Searcharxiv⌕ Search

SEARCH · Searcharxiv

Results for “patt-sol”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 55 records · Page 3Linked to original sources
arXiv1993

Solitons and Long Josephson Junctions

Magnetic flux quanta, of value h/2e, in long Josephson junctions behave as (quasi) solitons. Fluxon dynamical states are well described by a perturbed sine-Gordon equation model, with boundary conditions determined by the junction geometry and by externally applied magnetic fields, and they give rise to readily measurable physical phenomena, such as step structure in current-voltage characteristics and microwave radiation emission. Devices based on fluxon propagation offer potentially interesting applications as oscillators and amplifiers - as well as digital applica- tions, described elsewhere in this volume - in high performance integrated superconductive circuits.

patt-sol↗
arXiv1993

Threshold Growth Dynamics

We study the asymptotic shape of the occupied region for monotone deterministic dynamics in d-dimensional Euclidean space parametrized by a threshold theta, and a Borel set N with positive and finite Lebesgue measure. If A_n denotes the occupied set of the dynamics at integer time n, then A_n+1 is obtained by adjoining any point x for which the volume of overlap between x+N and A_n exceeds theta. Except in some degenerate cases, we prove that A_n converges to a unique limiting "shape" L starting from any bounded initial region that is suitably large. Moreover, L is computed as the polar transform for 1/w, where w is an explicit width function that depends on N and theta. It is further shown that L describes the limiting shape of wave fronts for certain cellular automaton growth rules related to lattice models of excitable media, as the threshold and range of interaction increase suitably. In the case of 2-d box neighborhoods, these limiting shapes are calculated and the dependence of their anisotropy on theta is examined. Other specific two- and three- dimensional examples are also discussed in some detail.

patt-sol↗
arXiv1993

Metastability in the Greenberg-Hastings Model

The Greenberg-Hastings Model (GHM) is a family of multitype cellular automata that emulate excitable media, exhibiting the nucleation and spiral formation characteristic of such complex systems. In this paper we study the asymptotic frequency of nucleation in 2-d GHM dynamics as the number k of types, or colors, becomes large. Starting from uniform product measure over colors, and assuming that the excitation threshold is not too large, the box size needed for formation of a spiral core is shown to grow exponentially in k. By exploiting connections with percolation theory we find that a box of size exp{(.23+-.06)k} is required in the nearest neighbor, threshold 1 case. In contrast, GHM rules obey power law nucleation scaling when started from a suitable non-uniform product measure over the k colors; this effect is driven by critical percolation. Finally, we present some analogous results for a Random GHM, an interacting Markovian system closely related to the epidemic with regrowth of Durrett and Neuhauser.

patt-sol↗
arXiv1993

Threshold-Range Scaling of Excitable Cellular Automata

Each cell of a two-dimensional lattice is painted one of k colors, arranged in a "color wheel." The colors advance (0 to k-1 mod k) either automatically or by contact with at least a threshold number of successor colors in a prescribed local neighborhood. Discrete-time parallel systems of this sort in which color 0 updates by contact and the rest update automatically are called Greenberg-Hastings (GH) rules. A system in which all colors update by contact is called a cyclic cellular automaton (CCA). Started from appropriate initial conditions these models generate periodic traveling waves. Started from random configurations the same rules exhibit complex self-organization, typically characterized by nucleation of locally periodic "ram's horns" or spirals. Corresponding random processes give rise to a variety of "forest fire" equilibria that display large-scale stochastic wave fronts. This article describes a framework, theoretically based, but relying on extensive interactive computer graphics experimentation, for investigation of the complex dynamics shared by excitable media in a broad spectrum of scientific contexts. By focusing on simple mathematical prototypes we obtain a better understanding of the basic organizational principles underlying spatially-distributed oscillating systems.

patt-sol↗
arXiv1993

Solitons and 1/f Noise in Molecular Chains

Davydov's model of solitons in alpha-helix protein chains is shown to display features of self-organized criticality (SOC), i.e., power law behaviour of correlations in space and 1/f-noise, as a consequence of considering random peptide group displacements from their (periodic) equilibrium positions along a chain. This may shed light on a basic mechanism leading to obtain flicker noise in alpha-helix protein chains and to predict a SOC regime in biomolecular structures from first principles. We believe our treatment of 1/f noise to be of some relevance to recent findings due to Voss on DNA [Phys. Rev. Lett. 68, 3805 (1992)].

