Existence and characterization of limit cycles in nearly symmetric neural networks

Abstract

It is known that additive neural networks with a symmetric interconnection matrix are completely stable, i.e., each trajectory converges toward some equilibrium point. This paper addresses the fundamental question of robustness of complete stability of additive neural networks with respect to small perturbations of the nominal symmetric interconnections. It is shown that in the general case, complete stability is not robust. More precisely, the paper considers a class of neural networks, and gives a necessary and sufficient condition for the existence of Hopf bifurcations (HBs) at the equilibrium point at the origin, arbitrarily close to symmetry. Such HBs originate stable limit cycles and hence cause the loss of complete stability. Furthermore, the paper highlights situations where the HBs are particularly critical, in the sense that the amplitude of the limit cycles is very sensitive to errors due to tolerances in the electronic implementation of the neuron interconnections. It is shown that sensitivity is crucially dependent on the neuron nonlinearity, and it is also significantly influenced by the features of the interconnection matrix and the network dimension. Finally, limitations of the obtained results are discussed and hints for future work are given.

On the Margin of Complete Stability for a Class of Cellular Neural Networks

“…Such nonrobust CNNs are able to display nonconvergent dynamics, including bifurcations, large-size periodic oscillations and even complex attractors, close to symmetry. Analogous results have been singled out for general neural networks with an additive neuron interconnecting structure [Di Marco et al, 2002]. On the other hand, in the CNNs community it is widely believed that nonrobust cases are expected to be exceptional in the class of symmetric CNNs.…”

Robustness of convergence in finite time for linear programming neural networks

“…If we choose N = {V 2 }, quantity E m deÿned in Property 1 is given by E m = 5:2%. It is also seen that for errors in the interconnections satisfying A¡E m = 5:2%, V 2 is not an equilibrium point of (5). Hence, on the basis of Theorem 1, we have that (5) is globally convergent to M = {x ? }…”

“…Indeed, recent work has shown that there are classes of nominally symmetric additive neural networks for which convergence is not robust with respect to perturbations due to tolerances [4,5]. Namely, even arbitrarily small perturbations cause the birth of large-size non-vanishing oscillations in the long-run behaviour of the trajectories, an highly undesirable situation which makes the networks not useful for solving signal processing tasks.…”

Optimal design of Neuro‐Mechanical Oscillators with stability constraints

“…As is well-known, RNNs may exhibit limit cycle behavior [44], a property which can be exploited for designing robot controllers [55,71]. However, a potential drawback of this approach is that if an RNN enters into a limit cycle, i.e., a self-excited periodic oscillation, it can become insensitive or react unpredictably to changes in the input signals; in this sense, external control of the system may be lost.…”