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At least 19 recordsLinked to original sources

arXiv1995

How birds fly together: Long-range order in a two-dimensional dynamical XY model

We propose a non-equilibrium continuum dynamical model for the collective motion of large groups of biological organisms (e.g., flocks of birds, slime molds, etc.) Our model becomes highly non-trivial, and different from the equilibrium model, for $d<d_c=4$; nonetheless, we are able to determine its scaling exponents {\it exactly} in $d=2$, and show that, unlike equilibrium systems, our model exhibits a broken continuous symmetry even in $d=2$. Our model describes a large universality class of microscopic rules, including those recently simulated by Viscek et. al.

adap-org↗

arXiv1998

The Evolutionary Design of Collective Computation in Cellular Automata

We investigate the ability of a genetic algorithm to design cellular automata that perform computations. The computational strategies of the resulting cellular automata can be understood using a framework in which ``particles'' embedded in space-time configurations carry information and interactions between particles effect information processing. This structural analysis can also be used to explain the evolutionary process by which the strategies were designed by the genetic algorithm. More generally, our goals are to understand how machine-learning processes can design complex decentralized systems with sophisticated collective computational abilities and to develop rigorous frameworks for understanding how the resulting dynamical systems perform computation.

adap-org↗

arXiv1993

Using Horseshoes to Create Coherent Structures

In this letter, we show that coherent structures are related to folds of horseshoes which are present in chaotic systems. We develop techniques that allow us to construct coherent structures by manipulating folds in three prototypical problems: a 1-D chaotic map, a 2-D chaotic map, and a chaotically advected fluid. The ability to construct such structures is of practical importance for the control of chaotic or turbulent extended systems such as fluids, plasmas, and coupled oscillator arrays.

chao-dyn↗

arXiv1993

Dependence of extensive chaos on the spatial correlation length (substantial revision)

We consider spatiotemporal chaotic systems for which spatial correlation functions decay substantially over a length scale xi (the spatial correlation length) that is small compared to the system size L. Numerical simulations suggest that such systems generally will be extensive, with the fractal dimension D growing in proportion to the system volume for sufficiently large systems (L >> xi). Intuitively, extensive chaos arises because of spatial disorder. Subsystems that are sufficiently separated in space should be uncorrelated and so contribute to the fractal dimension in proportion to their number. We report here the first numerical calculation that examines quantitatively how one important characterization of extensive chaos---the Lyapunov dimension density---depends on spatial disorder, as measured by the spatial correlation length xi. Surprisingly, we find that a representative extensively chaotic system does not act dynamically as many weakly interacting regions of size xi.

chao-dyn↗

arXiv1996

Double Phase Slips and Spatio-Temporal Chaos in a Model for Parametrically Excited Standing Waves

We present results of numerical simulations of coupled Ginzburg-Landau equations that describe parametrically excited waves. In one dimension we focus on a new regime in which the Eckhaus sideband instability does not lead to an overall change in the wavelength via the occurrence of a single phase slip but instead leads to ``double phase slips''. They are characterized by the phase slips occurring in sequential pairs, with the second phase slip quickly following and negating the first. The resulting dynamics range from transient excursions from a fixed point resembling those seen in excitable media, to periodic solutions of varying complexity and chaotic solutions. In larger systems we find in addition localized spatio-temporal chaos, where the solution consists of a chaotic region with quiescent regions on each side. We explain the localization using an effective phase diffusion equation which can be viewed as arising from a homogenization of the chaotic state. In two dimensions the double phase slips are replaced by fluctuating bound defect pairs.

chao-dyn↗

arXiv1997

Spatially Periodic Orbits in Coupled Sine Circle Maps

We study spatially periodic orbits for a coupled map lattice of sine circle maps with nearest neighbour coupling and periodic boundary conditions. The stability analysis for an arbitrary spatial period k is carried out in terms of the independent variables of the problem and the stability matrix is reduced to a neat block diagonal form. For a lattice of size kN, we show that the largest eigenvalue for the stability matrix of size $kN \times kN$ is the same as that for the basic spatial period k matrix of size $k \times k$. Thus the analysis for a kN lattice case can be reduced to that for a k lattice case. We illustrate this explicitly for a spatial period two case. Our formalism is general and can be extended to any coupled map lattice. We also obtain the stability regions of solutions which have the same spatial and temporal period numerically. Our analysis shows that such regions form a set of Arnold tongues in the $Ω-ε-K$ space. The tongues corresponding to higher spatial periods are contained within the tongues seen in the temporally periodic spatial period one or synchronised case. We find an interesting new bifurcation wherein the the spatially synchronised and temporal period one solution undergoes a bifurcation to a spatio-temporal period two travelling wave solution. The edges of the stability interval of this solution are analytically obtained.

