Exponential stability and spectral analysis of a delayed ring neural network with a small-world connection

This paper addresses the dynamical behavior of the linearized delayed ring neural network system with a small-world connection. The semigroup approach is adopted in investigation. The asymptotic eigenvalues of the system are presented. It shows that the spectrum of the system is located in the left half complex plane and its real part goes to -∞ when the connection weights between neurons are well-defined. The spectrum determined growth condition is held true and the exponential stability of the system is then established. Moreover, we present the necessary conditions for the neuron and feedback gains, for which the closed-loop system is delay-independent exponentially stable, and we further provide the sufficient and necessary conditions when the concrete number of neurons and the location of small-world connection are given. Finally, numerical simulations are presented to illustrate the convergence of the state for the system and demonstrate the effect of the feedback gain on stability.