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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 31 (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
A nonlinear dynamics perspective of Wolfram’s new kind of science. Part III: Predicting the unpredictable
 International Journal of Bifurcation and Chaos
, 2004
"... This tutorial provides a nonlinear dynamics perspective to Wolfram’s monumental work on A New Kind of Science. By mapping a Boolean local Rule, ortruth table, ontothepoint attractors of a specially tailored nonlinear dynamical system, we show how some of Wolfram’s empirical observations can be justi ..."
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Cited by 27 (1 self)
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This tutorial provides a nonlinear dynamics perspective to Wolfram’s monumental work on A New Kind of Science. By mapping a Boolean local Rule, ortruth table, ontothepoint attractors of a specially tailored nonlinear dynamical system, we show how some of Wolfram’s empirical observations can be justified on firm ground. The advantage of this new approach for studying Cellular Automata phenomena is that it is based on concepts from nonlinear dynamics and attractors where many fuzzy concepts introduced by Wolfram via brute force observations can be defined and justified via mathematical analysis. The main result of Part I is the introduction of a fundamental concept called linear separability and a complexity index κ for each local Rule which characterizes the intrinsic geometrical structure of an induced “Boolean cube ” in threedimensional Euclidean space. In particular, Wolfram’s seductive idea of a “threshold of
nDOUBLE SCROLL HYPERCUBES IN 1D CNNs
, 1997
"... Unidirectional and diffusive coupling of identical ndouble scroll cells in a onedimensional cellular neural network is studied. Weak coupling between the cells leads to hyperchaos, with ndouble scroll hypercube attractors observed in the common state subspace of the cells. Individually the cells ..."
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Cited by 13 (8 self)
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Unidirectional and diffusive coupling of identical ndouble scroll cells in a onedimensional cellular neural network is studied. Weak coupling between the cells leads to hyperchaos, with ndouble scroll hypercube attractors observed in the common state subspace of the cells. Individually the cells remain behaving as ndouble scrolls. The ndouble scroll hypercubes are filled with multiple scrolls. Their birth goes through the mechanism of intermittency.
Wave Phenomena in Cellular Neural Networks Using Two Kinds of Template Sets
"... Abstract—In this study, we propose cellular neural networks using two kinds of template sets. The computer simulations show that the proposed system can generate similar phenomena which is observed from the twolayer cellular neural networks. Investigating these phenomena contributes to underst ..."
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Abstract—In this study, we propose cellular neural networks using two kinds of template sets. The computer simulations show that the proposed system can generate similar phenomena which is observed from the twolayer cellular neural networks. Investigating these phenomena contributes to understand complex systems and to apply them to engineering systems. 1.
Intelligence and Cooperative Search by Coupled Local Minimizers
, 2001
"... this paper we propose a new methodology of coupled local minimizers (CLM) for solving continuous nonlinear optimization problems. We pose a somewhat similar challenge as for committee networks but within a di#erent and broader context of solving di#erentiable optimization problems. The aim is to (on ..."
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Cited by 9 (5 self)
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this paper we propose a new methodology of coupled local minimizers (CLM) for solving continuous nonlinear optimization problems. We pose a somewhat similar challenge as for committee networks but within a di#erent and broader context of solving di#erentiable optimization problems. The aim is to (online) combine the results from local optimizers in order to let the ensemble generate a local minimum that is better than the best result obtained from all individual local minimizers. We show how improved local minima can be obtained by having interaction and information exchange between the local search processes. This is realized through state synchronization constraints that are imposed between the local minimizers by incorporating principles of masterslave dynamics. Synchronization theory has been intensively studied within the area of chaotic systems and secure communications [Chen & Dong, 1998; Pecora & Carroll, 1990; Suykens et al., 1996, 1997, 1998; Wu & Chua, 1994]. The CLM method is related to Lagrange programming network approaches for chaos synchronization [Suykens & Vandewalle, 2000], where identical or generalized synchronization constraints are imposed on dynamical systems. CLMs also fit within the framework of Cellular Neural Networks (CNN) [Chua & Roska, 1993; Chua et al., 1995; Chua, 1998]. By considering the objective of minimizing the average cost of an ensemble of local minimizers subject to pairwise synchronization constraints, a continuoustime optimization algorithm is studied according to Lagrange programming networks [Cichocki & Unbehauen, 1994; Zhang & Constantinides, 1992]. The resulting continuoustime optimization algorithm is described by an array of coupled nonlinear cells or a onedimensional CNN with bidirectional coupling
Cluster Synchronization in Threedimensional Lattices of Diffusively Coupled Oscillators
 Int. J. of Bifurcation and Chaos
"... Cluster synchronization modes of continuous time oscillators that are diffusively coupled in a threedimensional (3D) lattice are studied in the paper via the corresponding linear invariant manifolds. Depending in an essential way on the number of oscillators composing the lattice in three volume d ..."