chao-dyn↗

arXiv1998

Low dimensional travelling interfaces in coupled map lattices

We study the dynamics of the travelling interface arising from a bistable piece-wise linear one-way coupled map lattice. We show how the dynamics of the interfacial sites, separating the two superstable phases of the local map, is finite dimensional and equivalent to a toral map. The velocity of the travelling interface corresponds to the rotation vector of the toral map. As a consequence, a rational velocity of the travelling interface is subject to mode-locking with respect to the system parameters. We analytically compute the Arnold's tongues where particular spatio-temporal periodic orbits exist. The boundaries of the mode-locked regions correspond to border-collision bifurcations of the toral map. By varying the system parameters it is possible to increase the number of interfacial sites corresponding to a border-collision bifurcation of the interfacial attracting cycle. We finally give some generalizations towards smooth coupled map lattices whose interface dynamics is typically infinite dimensional.

chao-dyn↗

arXiv1998

Lyapunov Exponents from Node-Counting Arguments

A conjecture connecting Lyapunov exponents of coupled map lattices and the node theorem is presented. It is based on the analogy between the linear stability analysis of extended chaotic states and the Schrödinger problem for a particle in a disordered potential. As a consequence, we propose an alternative method to compute the Lyapunov spectrum. The implications on the foundation of the recently proposed ``chronotopic approach'' are also discussed.

chao-dyn↗

arXiv1998

A solitary-wave representation of turbulence in the physical-plus-eddy space

A unique form of turbulent-transport equations is derived based on first principles.The role of nonequilibrium statistical mechanics employed to describe the phenomenology is that it enables to single out the unique form consistent with master equation of Liouville, a prerequisite not met with existing equations for turbulence modeling.The equation is variable-separated to yield a Navier-Stokes equation in 6D(physical-plus-eddy) space with homogeneous boundary conditions.Turbulent transports such as Reynolds' stress are calculated using a solution of this equation; a solitary-wave function.Satisfactory agreement is observed with existing experiment for mixing shear layer of incompressible flows although no empirical constants are involved.

chao-dyn↗

arXiv1998

Coherent solitary-wave of mixing layer turbulence in the physical-plus-eddy space

A six-dimensional Navier-Stokes equation derived by one of the authors(ST) is solved for a turbulent mixing layer to demonstrate that it has a solitary wave solution. Turbulence intensities and Reynolds' stress are calculated using this solution, showing satisfactory agreement with experiments although no emperical constants are involved in the theory.

chao-dyn↗

arXiv1993

Instabilities and Patterns (minor technical modifications)

Violation of (semi)-detailed balance conditions in lattice gas automata gives rise to unstable spatial fluctuations that lead to phase separation and pattern formation in spinodal decomposition, unstable propagating modes, driven diffusive systems and unstable uniform flows.

comp-gas↗

arXiv1997

A New Fast Method for Determining Local Properties of Striped Patterns

From the striped coats of zebras to the ripples in windblown sand, the natural world abounds with locally banded patterns. Such patterns have been of great interest throughout history, and, in the last twenty years, scientists in a wide variety of fields have been studying the patterns formed in well-controlled experiments that yield enormous quantities of high-precision data. These experiments involving phenomena as diverse as chemical reactions in shallow layers, thermal convection in horizontal fluid layers, periodically shaken layers of sand, and the growth of slime mold colonies often display patterns that appear qualitatively similar. Methods are needed to characterize in a reasonable amount of time the differences and similarities in patterns that develop in different systems, as well as in patterns formed in one system for different experimental conditions. In this Letter, we introduce a novel, fast method for determining local pattern properties such as wavenumber, orientation, and curvature as a function of position for locally striped patterns.

