CONTROLLING PATTERN FORMATION IN A CNN OF CHUA’S CIRCUITS

1996 ◽  
Vol 06 (11) ◽  
pp. 2127-2144 ◽  
Author(s):  
VLADIMIR D. SHALFEEV ◽  
ALEXEY S. KUZNETSOV
Keyword(s):  
Nonlinear Dynamics ◽  
Neural Networks ◽  
Pattern Formation ◽  
Formation Process ◽  
Initial Conditions ◽  
Cellular Neural Networks ◽  
External Signal ◽  
Nonlinear Coupling ◽  
Two Dimensional ◽  
Dimensional Array

In this letter we consider the nonlinear dynamics of a 2-dimensional CNN (cellular neural networks) made of a two-dimensional array of Chua’s oscillators, interconnected via nonlinear coupling. We focus our attention on the possibility of intelligently controlling the pattern formation process by applying an external signal from independent current sources to the cells of the CNN, or by an intelligent choice of initial conditions.

CHARACTERIZATION AND DYNAMICS OF PATTERN FORMATION IN CELLULAR NEURAL NETWORKS

1996 ◽  
Vol 06 (09) ◽  
pp. 1703-1724 ◽  
Author(s):  
K.R. CROUNSE ◽  
L.O. CHUA ◽  
P. THIRAN ◽  
G. SETTI
Keyword(s):  
Neural Networks ◽  
Pattern Formation ◽  
Spatial Frequency ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Temporal Frequency ◽  
Random Noise ◽  
Cellular Neural Networks ◽  
Unstable Equilibrium ◽  
Exact Results

We study some properties of pattern formation arising in large arrays of locally coupled first-order nonlinear dynamical systems, namely Cellular Neural Networks (CNNs). We will present exact results to analyze spatial patterns for symmetric coupling and to analyze spatio-temporal patterns for anti-symmetric coupling in one-dimensional lattices, which will then be completed by approximative results based on a spatial and/or temporal frequency approach. We will discuss the validity of these approximations, which bring a lot of insight. This spectral approach becomes very convenient for the two-dimensional lattice, as exact results get more complicated to establish. In this second part, we will only consider a symmetric coupling between cells. We will show what kinds of motifs can be found in the patterns generated by 3×3 templates. Then, we will discuss the dynamics of pattern formation starting from initial conditions which are a small random noise added to the unstable equilibrium: this can generally be well predicted by the spatial frequency approach. We will also study whether a defect in a pure pattern can propagate or not through the whole lattice, starting from initial conditions being a localized perturbation of a stable pattern: this phenomenon is no longer correctly predicted by the spatial frequency approach. We also show that patterns such as spirals and targets can be formed by “seed” initial conditions — localized, non-random perturbations of an unstable equilibrium. Finally, the effects on the patterns formed of a bias term in the dynamics are demonstrated.

CELLULAR NEURAL NETWORKS CAN MIMIC SMALL-WORLD NETWORKS

1999 ◽  
Vol 09 (10) ◽  
pp. 2105-2126 ◽  
Author(s):  
TAO YANG ◽  
LEON O. CHUA
Keyword(s):  
Neural Networks ◽  
Coupling Strength ◽  
Initial Conditions ◽  
Computing Time ◽  
Small World ◽  
Cellular Neural Networks ◽  
Clustering Coefficient ◽  
High Survival ◽  
Game Of Life ◽  
Hole Detection

Small-world phenomenon can occur in coupled dynamical systems which are highly clustered at a local level and yet strongly coupled at the global level. We show that cellular neural networks (CNN's) can exhibit "small-world phenomenon". We generalize the "characteristic path length" from previous works on "small-world phenomenon" into a "characteristic coupling strength" for measuring the average coupling strength of the outputs of CNN's. We also provide a simplified algorithm for calculating the "characteristic coupling strength" with a reasonable amount of computing time. We define a "clustering coefficient" and show how it can be calculated by a horizontal "hole detection" CNN, followed by a vertical "hole detection" CNN. Evolutions of the game-of-life CNN with different initial conditions are used to illustrate the emergence of a "small-world phenomenon". Our results show that the well-known game-of-life CNN is not a small-world network. However, generalized CNN life games whose individuals have strong mobility and high survival rate can exhibit small-world phenomenon in a robust way. Our simulations confirm the conjecture that a population with a strong mobility is more likely to qualify as a small world. CNN games whose individuals have weak mobility can also exhibit a small-world phenomenon under a proper choice of initial conditions. However, the resulting small worlds depend strongly on the initial conditions, and are therefore not robust.

