Results 1  10
of
150
Traveling wave solutions for systems of ODE's on a twodimensional spatial lattice
 SIAM J. APPL. MATH
, 1999
"... We consider infinite systems of ODE's on the twodimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, we obtain traveling wave solutions in each direction e i`, and we explore the rela ..."
Abstract

Cited by 36 (11 self)
 Add to MetaCart
We consider infinite systems of ODE's on the twodimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, we obtain traveling wave solutions in each direction e i`, and we explore the relation between the wave speed c, the angle `, and the detuning parameter a of the nonlinearity. Of particular interest is the phenomenon of "propagation failure, " and we study how the critical value a = a
The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems
 J. Dynam. Differential Equations
, 1997
"... We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c 6= 0, are also shown. More generally, the global structure of the set of all traveling wave solution ..."
Abstract

Cited by 34 (5 self)
 Add to MetaCart
We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c 6= 0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c 6= 0. Convergence results for solutions are obtained at the singular perturbation limit c ! 0. 1 Introduction We are interested in lattice differential equations, namely infinite systems of ordinary differential equations indexed by points on a spatial lattice, such as the Ddimensional integer lattice Z D . Our focus in this paper is the global structure of the set of traveling wave solutions for such systems. This entails results on existence and uniqueness, and on continuous (or smooth) dependence of traveling waves and their speeds on parameters, as well as some delicate convergence results in the singular perturbation case c ! 0 of the wav...
AER Image Filtering Architecture for VisionProcessing Systems
 IEEE Trans. Circuits Syst. I, Fundam. Theory Appl
, 1999
"... A VLSI architecture is proposed for the realization of realtime twodimensional (2D) image filtering in an addressevent representation (AER) vision system. The architecture is capable of implementing any convolutional kernel F (x; y) as long as it is decomposable into xaxis and yaxis components ..."
Abstract

Cited by 23 (7 self)
 Add to MetaCart
A VLSI architecture is proposed for the realization of realtime twodimensional (2D) image filtering in an addressevent representation (AER) vision system. The architecture is capable of implementing any convolutional kernel F (x; y) as long as it is decomposable into xaxis and yaxis components, i.e., F (x; y)=H(x)V (y), for some rotated coordinate system fx; yg and if this product can be approximated safely by a signed minimum operation. The proposed architecture is intended to be used in a complete vision system, known as the boundary contour system and feature contour system (BCSFCS) vision model, proposed by Grossberg and collaborators. The present paper proposes the architecture, provides a circuit implementation using MOS transistors operated in weak inversion, and shows behavioral simulation results at the system level operation and some electrical simulations. Index TermsAnalog integrated circuits, communication systems, convolution circuits, Gabor filters, image anal...
1995] “Autonomous cellular neural networks: A united paradigm for pattern formation and active wave propagation
 IEEE Trans. Circuits Syst.I
"... AbstractThis tutorial paper proposes a subclass of cellular neural networks (CNN) having no inputs (i.e., autonomous) as a universal active substrate or medium for modeling and generating many pattern formation and nonlinear wave phenomena from numerous disciplines, including biology, chemistry, ec ..."
Abstract

Cited by 23 (8 self)
 Add to MetaCart
AbstractThis tutorial paper proposes a subclass of cellular neural networks (CNN) having no inputs (i.e., autonomous) as a universal active substrate or medium for modeling and generating many pattern formation and nonlinear wave phenomena from numerous disciplines, including biology, chemistry, ecology, engineering, physics, etc. Each CNN is defined mathematically by its cell dynamics (e.g., state equations) and synaptic law, which specifies each cell’s interaction with its neighbors. We focus in this paper on reaction4iffusion CNNs having a linear synaptic law that approximates a spatial Laplacian operator. Such a synaptic law can be realized by one or more layers of linear resistor couplings. An autonomous CNN made of thirdorder universal cells and coupled to each other by only one layer of linear resistors provides a unified active medium for generating trigger (autowave) waves,
Traveling Waves in Lattice Dynamical Systems
"... In this paper, we study the existence and stability of traveling waves in lattice dynamical systems, in particular, in lattice ordinary differential equations (lattice ODE's) and in coupled map lattices (CML's). Instead of employing the moving coordinate approach as for partial differential equation ..."
Abstract

Cited by 22 (2 self)
 Add to MetaCart
In this paper, we study the existence and stability of traveling waves in lattice dynamical systems, in particular, in lattice ordinary differential equations (lattice ODE's) and in coupled map lattices (CML's). Instead of employing the moving coordinate approach as for partial differential equations, we construct a local coordinate system around a traveling wave solution of a lattice ODE, analogous to the local coordinate system around a periodic solution of an ODE. In this coordinate system the lattice ODE becomes a nonautonomous periodic differential equation, and the traveling wave corresponds to a periodic solution of this equation. We prove the asymptotic stability with asymptotic phase shift of the traveling wave solution under appropriate spectral conditions. We also show the existence of traveling waves in CML's which arise as timediscretizations of lattice ODE's. Finally, we show that these results apply to the discrete Nagumo equation. 1 Introduction This paper is concer...
Pattern Formation and Spatial Chaos in Spatially Discrete Evolution Equations
, 1995
"... We consider an array of scalar nonlinear dynamical systems u = \Gammaf (u), arranged on the sites of a spatial lattice, for example on the integer lattice ZZ 2 in the plane IR 2 . We impose a coupling between nearest neighbors, and also between nextnearest neighbors, in the form of discrete La ..."
Abstract