comp-gas↗

arXiv1993

Fast Low Fidelity Microsimulation of Vehicle Traffic on Supercomputers

A set of very simple rules for driving behavior used to simulate roadway traffic gives realistic results. Because of its simplicity, it is easy to implement the model on supercomputers (vectorizing and parallel), where we have achieved real time limits of more than 4~million~kilometers (or more than 53~million vehicle sec/sec). The model can be used for applications where both high simulation speed and individual vehicle resolution are needed. We use the model for extended statistical analysis to gain insight into traffic phenomena near capacity, and we discuss that this model is a good candidate for network routing applications. (Submitted to Transportation Research Board Meeting, Jan. 1994, Washington D.C.)

cond-mat↗

arXiv1993

An Aggregation Model for Electrochemical Depostition and its Application to Cop Dendritic Growth

We introduce an aggregation model for general electrochemical deposition experiments. Its most relevant feature is that it includes the overall effect of strong applied electric fields and therefore applies to non-equilibrium situations. We compare our model to experiments on CoP dendritic growth with very good agreement: The model accurately reproduces the dependence on the current of the alloy's composition, morphology, and growth time.

cond-mat↗

arXiv1994

Towards a theory of growing surfaces: Mapping two-dimensional Laplacian growth onto Hamiltonian dynamics and statistics

I show that the evolution of a two dimensional surface in a Laplacian field can be described by Hamiltonian dynamics. First the growing region is mapped conformally to the interior of the unit circle, creating in the process a set of mathematical zeros and poles that evolve dynamically as the surface grows. Then the dynamics of these quasi-particles is analysed. A class of arbitrary initial conditions is discussed explicitly, where the surface-tension-free Laplacian growth process is integrable. This formulation holds only as long as the singularities of the map are confined to within the unit circle. But the Hamiltonian structure further allows for surface tension to be introduced as an energetic term that effects repulsion between the quasi-particles and the surface. These results are used to formulate a first-principles statistical theory of pattern formation in stochastic growth, where noise is a key player.

cond-mat↗

arXiv1994

Current-Loop Model for the Intermediate State of Type-I Superconductors

A theory is developed of the intricately fingered patterns of flux domains observed in the intermediate state of thin type-I superconductors. The patterns are shown to arise from the competition between the long-range Biot-Savart interactions of the Meissner currents encircling each region and the superconducting-normal surface energy. The energy of a set of such domains is expressed as a nonlocal functional of the positions of their boundaries, and a simple gradient flow in configuration space yields branched flux domains qualitatively like those seen in experiment. Connections with pattern formation in amphiphilic monolayers and magnetic fluids are emphasized.

cond-mat↗

arXiv1995

Pattern Formation in Laplacian Growth: Theory

A first-principles statistical theory is constructed for the evolution of two dimensional interfaces in Laplacian fields. The aim is to predict the pattern that the growth evolves into, whether it becomes fractal and if so the characteristics of the fractal pattern. Using a time dependent map the growing region is conformally mapped onto the unit disk and the problem is converted to the dynamics of a many-body system. The evolution is argued to be Hamiltonian, and the Hamiltonian is shown to be the conjugate function of the real potential field. Without surface effects the problem is ill-posed, but the Hamiltonian structure of the dynamics allows introduction of surface effects as a repulsive potential between the particles and the interface. This further leads to a field representation of the problem, where the field's vacuum harbours the zeros and the poles of the conformal map as particles and antiparticles. These can be excited from the vacuum either by fluctuations or by surface effects. Creation and annihilation of particles is shown to be consistent with the formalism and lead to tip-splitting and side-branching. The Hamiltonian further allows to make use of statistical mechanical tools to analyse the statistics of the many-body system. I outline the way to convert the distribution of the particles into the morphology of the interface. In particular, I relate the particles statistics to the distributions of the curvature and the growth probability along the physical interface and to the fractal dimension. If the pattern turns fractal the latter distribution gives rise to a multifractal spectrum, which can be explicitly calculated for a given particles distribution. A `dilute boundary layer approximation' is discussed, which allows explicit calculations and shows emergence of an algebraically long tail

cond-mat↗

arXiv1997

Universality in Dynamic Coarsening of a Fractal Cluster

Dynamics of coarsening of a statistically homogeneous fractal cluster, created by a morphological instability of diffusion-controlled growth, is investigated theoretically. An exact mathematical setting of the problem is presented that obeys a global conservation law. A statistical mean field theory is developed that accounts for shadowing during the growth instability and assumes that the total mass and fractal dimension of the cluster remain constant. The coarsening dynamics are shown to be self-similar, and the dynamic scaling exponents are calculated for any Euclidean dimension.

cond-mat.stat-mech↗