patt-sol↗
arXiv1993

Pattern formation during Rayleigh-Bénard convection in non-Boussinesq fluids

Motivated by recent experimental studies of Bodenschatz et al. [E. Bodenschatz, J.R. de Bruyn, G. Ahlers and D.S. Cannell, Phys. Rev. Lett. {\bf 67}, 3078 (1991) ], we present a numerical study of a generalized two dimensional Swift-Hohenberg equation to model pattern formation in Rayleigh-Bénard convection in a non-Boussinesq fluid. It is shown that many of the features observed in these experiments can be reproduced by this generalized model that explicitly includes non-Boussinesq and mean flow effects. The spontaneous formation of hexagons, rolls, and a rotating spiral pattern is studied, as well as the transitions and competition among them. Mean flow, non-Boussinesq effects, the geometric shape of the lateral wall, and sidewall forcing are all shown to be crucial in the formation of the rotating spirals. We also study nucleation and growth of hexagonal patterns and find that the front velocity in this two dimensional model is consistent with the prediction of marginal stability theory for one dimensional fronts.

patt-sol↗
arXiv1993

Phase Transitions In Two-Dimensional Traffic Flow Models

We introduce two simple two-dimensional lattice models to study traffic flow in cities. We have found that a few basic elements give rise to the characteristic phase diagram of a first-order phase transition from a freely moving phase to a jammed state, with a critical point. The jammed phase presents new transitions corresponding to structural transformations of the jam. We discuss their relevance in the infinite size limit.

patt-sol↗
arXiv1993

Nonequilibrium Phenomena in Liquid Crystals

This paper summarizes a talk presented at the April NATO ASI on Spatiotemporal Chaos in Complex Fluids, in Santa Fe, NM. The paper gives reasons that make complex fluids good material systems for conducting experiments on pattern formation and other nonequilibrium phenomena. Much of the discussion focuses on the different phenomena observed in solidification and how the increasing complexity of fluid systems decreases the velocity scale for achieving "rapid" solidification. Five systems are compared to illustrate this point: simple fluids, simple alloys, thermotropic liquid crystals, lyotropic liquid crystals, and diblock copolymers. Finally, an example is given of the kinds of transitions that may be observed in rapid solidification.

patt-sol↗
arXiv1993

Discrete Breathers and Energy Localization in Nonlinear Lattices

We discuss the process by which energy, initially evenly distributed in a nonlinear lattice, can localize itself into large amplitude excitations. We show that, the standard modulational instability mechanism, which can initiate the process by the formation of small amplitude breathers, is completed efficiently, in the presence of discreteness, by energy exchange mechanisms between the nonlinear excitations which favor systematically the growth of the larger excitations. The process is however self regulated because the large amplitude excitations are finally trapped by the Peierls-Nabarro potential.

patt-sol↗
arXiv1993

A quantitative measurement of spatial order in ventricular fibrillation

As an objective measurement of spatial order in ventricular fibrillation (VF), spatial correlation functions and their characteristic lengths were estimated from epicardial electrograms of pigs in VF. The correlation length of the VF in pigs was found to be approximately 4-10 mm, varying as fibrillation progressed. The degree of correlation decreased in the first 4 seconds after fibrillation then increased over the next minute. The correlation length is much smaller than the scale of the heart, suggesting that many independent regions of activity exist on the epicardium at any one time. On the other hand, the correlation length is 4 to 10 times the interelectrode spacing, indicating that some coherence is present. These results imply that the heart behaves during VF as a high-dimensional, but not random, system involving many spatial degrees of freedom, which may explain the lack of convergence of fractal dimension estimates reported in the literature. Changes in the correlation length also suggest that VF reorganizes slightly in the first minute after an initial breakdown in structure.

patt-sol↗
arXiv1993

Nonradial Solutions of a Semilinear Elliptic Equation in Two Dimensions

: We establish existence of an infinite family of exponentially-decaying non-radial $C^2$ solutions to the equation $Δu + f(u) = 0$ on $R^2$ for a large class of nonlinearities $f$. These solutions have the form $u(r,θ)=e^{i mθ}w(r)$, where $r$ and $θ$ are polar coordinates, $m$ is an integer, and $w:[0,\infty ) \to R$ is exponentially decreasing far from the origin. We prove there is a solution with each prescribed number of nodes.

patt-sol↗
arXiv1993

D4-symmetric Maps with Hidden Euclidean Symmetry

Bifurcation problems in which periodic boundary conditions (PBC) or Neumann boundary conditions (NBC) are imposed often involve partial differential equations that have Euclidean symmetry. In this case posing the bifurcation problem with either PBC or NBC on a finite domain can lead to a symmetric bifurcation problem for which the manifest symmetries of the domain do not completely characterize the constraints due to symmetry on the bifurcation equations. Additional constraints due to the Euclidean symmetry of the equations can have a crucial influence on the structure of the bifurcation equations. An example is the bifurcation of standing waves on the surface of fluid layer. The Euclidean symmetry of an infinite fluid layer constrains the bifurcation of surface waves in a finite container with square cross section because the waves satisfying PBC or NBC can be shown to lie in certain finite-dimensional fixed point subspaces of the infinite-dimensional problem. These constraints are studied by analyzing the finite-dimensional vector fields obtained on these subspaces by restricting the bifurcation equations for the infinite layer. Particular emphasis is given to determining which bifurcations might reveal observable effects of the rotational symmetry of the infinite layer. It turns out that a necessary condition for this possibility to arise is that the subspace for PBC must carry a reducible representation of the normalizer subgroup acting on the subspace. This condition can be met in different ways in both codimension-one and codimension-two bifurcations.

