information, chaos, and thermodynamic notions in quantum systems(2022-07-29) Guglielmo, Tyler; Fischler, Willy; Paban, Sonia; Caceres, Elena; Kilic, Can; Flauger, RaphaelShow more This dissertation consists of four chapters. The first broadly and briefly orients the reader through an introduction to information spreading in quantum Ising systems, classical chaos, and entanglement in holographic settings. The next three chapters correspond to distinct lines of research conducted during my time as a graduate student, chosen for their thematic relation to information, information dynamics, chaos, and entanglement entropy. Chapter two is based on work conducted with Stefan Eccles, Willy Fischler, Juan Pedraza, and Sarah Racz, studying the spread of information in non-local quantum Ising systems [23]. Chapter three is based on work with Willy Fischler and Phuc Nugyen, studying chaos in a non-commutative field theory [29]. Chapter four is based on work with Phuc Nugyen, testing volume law entanglement in a holographic boosted black hole setting [38].Show more Item Modeling highly symmetric quantum systems with regularized [delta]-function potentials in two and three dimensions(2022-05-02) Furman, Walter Ace; Reichl, L. E.; MacDonald, Allan H; Downer, Michael C; Akinwande, DejiShow more The regularized [delta]-function potential model is used to compute optical and electronic properties of highly symmetric quantum systems. Dirac [delta]-functions allow us to very easily simulate complex quantum systems. In particular, when investigating effects due to symmetric properties of the system, other functional models -- like, for example, elliptic-[theta] functions -- prove to be more computationally expensive than necessary. In this dissertation, we demonstrate the use of regularized [delta]-function potential models in the simulation of a single-walled carbon nanotube and a tetrahedral molecule. Specifically, we show that, when a nanotube with armchair chirality is driven by a circularly polarized optical field, it generates selective high-order harmonic radiation. Using Floquet-Bloch theory, we compute the quasienergy and average energy band structure, as well as the non-linear electron current and emitted high-harmonic power spectrum. Furthermore, we compute and visualize the quasibound state wavefunctions of electrons in a tetrahedrally symmetric molecule. Quasibound states live in the positive energy continuum of molecules, and, due to their relatively long lifetimes, they can have a great impact on dynamics of chemical reactions. Using Wigner-Eisenbud theory and a basis of tetrahedral harmonic functions, we compute the electron scattering dynamics of such a system and show correspondence between the quasibound state energies and the complex poles of the S-matrix.Show more Item Quantum chaos, scattering, and BICs on smooth potentials(2019-08-08) Porter, Maxwell Dare; Reichl, L. E.; Morrison, Philip J.; Niu, Qian; Dicus, Duane A.; La Cour, Brian R.Show more In this dissertation, quantum signatures of chaos and scattering dynamics on several smooth potential models are investigated. In the second chapter, we study chaos in a honeycomb lattice reminiscent of graphene [Phys. Rev. E 93, 012204 (2016)].We show classical chaos to exist in a wide energy range, identify key stable and unstable orbits, and find a quantum eigenstate with irregular nodal structure that’s suggestive of quantum chaos. Newer work for this dissertation shows this model can approximately reproduce the band structure of graphene with only two tuning parameters at low computational cost. In the third chapter, we study a square lattice of Gaussians with variable width [Phys. Rev. E 95, 052213 (2017)]. We find underlying classical chaos correlates with more avoided crossings in band structure, causing frequent mixing of eigenstates. We also find hints of eigenstate scarring. Newer work for this dissertation shows some similarity to the classic Sinai billiard, but with stronger chaos due to the smooth Gaussian peaks, and mixed dynamics due to the potential saddles. In the fourth chapter, we further study quantum chaos in a square lattice [Chaos 27, 104604 (2017)]. We give stronger evidence that underlying chaos causes avoided crossings in the band structure, both on and off the Brillouin zone symmetry lines, and give further examples of eigenstate mixing. We then provide stronger evidence of quantum chaos via energy level spacing distributions. In the fifth chapter, we study scattering from a chaotic triple Gaussian potential [Phys. Rev. E 97, 042206 (2018)]. We use Wigner-Eisenbud (W-E) reaction matrix scattering theory to find quasibound state resonances, and corroborate them with Wigner-Smith delay time plots, reaction region eigenstate plots, and the non-Hermitian Hamiltonian method. We also give evidence of classical and quantum chaos. In the sixth chapter, we use W-E scattering theory to demonstrate that quantum bound states called BICs and long-lived quasibound states exist in the scattering continuum of a quasi-1D Gaussian well lattice [Physica B 571, 15 (2019)]. The BICs come in two types: one protected from decaying by reflection symmetry, the other protected by discrete translational symmetry (conservation of Bloch momentum). The long-lived quasibound states come in odd-even pairs and exist to high energies. The reaction region eigenstates corresponding to these special scattering states are shown to be localized and largely independent of the reaction region boundary (often called the "channel radius").Show more Item The vibrational dynamics of 3D HOCl above dissociation(2015-08) Lin, Yi-Der; Reichl, L. E.; Bohm, Arno; Sitz, Greg O.; Stanton, John F.; Morrison, Philip J.Show more We have analyzed the vibrational dynamics of HOCl above dissociation using a 3D energy surface which governs the vibrational dynamics of HOCl above dissociation. The dynamics is dominated by an invariant manifold which is trasversally unstable for small spacing between Cl and HO complex, and stable for large spacing. Above dissociation, the InM separates two mirror image periodic orbits, embedded in a large chaotic sea, that can hold a large number of quantum states. The periodic orbits have the capability of forming significant quasibound states of the molecule above dissociation.Show more