Browsing by Subject "physics, mathematical"

Now showing 1 - 14 of 14

Boundaries of Siegel Disks: Numerical Studies of their Dynamics and Regularity
(2008-09) de la Llave, Rafael; Petrov, Nikola P.; de la Llave, Rafael
Siegel disks are domains around fixed points of holomorphic maps in which the maps are locally linearizable (i.e., become a rotation under an appropriate change of coordinates which is analytic in a neighborhood of the origin). The dynamical behavior of the iterates of the map on the boundary of the Siegel disk exhibits strong scaling properties which have been intensively studied in the physical and mathematical literature. In the cases we study, the boundary of the Siegel disk is a Jordan curve containing a critical point of the map (we consider critical maps of different orders), and there exists a natural parametrization which transforms the dynamics on the boundary into a rotation. We compute numerically this parameterization and use methods of harmonic analysis to compute the global Holder regularity of the parametrization for different maps and rotation numbers. We obtain that the regularity of the boundaries and the scaling exponents are universal numbers in the sense of renormalization theory (i.e., they do not depend on the map when the map ranges in an open set), and only depend on the order of the critical point of the map in the boundary of the Siegel disk and the tail of the continued function expansion of the rotation number. We also discuss some possible relations between the regularity of the parametrization of the boundaries and the corresponding scaling exponents. (C) 2008 American Institute of Physics.
Combinatorial Games with a Pass: A Dynamical Systems Approach
(2011-12) Morrison, Rebecca E.; Friedman, Eric J.; Landsberg, Adam S.; Morrison, Rebecca E.
By treating combinatorial games as dynamical systems, we are able to address a longstanding open question in combinatorial game theory, namely, how the introduction of a "pass" move into a game affects its behavior. We consider two well known combinatorial games, 3-pile Nim and 3-row Chomp. In the case of Nim, we observe that the introduction of the pass dramatically alters the game's underlying structure, rendering it considerably more complex, while for Chomp, the pass move is found to have relatively minimal impact. We show how these results can be understood by recasting these games as dynamical systems describable by dynamical recursion relations. From these recursion relations, we are able to identify underlying structural connections between these "games with passes" and a recently introduced class of "generic (perturbed) games." This connection, together with a (non-rigorous) numerical stability analysis, allows one to understand and predict the effect of a pass on a game. (C) 2011 American Institute of Physics. [doi:10.1063/1.3650234]
A Dynamical Systems Approach to Spiral Wave Dynamics
(1994-09) Barkley, Dwight; Kevrekidis, Ioannis G.; Barkley, Dwight
A simple system of five nonlinear ordinary differential equations is shown to reproduce many dynamical features of spiral waves in two-dimensional excitable media.
Instability Criteria for Steady Flows of a Perfect Fluid
(1992-07) Friedlander, Susan; Vishik, Misha M.; Vishik, Misha M.
An instability criterion based on the positivity of a Lyapunov-type exponent is used to study the stability of the Euler equations governing the motion of an inviscid incompressible fluid. It is proved that any flow with exponential stretching of the fluid particles is unstable. In the case of an arbitrary axisymmetric steady integrable flow, a sufficient condition for instability is exhibited in terms of the curvature and the geodesic torsion of a stream line and the helicity of the flow.
Lagrangian Based Methods for Coherent Structure Detection
(2015-09) Allshouse, Michael R.; Peacock, Thomas; Allshouse, Michael R.
There has been a proliferation in the development of Lagrangian analytical methods for detecting coherent structures in fluid flow transport, yielding a variety of qualitatively different approaches. We present a review of four approaches and demonstrate the utility of these methods via their application to the same sample analytic model, the canonical double-gyre flow, highlighting the pros and cons of each approach. Two of the methods, the geometric and probabilistic approaches, are well established and require velocity field data over the time interval of interest to identify particularly important material lines and surfaces, and influential regions, respectively. The other two approaches, implementing tools from cluster and braid theory, seek coherent structures based on limited trajectory data, attempting to partition the flow transport into distinct regions. All four of these approaches share the common trait that they are objective methods, meaning that their results do not depend on the frame of reference used. For each method, we also present a number of example applications ranging from blood flow and chemical reactions to ocean and atmospheric flows. (C) 2015 AIP Publishing LLC.
