Sannay Mohamad

Global exponential stability in continuous-time and discrete-time delayed bidirectional neural networks

Convergence dynamics of continuous-time bidirectional neural networks with constant transmission delays are studied. Without assuming the symmetry of synaptic connection weights and the monotonicity and differentiability of activation functions, Lyapunov functionals and Halanay-type inequalities are constructed and employed to derive delay independent sufficient conditions under which the continuous-time networks converge exponentially to the equilibria associated with temporally uniform external inputs to the networks. Discrete-time analogues of the continuous-time networks are formulated and we study their dynamical characteristics. It is shown that the convergence dynamics of the continuous-time networks are preserved by the discrete-time analogues without any restriction on the discretization step-size. Several examples are given to illustrate the advantages of the discrete-time analogues in numerically simulating the continuous-time networks.

Discrete-time analogues of integrodifferential equations modelling bidirectional neural networks

We formulate discrete-time analogues of integrodifferential equations modelling bidirectional neural networks studied by Gopalsamy and He. The discrete-time analogues are considered to be numerical discretizations of the continuous-time networks and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the equilibria of the continuous-time networks. By constructing a Lyapunov-type sequence, we obtain easily verifiable sufficient conditions under which every solution of the discrete-time analogue converges exponentially to the unique equilibrium. The sufficient conditions are identical to those obtained by Gopalsamy and He for the uniqueness and global asymptotic stability of the equilibrium of the continuous-time network. By constructing discrete-time versions of Halanay-type inequalities, we obtain another set of easily verifiable sufficient conditions for the global exponential stability of the unique equilibrium of the discrete-time analogue. The latter sufficient conditions have not been obtained in the literature of continuous-time bidirectional neural networks. Several computer simulations are provided to illustrate the advantages of our discrete-time analogue in numerically simulating the continuous-time network with distributed delays over finite intervals.

Multistability analysis for a general class of delayed Cohen-Grossberg neural networks

In this paper, by discussing parameter conditions based on properties of activation functions, we decompose state space into positively invariant sets and establish sufficient conditions for the existence of locally stable equilibria for delayed Cohen-Grossberg neural networks (CGNNs) through Cauchy convergence principle. Some new criteria are derived for ensuring equilibria (periodic orbits) to be locally or globally exponentially stable in any designated region. Finally, our results are demonstrated by four numerical simulations.

Convergence Dynamics of Delayed Hopfield-Type Neural Networks Under Almost Periodic Stimuli

Convergence dynamics of Hopfield-type neural networks subjected to almost periodic external stimuli are investigated. In this article, we assume that the network parameters vary almost periodically with time and we incorporate variable delays in the processing part of the network architectures. By employing Halanay inequalities, we obtain delay independent sufficient conditions for the networks to converge exponentially toward encoded patterns associated with the external stimuli. The networks are guaranteed to have exponentially hetero-associative stable encoding of the external stimuli.

New results on exponential attractivity of multiple almost periodic solutions of cellular neural networks with time-varying delays

We present new results on multi-almost periodicity and attractivity of delayed cellular neural networks (DCNNs). Some criteria are derived for ensuring locally or globally exponential attractivity of multiple almost periodic solutions in designated regions. It is shown that N-dimensional DCNNs can have a coexistence of 2^N locally attractive almost periodic solutions. Furthermore, the obtained conclusions have a wider applicable range, improve and complement the existing ones. Finally, computer simulations are given to show the effectiveness of the theoretical results.

Multistability of HNNs with almost periodic stimuli and continuously distributed delays

In this article, we investigate multistability of Hopfield neural networks (HNNs) with almost periodic stimuli and continuously distributed delays. By employing the theory of exponential dichotomy and Schauders fixed point theorem, sufficient conditions are gained for the existence of 2 N almost periodic solutions which lie in invariant regions. Meanwhile, we derive some new criteria for the networks to converge toward these 2 N almost periodic solutions and the domain of attraction is also given. The obtained results are new, general and improve corresponding results existing in previous literature.

2N almost periodic attractors for CNNs with variable and distributed delays

Abstract In this paper, we investigate dynamics of 2 N almost periodic attractors for cellular neural networks (CNNs) with variable and distributed delays. By imposing some new assumptions on activation functions and system parameters, we split invariant basin of CNNs into 2 N compact convex subsets. Then the existence of 2 N almost periodic solutions lying in compact convex subsets is attained due to employment of the theory of exponential dichotomy and Schauders fixed point theorem. Meanwhile, we derive some new criteria for the networks to converge toward these 2 N almost periodic solutions and exponential attracting domains are also given correspondingly. The obtained results are new and can be applied to a large class of neural networks.

Computer simulations of exponentially convergent networks with large impulses

This paper demonstrates the use of a semi-discretization technique for obtaining a discrete-time analogue of an exponentially convergent network that is subject to impulses with large magnitude. Prior to implementing the analogue for computer simulations, we investigate its exponential convergence towards a unique equilibrium state and thereby obtain a family of sufficiency conditions governing the network parameters and the impulse magnitude and frequency. Although the time-step does not appear in the conditions that govern the network parameters, its value needs to be sufficiently small in order for the analogue displays correct convergence behaviour of the network when subjected particularly to large impulses.

A unified treatment for stability preservation in computer simulations of impulsive BAM networks

This paper is concerned with the stability preservation in computer simulations of an impulsive bidirectional associative memory (BAM) network. The simulations are provided by difference equations formulated from a semi-discretisation technique and impulsive maps as discrete-time representations of the nonlinear impulses which attempt to destabilise the BAM network at fixed moments of time. Prior to producing the computer simulations, the analogue is analysed for its exponential convergence towards a unique equilibrium state. The analysis exploits the method of Lyapunov sequences to derive several sufficient conditions that govern the network parameters and the impulse magnitude and frequency. As special cases, one can obtain from our results, those corresponding to the non-impulsive discrete-time BAM networks and also those corresponding to continuous-time (impulsive and non-impulsive) systems. The treatment of the analysis leads us to a relation between the Lyapunov exponent of the non-impulsive system and that of the impulsive system involving the size of the impulses and the inter-impulse intervals.

EXTREME STABILITY AND ALMOST PERIODICITY IN CONTINUOUS AND DISCRETE NEURONAL MODELS WITH FINITE DELAYS

We consider the dynamical characteristics of a continuous-time isolated Hopfield-type neuron subjected to an almost periodic external stimulus. The model neuron is assumed to be dissipative having finite time delays in the process of encoding the external input stimulus and recalling the encoded pattern associated with the external stimulus. By using non-autonomous Halanay-type inequalities we obtain sufficient conditions for the hetero-associative stable encoding of temporally non-uniform stimuli. A brief study of a discrete-time model derived from the continuous-time system is given. It is shown that the discrete-time model preserves the stability conditions of the continuous-time system.