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Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on

Issue 10 • Date Oct 1995

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Displaying Results 1 - 25 of 29
  • Pattern formation and spatial chaos in lattice dynamical systems. I

    Page(s): 746 - 751
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    We survey some recent advances in the theory of lattice dynamical systems, with both discrete-time and continuous-time problems being considered. Among the topics discussed, are pattern formation and spatial chaos, bifurcation of regular patterns (checkerboards and stripes), traveling waves and propagation failure, and homoclinic orbits of Zd actions View full abstract»

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  • Homoclinic orbits and solitary waves in a one-dimensional array of Chua's circuits

    Page(s): 785 - 801
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    The possible propagation of solitary waves in a one-dimensional array of inductively coupled Chua's circuits is considered. We show that in the long-wave limit, the problem can be reduced to the analysis of the homoclinic orbits of a dynamical system described by three coupled nonlinear ordinary differential equations modeling the individual dynamics of a single Chua's circuit. Analytical, numerical, and experimental results concerning the bifurcations associated with the appearance of homoclinic orbits and thus with the propagation of solitary waves are provided View full abstract»

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  • Pattern formation and spatial chaos in lattice dynamical systems. II

    Page(s): 752 - 756
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    For part I see ibid., vol.42, no.10, pp.746-51 (1995). We survey a class of continuous-time lattice dynamical systems, with an idealized nonlinearity. We introduce a class of equilibria called mosaic solutions, which are composed of the elements 1, -1, 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 formation), or h>0 (termed spatial chaos). Numerical techniques for calculating h are described View full abstract»

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  • Simulating nonlinear waves and partial differential equations via CNN. I. Basic techniques

    Page(s): 807 - 815
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    Cellular neural networks (CNNs)-a paradigm for locally connected analog array-computing structures-are considered for solving partial differential equations (PDE's) and systems of ordinary differential equations (ODE's). The relationship between various implementations of nonanalytical PDE solvers is discussed. The applicability of CNNs is shown by three examples of nonlinear PDE implementations: a reaction-diffusion type system, Burgers' equation, and a form of the Navier-Stokes equation in a two-dimensional setting View full abstract»

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  • Autowaves and spatio-temporal chaos in CNNs. I. A tutorial

    Page(s): 638 - 649
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    This paper presents a summary of the most commonly observed spatio-temporal phenomena in discrete cellular neural networks (CNNs) of dimension one and two. Among the phenomena discussed are traveling wave phenomena in chains and 2-D arrays, and spiral waves and target waves in both excitable and fluctuating media. Chua's circuit is used as the basic cell in the CNN arrays. Parameter values and initial conditions for the corresponding simulations are given so they can be reproduced with different simulators View full abstract»

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  • Pattern formation properties of autonomous Cellular Neural Networks

    Page(s): 757 - 774
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    We use the Cellular Neural Network (CNN) to study the pattern formation properties of large scale spatially distributed systems. We have found that the Cellular Neural Network can produce patterns similar to those found in Ising spin glass systems, discrete bistable systems, and the reaction-diffusion system. A thorough analysis of a 1-D CNN whose cells are coupled to immediate neighbors allows us to completely characterize the patterns that can exist as stable equilibria, and to measure their complexity thanks to an entropy function. In the 2-D case, we do not restrict the symmetric coupling between cells to be with immediate neighbors only or to have a special diffusive form. When larger neighborhoods and generalized diffusion coupling are allowed, it is found that some new and unique patterns can be formed that do not fit the standard ferro-antiferromagnetic paradigms. We have begun to develop a theoretical generalization of these paradigms which can be used to predict the pattern formation properties of given templates. We give many examples. It is our opinion that the Cellular Neural Network model provides a method to control the critical instabilities needed for pattern formation without obfuscating parameterizations, complex nonlinearities, or high-order cell states, and which will allow a general and convenient investigation of the essence of the pattern formation properties of these systems View full abstract»

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  • Simulating nonlinear waves and partial differential equations via CNN. II. Typical examples

