Long transient oscillations in a class of cooperative cellular neural networks

The real-time processing capabilities of cellular neural networks (CNNs) are inherently related to the fast convergence time of the solutions toward the asymptotically stable equilibrium points. A typical requirement is that the settling time should not exceed a few (or at most 10) cell time constants. This paper introduces a class of completely stable nonsymmetric cooperative CNN rings whose solutions display unexpectedly long transient oscillations for a wide set of initial conditions and for a wide set of interconnection parameters. Numerical simulations show that the oscillations can easily last hundreds of cycles, and thousands of cell time constants, before settling to a steady state, thus possibly impairing their real-time processing capabilities. Goal of the paper is also to show, by means of laboratory experiments on a discrete component prototype of the CNN ring, that the long oscillation phenomenon is physically robust with respect to the non-idealities of the circuit implementation. The experiments show some other peculiar features of the long lasting oscillations as the metamorphosis between different periodic behaviors during the transient. Finally, analytic asymptotic estimates on the duration of the transient oscillations are provided.