patt-sol↗
arXiv1993

Multidimensional Localized Solitons

Recently it has been discovered that some nonlinear evolution equations in 2+1 dimensions, which are integrable by the use of the Spectral Transform, admit localized (in the space) soliton solutions. This article briefly reviews some of the main results obtained in the last five years thanks to the renewed interest in soliton theory due to this discovery. The theoretical tools needed to understand the unexpected richness of behaviour of multidimensional localized solitons during their mutual scattering are furnished. Analogies and especially discrepancies with the unidimensional case are stressed.

patt-sol↗
arXiv1993

Properties Of Solutions Of The Kpi Equation

The Kadomtsev--Petviashvili I (KPI) is considered as a useful laboratory for experimenting new theoretical tools able to handle the specific features of integrable models in $2+1$ dimensions. The linearized version of the KPI equation is first considered by solving the initial value problem for different classes of initial data. Properties of the solutions in different cases are analyzed in details. The obtained results are used as a guideline for studying the properties of the solution $u(t,x,y)$ of the Kadomtsev--Petviashvili I (KPI) equation with given initial data $u(0,x,y)$ belonging to the Schwartz space. The spectral theory associated to KPI is studied in the space of the Fourier transform of the solutions. The variables $p=\{p_1,p_2\}$ of the Fourier space are shown to be the most convenient spectral variables to use for spectral data. Spectral data are shown to decay rapidly at large $p$ but to be discontinuous at $p=0$. Direct and inverse problems are solved with special attention to the behaviour of all the quantities involved in the neighborhood of $t=0$ and $p=0$. It is shown in particular that the solution $u(t,x,y)$ has a time derivative discontinuous at $t=0$ and that at any $t\not=0$ it does not belong to the Schwartz space no matter how small in norm and rapidly decaying at large distances the initial data are chosen.

patt-sol↗
arXiv1994

Domain Structures and Zig-Zag Patterns Modeled by a Fourth-Order Ginzburg-Landau Equation

Domain walls between spatially periodic patterns with different wave numbers, can arise in pattern-forming systems with a neutral curve that has a double minimum. Within the framework of the phase equation, the interaction of such walls is purely attractive. Thus, they annihilate each other and, if the total phase is not conserved, the final state will be periodic with a single wave number. Here we study the stability of arrays of domain walls (domain structures) in a fourth-order Ginzburg-Landau equation. We find a discrete set of domain structures which differ in the quantized lengths of the domains. They are stable even without phase conservation, as, for instance, in the presence of a subcritical ramp in the control parameter. We attribute this to the spatially oscillatory behavior of the wave number which should lead to an oscillatory interaction between the domain walls. These results are expected to shed also some light on the stability of two-dimensional zig-zag structures.

patt-sol↗
arXiv1994

Asymptotic expansion for layer solutions of a singularly perturbed reaction-diffusion system

For a singularly perturbed system of reaction--diffusion equations, assuming that the 0th order solutions in regular and singular regions are all stable, we construct matched asymptotic expansions for formal solutions to any desired order in $ε$. The formal solution shows that there is an invariant manifold of wave-front-like solutions that attracts other nearby solutions. With an additional assumption on the sign of the wave speed, the wave-front-like solutions converge slowly to stable stationary solutions on that manifold.

patt-sol↗
arXiv1994

Kinks in the Presence of Rapidly Varying Perturbations

Dynamics of sine-Gordon kinks in the presence of rapidly varying periodic perturbations of different physical origins is described analytically and numerically. The analytical approach is based on asymptotic expansions, and it allows to derive, in a rigorous way, an effective nonlinear equation for the slowly varying field component in any order of the asymptotic procedure as expansions in the small parameter $ω^{-1}$, $ω$ being the frequency of the rapidly varying ac driving force. Three physically important examples of such a dynamics, {\em i.e.}, kinks driven by a direct or parametric ac force, and kinks on rotating and oscillating background, are analysed in detail. It is shown that in the main order of the asymptotic procedure the effective equation for the slowly varying field component is {\em a renormalized sine-Gordon equation} in the case of the direct driving force or rotating (but phase-locked to an external ac force) background, and it is {\em the double sine-Gordon equation} for the parametric driving force. The properties of the kinks described by the renormalized nonlinear equations are analysed, and it is demonstrated analytically and numerically which kinds of physical phenomena may be expected in dealing with the renormalized, rather than the unrenormalized, nonlinear dynamics. In particular, we predict several qualitatively new effects which include, {\em e.g.}, the perturbation-induced internal oscillations of the $2π-$kink in a parametrically driven sine-Gordon model, and generation of kink motion by a pure ac driving force on a rotating background.

patt-sol↗