    Page(s): 816 - 820
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    For part I see ibid., vol.42, no.10, pp.807-15 (1995). Application of cellular neural network (CNN) paradigm of locally connected analog array-computing structures is considered for solving partial differential equations (PDE's) and systems of ordinary differential equations (ODE). Three examples are presented: a chain of particles with nonlinear interactions, solitons in a nonlinear Klein-Gordon equation, and an application of a reaction-diffusion CNN for fingerprint enhancement View full abstract»

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  • Circuit models for linear and nonlinear waves

    Page(s): 578 - 582
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    Physical fields supporting wave propagation and/or diffusion are conveniently described by partial differential equations. The latter can be solved numerically by various discretization schemes. Some of these describe the performance of lumped networks imbedding some or all of the properties of the field that reflect in those of the algorithm. Moreover, the approximating lumped networks are locally connected, thus representing cellular neural networks that can be considered as analog processors describing the field features. The concepts above are illustrated with the example of the continuity equation and are applied to the analysis of hydrokinetic problems View full abstract»

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  • Autowaves and spatio-temporal chaos in CNNs. II. A tutorial

    Page(s): 650 - 664
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    For part I, see ibid., vol. 42, no. 10, pp. 638-49 (Oct. 1995). In this paper, we reproduce and discuss some of the spatio-temporal phenomena, recently simulated in discrete CNNs of dimension one, two, and three. We show how target, or concentric waves, can be generated in both excitable and fluctuating 2-D media, and how several types of scroll waves can be simulated in 3-D arrays. The basic property of autowaves exhibited in interactions-mutual annihilation-is demonstrated through examples. Also included is a discussion of the coexistence of low- and high-dimensional attractors in a CNN ring. Chua's circuit is used as the basic cell in the CNN arrays. Parameter values and initial conditions for the corresponding simulations are given so they can be reproduced with different simulators View full abstract»

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  • Effects of square-wave modulation on CNN patterns

    Page(s): 700 - 705
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    By driving each cell (Chua's circuit) of a CNN with a nonuniform square-wave current source, different patterns are observed and reported in this paper. The dynamics of such a modulation signal could find useful applications in controlling certain features in the patterns View full abstract»

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  • Turing patterns in CNNs. III. Computer simulation results

    Page(s): 627 - 637
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    For part II, see ibid., vol. 42, no. 10, pp. 612-26 (Oct. 1995). In this paper, various patterns obtained through computer simulations are presented. It is shown through simple examples that the four inequalities known as the Turing instability conditions are only necessary but not sufficient conditions for Turing patterns to exist. The influence of various parameters, initial conditions as well as sidewall forcing on the final patterns and the possibilities of eliminating defects are studied by means of computer simulations. The possibility of generating perfectly regular patterns or patterns with few defects through the application of small and short controlling signals at one boundary suggests the intriguing possibility of producing high-purity high-tech materials, such as crystals, which was not possible with current technology. In addition, novel applications in image processing, pattern recognition, and other areas can be expected View full abstract»

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  • Study of reentry initiation in coupled parallel fibers [cardiology]

    Page(s): 665 - 671
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    Initiation of reentries in coupled parallel fibers is studied as a function of several control parameters intrinsic to those fibers. The influence of inhomogeneities in the fibers leading to drift of the vortices and the interaction between them is also analyzed numerically and experimentally View full abstract»

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  • Chaotic waves and spatio-temporal patterns in large arrays of doubly-coupled Chua's circuits

    Page(s): 706 - 714
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    We investigate complex dynamic phenomena in arrays composed of interacting chaotic circuits. Such arrays can be thought of as a model of nonlinear phenomena in spatially extended (high-dimensional or infinite-dimensional) systems and active media with potential applications in signal processing. In this paper, we consider a particular structure of the network in which there exists double diffusive interactions between the cells. Such a double interaction can be considered as a paradigm and means for understanding very complex interactions existing in real systems where separate cells can communicate in various ways. We consider two basic cases where separate cells without coupling exhibit two different types of chaotic behavior. Depending on the connection structure, initial conditions imposed in the cells, the array exhibits various kinds of spatially ordered chaotic waves. Patterns of behavior depending on the excitation of the array and the connection structure are studied in this paper. Chua's circuits are taken as standard chaotic cells View full abstract»