PIECEWISE TWO-DIMENSIONAL MAPS AND APPLICATIONS TO CELLULAR NEURAL NETWORKS

2004 ◽  
Vol 14 (07) ◽  
pp. 2223-2228 ◽  
Author(s):  
HSIN-MEI CHANG ◽  
JONG JUANG
Keyword(s):  
Neural Networks ◽  
Cellular Neural Networks ◽  
Two Dimensional ◽  
Smale Horseshoe ◽  
Linear Map ◽  
One Dimensional ◽  
Spatial Chaos

Of concern is a two-dimensional map T of the form T(x,y)=(y,F(y)-bx). Here F is a three-piece linear map. In this paper, we first prove a theorem which states that a semiconjugate condition for T implies the existence of Smale horseshoe. Second, the theorem is applied to show the spatial chaos of one-dimensional Cellular Neural Networks. We improve a result of Hsu [2000].

ON THE TEMPLATES CORRESPONDING TO CYCLE-SYMMETRIC CONNECTIVITY IN CELLULAR NEURAL NETWORKS

2002 ◽  
Vol 12 (12) ◽  
pp. 2957-2966 ◽  
Author(s):  
CHIH-WEN SHIH ◽  
CHIH-WEN WENG
Keyword(s):  
Neural Networks ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Cellular Neural Networks ◽  
Necessary And Sufficient Conditions ◽  
Local Interaction ◽  
Two Dimensional ◽  
Linear Coupling ◽  
Symmetric Connectivity ◽  
Necessary And Sufficient

In the architecture of cellular neural networks (CNN), connections among cells are built on linear coupling laws. These laws are characterized by the so-called templates which express the local interaction weights among cells. Recently, the complete stability for CNN has been extended from symmetric connections to cycle-symmetric connections. In this presentation, we investigate a class of two-dimensional space-invariant templates. We find necessary and sufficient conditions for the class of templates to have cycle-symmetric connections. Complete stability for CNN with several interesting templates is thus concluded.

COMPLETE STABILITY FOR STANDARD CELLULAR NEURAL NETWORKS

1999 ◽  
Vol 09 (05) ◽  
pp. 909-918 ◽  
Author(s):  
SONG-SUN LIN ◽  
CHIH-WEN SHIH
Keyword(s):  
Neural Networks ◽  
Boundary Condition ◽  
Symmetric Space ◽  
Dirichlet Boundary Condition ◽  
Periodic Boundary ◽  
Dirichlet Boundary ◽  
Periodic Boundary Conditions ◽  
Cellular Neural Networks ◽  
Two Dimensional ◽  
Space Variant

We consider cellular neural networks with symmetric space-variant feedback template. The complete stability is proved via detailed analysis on the energy function. The proof is presented for the two-dimensional case with Dirichlet boundary condition. It can be extended to other dimensions with minor adjustments. Modifications to the cases of Neumann and periodic boundary conditions are also mentioned.

SMALE HORSESHOE OF CELLULAR NEURAL NETWORKS

2000 ◽  
Vol 10 (09) ◽  
pp. 2119-2127 ◽  
Author(s):  
CHENG-HSIUNG HSU
Keyword(s):  
Neural Networks ◽  
Cellular Neural Networks ◽  
Two Dimensional ◽  
Smale Horseshoe ◽  
One Dimensional

The paper shows the spatial disorder of one-dimensional Cellular Neural Networks (CNN) using the iteration map method. Under certain parameters, the map is two-dimensional and the Smale horseshoe is constructed. Moreover, we also illustrate the variant of CNN, closely related to Henón-type and Belykh maps, and discrete Allen–Cahn equations.

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