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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 ..."
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Cited by 35 (5 self)
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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...
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 ..."
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Cited by 23 (8 self)
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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 differentia ..."
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Cited by 22 (2 self)
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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 ..."
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Cited by 20 (9 self)
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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 ..."
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Cited by 18 (6 self)
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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 Family of nScroll Attractors from a Generalized Chua's Circuit
"... Previously, ndouble scroll attractors have been introduced by Suykens & Vandewalle. A generalized Chua's circuit was considered with additional breakpoints in the nonlinear characteristic. A piecewiselinear implementation and experimental confirmation has been given by Arena et al. In thi ..."
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Cited by 17 (12 self)
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Previously, ndouble scroll attractors have been introduced by Suykens & Vandewalle. A generalized Chua's circuit was considered with additional breakpoints in the nonlinear characteristic. A piecewiselinear implementation and experimental confirmation has been given by Arena et al. In this paper we present a more complete family of nscroll attractors generated from the latter circuit. The new family contains both an even and odd number of scrolls, while the previous work considered only an even number. A Lur'e representation of the generalized Chua's circuit is also given. 1
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 ..."
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Cited by 15 (1 self)
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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.
Dynamics Of Lattice Differential Equations
"... . In this paper recent work on the dynamics of lattice differential equations is surveyed. In particular, results on propagation failure and lattice induced anisotropy for traveling wave or plane wave solutions in higher space dimensions spatially discrete bistable reactiondiffusion systems are con ..."
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Cited by 14 (6 self)
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. In this paper recent work on the dynamics of lattice differential equations is surveyed. In particular, results on propagation failure and lattice induced anisotropy for traveling wave or plane wave solutions in higher space dimensions spatially discrete bistable reactiondiffusion systems are considered. In addition, analysis of and spatial chaos in the equilibrium states of spatially discrete reactiondiffusion systems are discussed. Key words. lattice differential equations, traveling wave solutions, propogation failure, lattice anisotropy, equilibrium solutions, stability, spatial entropy Abbreviated title. Lattice Differential Equations 1 School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332 USA. The work of this author was supported in part by ARO Contract DAAH0493G0199 and by NSF Grant DMS9005420. 2 Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912 USA. The work of this author was supported in part by NSF Grant D...
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 ..."
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Cited by 14 (6 self)
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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.
Robust nonlinear H∞ synchronization of chaotic Lur’e systems
 IEEE TRANS. CIRCUITS SYST. I
, 1997
"... In this paper, we propose a method of robust nonlinear H∞ master–slave synchronization for chaotic Lur’e systems with applications to secure communication. The scheme makes use of vector field modulation and either full static state or linear dynamic output error feedback control. The master–slave ..."
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Cited by 12 (5 self)
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In this paper, we propose a method of robust nonlinear H∞ master–slave synchronization for chaotic Lur’e systems with applications to secure communication. The scheme makes use of vector field modulation and either full static state or linear dynamic output error feedback control. The master–slave systems are assumed to be nonidentical and channel noise is taken into account. Binary valued continuous time message signals are recovered by minimizing the vPgain from the exogenous input to the tracking error for the standard plant representation of the scheme. The exogenous input takes into account the message signal, channel noise and parameter mismatch. Matrix inequality conditions for dissipativity with finite vPgain of the standard plant form are derived based on a quadratic storage function. The controllers are designed by solving a nonlinear optimization problem which takes into account both channel noise and parameter mismatch. The method is illustrated on Chua’s circuit.