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  • Synchronization of a one-dimensional array of Chua's circuits by feedback control and noise

    Page(s): 736 - 740
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    A one-dimensional array of coupled Chua's circuits is investigated. Without control, the synchronization between sites is poor, and the total output is very weak due to the spatial disorder. By feedback injections at certain fraction of sites (i.e., pinnings), synchronization can be established between the sites. If the pinning density is low, some sites may be left unsynchronized. In this case, by properly applying noise for certain time interval, spatial disorder can be perfectly excluded. The optimal noise intensity for synchronization (or say, stochastic resonance for synchronization) is briefly discussed View full abstract»

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  • Investigation of chaos in large arrays of Chua's circuits via a spectral technique

    Page(s): 802 - 806
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    A spectral technique is proposed for studying and predicting chaos in a one-dimensional array of Chua's circuits. By use of a suitable double transform, the network is reduced to a scalar Lur'e system to which the describing function technique is applied for discovering the existence and the characteristics of periodic waves. Finally, by the computation of the distortion index, an approximate tool is given for detecting the occurrence of chaos View full abstract»

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  • Application of Kronecker products to the analysis of systems with uniform linear coupling

    Page(s): 775 - 778
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    In this paper, we show how the Kronecker product can be useful in simplifying the notation encountered in the analysis of linearly coupled systems where the cells are identical and the coupling is uniform. We give two applications where this is useful. First we use it in the analysis of Turing patterns in reaction-diffusion systems to obtain conditions for the coupling to destabilize the uniform equilibrium point. In the second application we use the Kronecker product to obtain simple sufficient conditions for an array of linearly coupled dynamical systems to synchronize. We discuss briefly extensions to additive nonlinear coupling View full abstract»

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  • Spatio-temporal chaos in simple coupled chaotic circuits

    Page(s): 678 - 686
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    In this paper, simple autonomous chaotic circuits coupled by resistors are investigated. By carrying out computer calculations and circuit experiments, irregular self-switching phenomenon of three spatial patterns characterized by the phase states of quasi-synchronization of chaos can be observed from only four simple chaotic circuits. This is the same phenomenon as chaotic wandering of spatial patterns observed very often from systems with a large number of degrees of freedom. Spatial-temporal chaos observed from systems of large size can be also generated in the proposed system consisting of only four chaotic circuits. A six subcircuits case and a coupled chaotic circuits network are also studied, and such systems are confirmed to produce more complicated spatio-temporal phenomena View full abstract»

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  • Turing patterns in CNNs. II. Equations and behaviors

    Page(s): 612 - 626
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    For part I, see ibid., vol. 42, no. 10, pp. 602-11 (Oct. 1995). The general state equations describing two-grid coupled CNNs based on the reduced Chua's circuit are derived, and the analysis of Turing pattern formation is approached from a specific point of view: spatial-eigenfunction based equation decoupling. Discrete spatial eigenfunctions for two common types of boundary conditions are presented, and the way the dynamics is influenced by the shape and position of the dispersion curve is analyzed View full abstract»

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  • Autonomous cellular neural networks: a unified paradigm for pattern formation and active wave propagation

    Page(s): 559 - 577
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    This 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, and physics. 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 on reaction-diffusion 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 third-order universal cells and coupled to each other by only one layer of linear resistors provides a unified active medium for generating trigger (autowave) waves, target (concentric) waves, spiral waves, and scroll waves. When a second layer of linear resistors is added to couple a second capacitor voltage in each cell to its neighboring cells, the resulting CNN can be used to generate various turing patterns View full abstract»