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Cited by 5 (0 self)
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Cluster synchronization modes of continuous time oscillators that are diffusively coupled in a threedimensional (3D) lattice are studied in the paper via the corresponding linear invariant manifolds. Depending in an essential way on the number of oscillators composing the lattice in three volume directions, the set of possible regimes of spatiotemporal synchronization is examined. Sufficient conditions of the stability of cluster synchronization are obtained analytically for a wide class of coupled dynamical systems with complicated individual behavior. Dependence of the necessary coupling strengths for the onset of global synchronization on the number of oscillators in each lattice direction is discussed and an approximative formula is proposed. The appearance and order of stabilization of the cluster synchronization modes with increasing coupling between the oscillators are revealed for 2D and 3D lattices of coupled Lur’e systems and of coupled Rössler oscillators.
Oscillatory Phenomena in Cellular Neural Network Using Two Kinds of Templates
"... Abstract—In this study, oscillatory phenomena in a cellular neural network using two kinds of templates are investigated. The number of cells is limited to 3 × 3. The relationship among oscillatory phenomena and two parameters which are corresponding to coupling factors and biases. We consider t ..."
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Abstract—In this study, oscillatory phenomena in a cellular neural network using two kinds of templates are investigated. The number of cells is limited to 3 × 3. The relationship among oscillatory phenomena and two parameters which are corresponding to coupling factors and biases. We consider that investigating this system is important to understanding coupled oscillatory systems. Because this system is a new class of coupled oscillatory systems. 1.
On Global Dynamic Behavior of Weakly Connected Oscillatory Networks
"... The global dynamics of weakly connected oscillatory networks is investigated: as a case study onedimensional arrays of third order oscillators are considered. Through the joint application of the describing function technique and of Malkin’s Theorem a very accurate analytical expression of the phas ..."
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The global dynamics of weakly connected oscillatory networks is investigated: as a case study onedimensional arrays of third order oscillators are considered. Through the joint application of the describing function technique and of Malkin’s Theorem a very accurate analytical expression of the phase deviation equation (i.e. the equation that describes the phase deviation due to the weak coupling) is derived. The total number of limit cycles and their stability properties are estimated via the analytical study of the phase deviation equation. The proposed technique significantly extends the results available in the literature and can be applied to almost all complex networks of oscillators. In particular twodimensional, space variant and fully connected networks can be dealt with. 1
2000b] “Analytical criteria for local activity of reaction–diffusion CNN with four state variables and applications to the Hodgkin–Huxley equation,” Int
 J. Bifurcation and Chaos
"... This paper presents analytical criteria for local activity in reaction–diffusion Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999] with four local state variables. As a first application, we apply the criteria to a Hodgkin–Huxley CNN, which has cells defined by the equations of the cardiac P ..."
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This paper presents analytical criteria for local activity in reaction–diffusion Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999] with four local state variables. As a first application, we apply the criteria to a Hodgkin–Huxley CNN, which has cells defined by the equations of the cardiac Purkinje fiber model of morphogenesis that was first introduced in [Noble, 1962] to describe the longlasting action and pacemaker potentials of the Purkinje fiber of the heart. The bifurcation diagrams of the Hodgkin–Huxley CNN’s supply a possible explanation for why a heart with a normal heartrate may stop beating suddenly: The cell parameter of a normal heart is located in a locally active unstable domain and just nearby an edge of chaos. The membrane potential along a fiber is simulated in a Hodgkin–Huxley CNN by a computer. As a second application, we present a smoothed Chua’s circuit (SCC) CNN. The bifurcation diagrams of the SCC CNN’s show that there does not exist a locally passive domain, and the edges of chaos corresponding to different fixedcell parameters are significantly different. Our computer simulations show that oscillatory patterns, chaotic patterns, or divergent patterns may emerge if the selected cell parameters are located in locally active domains but nearby the edge of chaos. This research demonstrates once again the effectiveness of the local activity theory in choosing the parameters for the emergence of complex (static and dynamic) patterns in a homogeneous lattice formed by coupled locally active cells. 1.
1 CNN Dynamics Represents a Broader Class than PDEs
"... The relationship between Cellular Nonlinear Networks (CNNs) and Partial Differential Equations (PDEs) is investigated. The equivalence between discretespace CNN models and continuousspace PDE models is rigorously defined. The key role of space discretization is explained. The problem of the equival ..."
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The relationship between Cellular Nonlinear Networks (CNNs) and Partial Differential Equations (PDEs) is investigated. The equivalence between discretespace CNN models and continuousspace PDE models is rigorously defined. The key role of space discretization is explained. The problem of the equivalence is split into two subproblems: approximation and topological equivalence, that can be explicitly studied for any CNN models. It is known that each PDE can be approximated by a space difference scheme, i.e. a CNN model, that presents a similar dynamic behavior. It is shown, through several examples, that there exist CNN models that are not equivalent to any PDEs, either because they do not approximate any PDE models, or because they have a qualitatively different dynamic behavior (i.e they are not topologically equivalent to the PDE, that approximate). This proves that the spatiotemporal CNN dynamics is broader than that described by PDEs.