Space-invariant Templates Research Articles

BIFURCATIONS AND CHAOS IN CELLULAR NEURAL NETWORKS

The dynamic behavior of first-order autonomous space invariant cellular neural networks (CNNs) is investigated. It is shown that complex dynamics may occur in very simple CNN structures, described by two-dimensional templates that present only vertical and horizontal couplings. The bifurcation processes are analyzed through the computation of the limit cycle Floquet's multipliers, the evaluation of the Lyapunov exponents and of the signal spectra. As a main result a detailed and accurate two-dimensional bifurcation diagram is reported. The diagram allows one to distinguish several regions in the parameter space of a single CNN. They correspond to stable, periodic, quasi-periodic, and chaotic behavior, respectively. In particular it is shown that chaotic regions can be reached through two different routes: period doubling and torus breakdown. We remark that most practical CNN implementations exploit first order cells and space-invariant templates: so far only a few examples of complex dynamics and no complete bifurcation analysis have been presented for such networks.

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

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.

Complex dynamic phenomena in space-invariant cellular neural networks

It is shown that first-order autonomous space-invariant cellular neural networks (CNNs) may exhibit a complex dynamic behavior (i.e., equilibrium point and limit cycle bifurcation, strange and chaotic attractors). The most significant limit cycle bifurcation processes, leading to chaos, are investigated through the computation of the corresponding Floquet's multipliers and Lyapunov exponents. It is worth noting that most practical CNN implementations exploit first-order cells and space-invariant templates: so far no example of complex dynamics has been shown in first-order autonomous space-invariant CNNs.

On Stability of Cellular Neural Networks

The main results about stability of cellular neural networks (CNNs) are reviewed. Some of them are extended and reformulated, with the purpose of providing to the CNN designer simple criteria for checking the stability properties. A particular emphasis is given to the conditions for the stability of CNNs described by space-invariant templates.

Analysis of time-varying cellular neural networks for quadratic global optimization

The algorithm for quadratic global optimization performed by a cellular neural network (CNN) with a slowly varying slope of the output characteristic (see References 1 and 2) is analysed. It is shown that the only CNN which finds the global minimum of a quadratic function for any values of the input parameters is the network composed by only two cells. If the dimension is higher than two, even the CNN described by the simplest one-dimensional space-invariant template Â=[A1, A0, A1], fails to find the global minimum in a subset of the parameter space. Extensive simulations show that the CNN described by the above three-element template works correctly within several parameter ranges; however, if the parameters are chosen according to a random algorithm, the error rate increases with the number of cells. © 1998 John Wiley & Sons, Ltd.

Practical stability criteria for cellular neural networks

New sufficient conditions for the existence of stable equilibrium points in cellular neural networks (CNNs), described by space-invariant templates, are presented. Extensive simulations have shown that such conditions also guarantee the complete stability of the network.