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  • Cycling chaos

    Page(s): 821 - 823
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    Saddle connections between equilibria can occur structurally stable in systems with symmetry, and these saddle connections can cycle so that a given equilibrium is connected to itself by a sequence of connections. These cycles provide a way of generating intermittency, as a trajectory will spend some time near each saddle before quickly moving to the next saddle. Guckenheime and Holmes (1988) showed that cycles of saddle connections can appear via bifurcation. In this paper, we show numerically that the equilibria in the Guckenheimer-Holmes example can be replaced by chaotic sets, such as those that appear in a Chua circuit or a Lorenz attractor. Consequently, there are trajectories that behave chaotically, but where the spatial location of the chaos cycles. We call this phenomenon cycling chaos View full abstract»

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  • A computer-assisted investigation of a 2-D array of Chua's circuits

    Page(s): 721 - 735
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    This paper presents simulation results on a 2-D array of coupled Chua's circuits called an f(x)-x˙ coupled chaotic CNN in order to distinguish it from other recently proposed chaotic CNN architectures. Each isolated cell with unity self-feedback in the network is a Chua's circuit and is connected only to its nearest neighbors defined by a metric and a neighborhood size. The network is designed with a 2-D torus like connection topology having cyclic boundary conditions that play an important role in complete phase synchronization. It is observed in our computer simulations that i) depending on the choice of the intracell parameters and the connection weights, the cells of the network appear to be operating in a double-scroll Chua's attractor, in spiral Chua's attractors, in stable equilibria, in a period-1, a period-2, a large limit-cycle, or in a large chaotic regime, ii) complete phase synchronization in the network with all cells operating in the double scroll regime ran be obtained by a suitable choice of the intracell parameters and the feedback connection weights, iii) there is a set of the intracell parameters and connection weights resulting in a chaotic regime such that each cell depending on its constant external input falls into one of the three attractors; namely, the double-scroll, P+ spiral, or P- spiral Chua's attractor. As a new phenomenon, a close relation between phase synchronization settling-time and input pattern is observed that offers new potentials of Chua's circuit arrays for pattern recognition applications View full abstract»

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  • Application of cellular neural networks to model population dynamics

    Page(s): 715 - 720
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    In this paper, we show an application of the CNN paradigm to model spatial population dynamics. In particular, we present the procedure for determining the CNN template coefficients for implementing Schelling's model of social segregation. Simulation results are given View full abstract»

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  • Turing patterns in CNNs. I. Once over lightly

    Page(s): 602 - 611
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    The aim of this three part tutorial is to focus the reader's attention to a new exciting behavior of a particular class of cellular neural networks (CNNs): Turing pattern formation in two-grid coupled CNNs. We first analyze the reduced Chua's circuit as the basic cell for the two-grid coupled CNNs capable of producing Turing patterns. We use a nonstandard normalization to derive a dimensionless state equation of the individual cell. Then, we present an intuitive explanation of Turing pattern formation mechanism for a 1-D two-grid coupled array in relation to the original mechanism proposed by Turing. Finally, we derive the first two conditions for Turing pattern formation, discuss the boundary conditions, and illustrate via an example on how the number of the equilibrium points of a CNN increases rapidly even though each isolated cell has only one equilibrium point View full abstract»

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  • Coexistence of excitability, Hopf and Turing modes in a one-dimensional array of nonlinear circuits

    Page(s): 672 - 677
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    The behavior in the vicinity of a point where excitable, Hopf and Turing modes coexisting is analyzed within the framework of a simplified version of Chua's nonlinear circuits. The different factors (e.g., initial and boundary conditions and small changes in circuit parameters) which determine the final state of the system are numerically studied View full abstract»

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  • Chaos and structures in a chain of mutually-coupled Chua's circuits

    Page(s): 693 - 699
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    The formation and interaction of structures in a chain of coupled Chua's circuits are investigated. Primary attention is focused on the control of spatio-temporal structures by choosing initial conditions and the values of coupling View full abstract»

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