Cited by 20 (9 self)
 Add to MetaCart
We consider an array of scalar nonlinear dynamical systems u = \Gammaf (u), arranged on the sites of a spatial lattice, for example on the integer lattice ZZ 2 in the plane IR 2 . We impose a coupling between nearest neighbors, and also between nextnearest neighbors, in the form of discrete Laplacians with + and \Thetashaped stencils. These couplings can be of any strength, and of either sign (positive or negative), and the resulting infinite systems of ODE's need not be near a PDE continuum limit. We study stable equilibria for such systems, from the point of view of pattern formation and spatial chaos, where these terms mean that the spatial entropy of the set of stable equilibria is zero, respectively, positive. In particular, for an idealized class of nonlinearities f corresponding to a "double obstacle" at u = \Sigma1 with f(u) = flu in between, it is natural to consider "mosaic solutions," namely equilibria which assume only the values u i;j 2 f\Gamma1; 0; 1g at each (i; ...
Pattern Formation and Spatial Chaos in Lattice Dynamical Systems: II
"... We survey a class of continuoustime lattice dynamical systems, with an idealized nonlinear. We introduce a class of equilibria called mosaic solutions, which are composed of the elements 1, \Gamma1, and 0, placed at each lattice point. A stability criterion for such solutions is given. The spatial ..."
Abstract

Cited by 18 (6 self)
 Add to MetaCart
We survey a class of continuoustime lattice dynamical systems, with an idealized nonlinear. We introduce a class of equilibria called mosaic solutions, which are composed of the elements 1, \Gamma1, and 0, placed at each lattice point. A stability criterion for such solutions is given. The spatial entropy h of the set of all such stable solutions is defined, and we study how this quantity varies with parameters. Systems are qualitatively distinguished according to whether h = 0 (termed pattern formatio), or h ? 0 (termed spatial chaos). Numerical techniques for calculating h are described. 1. Mosaic Solutions As described in the companion paper [4], we study the phenomenon of pattern formation and spatial chaos in lattice dynamical systems. In order for us to see these phenomena globally, we consider a special class of equilibrium solutions, called mosaic solutions, introduced in [5], and studied there and in [6]. We work here with the system (1:1) u i;j = \Gammafi + \Delta + ...
A Survey on Cellular Automata
, 2003
"... A cellular automaton is a decentralized computing model providing an excellent platform for performing complex computation with the help of only local information. Researchers, scientists and practitioners from different fields have exploited the CA paradigm of local information, decentralized contr ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
A cellular automaton is a decentralized computing model providing an excellent platform for performing complex computation with the help of only local information. Researchers, scientists and practitioners from different fields have exploited the CA paradigm of local information, decentralized control and universal computation for modeling different applications. This article provides a survey of available literature of some of the methodologies employed by researchers to utilize cellular automata for modeling purposes. The survey introduces the different types of cellular automata being used for modeling and the analytical methods used to predict its global behavior from its local configurations. It further gives a detailed sketch of the efforts undertaken to configure the local settings of CA from a given global situation; the problem which has been traditionally termed as the inverse problem. Finally, it presents the different fields in which CA have been applied. The extensive bibliography provided with the article will be of help to the new entrant as well as researchers working in this field.
Neural Network Adaptations to Hardware Implementations
, 1997
"... In order to take advantage of the massive parallelism offered by artificial neural networks, hardware implementations are essential. However, most standard neural network models are not very suitable for implementation in hardware and adaptations are needed. In this section an overview is given of t ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
In order to take advantage of the massive parallelism offered by artificial neural networks, hardware implementations are essential. However, most standard neural network models are not very suitable for implementation in hardware and adaptations are needed. In this section an overview is given of the various issues that are encountered when mapping an ideal neural network model onto a compact and reliable neural network hardware implementation, like quantization, handling nonuniformities and nonideal responses, and restraining computational complexity. Furthermore, a broad range of hardwarefriendly learning rules is presented, which allow for simpler and more reliable hardware implementations. The relevance of these neural network adaptations to hardware is illustrated by their application in existing hardware implementations.
Analysis and Computation of Traveling Wave Solutions of Bistable DifferentialDifference Equations
 Nonlinearity
, 1999
"... We consider traveling wave solutions to a class of differentialdifference equations. Our interest is in understanding propagation failure, directional dependence due to the discrete Laplacian, and the relationship between traveling wave solutions of the spatially continuous and spatially discrete ..."
Abstract

Cited by 14 (6 self)
 Add to MetaCart
We consider traveling wave solutions to a class of differentialdifference equations. Our interest is in understanding propagation failure, directional dependence due to the discrete Laplacian, and the relationship between traveling wave solutions of the spatially continuous and spatially discrete limits of this equation. The differentialdifference equations that we study include damped and undamped nonlinear wave and reactiondiffusion equations as well as their spatially discrete counterparts. Both analytical and numerical results are given.