Propagation of a Solitary Fission Wave
(2012-06) Osborne, A. G.; Recktenwald, G. D.; Deinert, M. R.; Osborne, A. G.; Recktenwald, G. D.; Deinert, M. R.; Osborne, A. G.; Recktenwald, G. D.; Deinert, M. R.
Reaction-diffusion phenomena are encountered in an astonishing array of natural systems. Under the right conditions, self stabilizing reaction waves can arise that will propagate at constant velocity. Numerical studies have shown that fission waves of this type are also possible and that they exhibit soliton like properties. Here, we derive the conditions required for a solitary fission wave to propagate at constant velocity. The results place strict conditions on the shapes of the flux, diffusive, and reactive profiles that would be required for such a phenomenon to persist, and this condition would apply to other reaction diffusion phenomena as well. Numerical simulations are used to confirm the results and show that solitary fission waves fall into a bistable class of reaction diffusion phenomena. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4729927]
Quasi-Two-Dimensional Dynamics of Plasmas and Fluids
(1994-06) Horton, Wendell; Hasegawa, Akira; Horton, Wendell
In the lowest order of approximation quasi-twa-dimensional dynamics of planetary atmospheres and of plasmas in a magnetic field can be described by a common convective vortex equation, the Charney and Hasegawa-Mirna (CHM) equation. In contrast to the two-dimensional Navier-Stokes equation, the CHM equation admits "shielded vortex solutions" in a homogeneous limit and linear waves ("Rossby waves" in the planetary atmosphere and "drift waves" in plasmas) in the presence of inhomogeneity. Because of these properties, the nonlinear dynamics described by the CHM equation provide rich solutions which involve turbulent, coherent and wave behaviors. Bringing in non ideal effects such as resistivity makes the plasma equation significantly different from the atmospheric equation with such new effects as instability of the drift wave driven by the resistivity and density gradient. The model equation deviates from the CHM equation and becomes coupled with Maxwell equations. This article reviews the linear and nonlinear dynamics of the quasi-two-dimensional aspect of plasmas and planetary atmosphere starting from the introduction of the ideal model equation (CHM equation) and extending into the most recent progress in plasma turbulence.
Refining Finite-Time Lyapunov Exponent Ridges and the Challenges of Classifying Them
(2015-08) Allshouse, Michael R.; Peacock, Thomas; Allshouse, Michael R.
While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of a FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by a FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model. (C) 2015 AIP Publishing LLC.
Rhombic Patterns: Broken Hexagonal Symmetry
(1993-10) Ouyang, Qi; Gunaratne, Gemunu H.; Swinney, Harry L.; Ouyang, Qi; Swinney, Harry L.
Landau-Ginzburg equations derived to conserve two-dimensional spatial symmetries lead to the prediction that rhombic arrays with characteristic angles slightly differ from 60 degrees should form in many systems. Beyond the bifurcation from the uniform state to patterns, rhombic patterns are linearly stable for a band of angles near the 60 degrees angle of regular hexagons. Experiments conducted on a reaction-diffusion system involving a chlorite-iodide-malonic acid reaction yield rhombic patterns in good accord with the theory.
Rotational Invariance, The Spin-Statistics Connection And The TCP Theorem
(2000-11) Sudarshan, E. C. G.; Sudarshan, E. C. G.
Quantum Field Theory formulated in terms of hermitian fields automatically leads to a spin-statistics connection when invariance under rotations is required. In three (or more) dimensions of space this implies Bose statistics for integer spin fields and Fermi statistics for half-integer spin fields. One should recall that spin-1/2 fields in three dimensions have two nonhermitian or four hermitian components. This automatic doubling of the number of components enables one to define a pseudoscalar matrix, and this in turn allows one to prove the TCP theorem for rotationally invariant field theories. In two space dimensions one obtains anyon statistics independent of the >spin>. For the quantum mechanics of identical particles we obtain only the possibility of either statistics for either spin as long as the spatial dimension is three (or highs). For two space dimensions we get anyon statistics. This difference is due to the contractibility of closed loops in three or more dimensions. The relation to the arguments of Broyles, of Bacry and of Berry and Robbins is discussed.
Scaling of Geochemical Reaction Rates via Advective Solute Transport
(2015-07) Hunt, A. G.; Ghanbarian, B.; Skinner, T. E.; Ewing, R. P.; Ghanbarian, B.
Transport in porous media is quite complex, and still yields occasional surprises. In geological porous media, the rate at which chemical reactions (e.g., weathering and dissolution) occur is found to diminish by orders of magnitude with increasing time or distance. The temporal rates of laboratory experiments and field observations differ, and extrapolating from laboratory experiments (in months) to field rates (in millions of years) can lead to order-of-magnitude errors. The reactions are transport-limited, but characterizing them using standard solute transport expressions can yield results in agreement with experiment only if spurious assumptions and parameters are introduced. We previously developed a theory of non-reactive solute transport based on applying critical path analysis to the cluster statistics of percolation. The fractal structure of the clusters can be used to generate solute distributions in both time and space. Solute velocities calculated from the temporal evolution of that distribution have the same time dependence as reaction-rate scaling in a wide range of field studies and laboratory experiments, covering some 10 decades in time. The present theory thus both explains a wide range of experiments, and also predicts changes in the scaling behavior in individual systems with increasing time and/or length scales. No other theory captures these variations in scaling by invoking a single physical mechanism. Because the successfully predicted chemical reactions include known results for silicate weathering rates, our theory provides a framework for understanding changes in the global carbon cycle, including its effects on extinctions, climate change, soil production, and denudation rates. It further provides a basis for understanding the fundamental time scales of hydrology and shallow geochemistry, as well as the basis of industrial agriculture. (C) 2015 AIP Publishing LLC.
Transition to Chemical Turbulence
(1991-12) Ouyang, Q.; Swinney, Harry L.; Ouyang, Q.; Swinney, Harry L.
Experiments have been conducted on Turing-type chemical spatial patterns and their variants in a quasi-two-dimensional open spatial reactor with a chlorite-iodide-malonic acid reaction. A variety of stationary spatial structures-hexagons, stripes, and mixed states-were observed, and transitions to these states were studied. For conditions beyond those corresponding to the emergence of patterns, a transition was observed from stationary spatial patterns to chemical turbulence, which is marked by a continuous motion of the pattern within a domain and of the grain boundaries between domains. The transition to chemical turbulence was analyzed by measuring the correlation length, the average pattern speed, and the total length of the domain boundaries. The emergence of chemical turbulence is accompanied by a large increase in the defects in the pattern, which suggests that this is an example of defect-mediated turbulence.
Transport Properties in Nontwist Area-Preserving Maps
(2009-12) Szezech, J. D.; Caldas, I. L.; Lopes, S. R.; Viana, R. L.; Morrison, P. J.; Morrison, P. J.
Nontwist systems, common in the dynamical descriptions of fluids and plasmas, possess a shearless curve with a concomitant transport barrier that eliminates or reduces chaotic transport, even after its breakdown. In order to investigate the transport properties of nontwist systems, we analyze the barrier escape time and barrier transmissivity for the standard nontwist map, a paradigm of such systems. We interpret the sensitive dependence of these quantities upon map parameters by investigating chaotic orbit stickiness and the associated role played by the dominant crossing of stable and unstable manifolds. (C) 2009 American Institute of Physics. [doi: 10.1063/1.3247349]
Using Scattering Theory to Compute Invariant Manifolds and Numerical Results for the Laser-Driven Henon-Heiles System
(2012-12) Blazevski, Daniel; Franklin, Jennifer; Blazevski, Daniel; Franklin, Jennifer
Scattering theory is a convenient way to describe systems that are subject to time-dependent perturbations which are localized in time. Using scattering theory, one can compute time-dependent invariant objects for the perturbed system knowing the invariant objects of the unperturbed system. In this paper, we use scattering theory to give numerical computations of invariant manifolds appearing in laser-driven reactions. In this setting, invariant manifolds separate regions of phase space that lead to different outcomes of the reaction and can be used to compute reaction rates. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4767656]