This paper is concerned with a class of essentially strongly order-preserving semiflows, which are defined on an ordered metric space and are generalizations of strongly order-preserving semiflows. For essentially strongly order-preserving semiflows, we prove several principles, which are analogues of the nonordering principle for limit sets, the limit set dichtomy and the sequential limit set trichotomy for strongly order-preserving semiflows. Then, under certain compactness hypotheses, we obtain some results on convergence, quasiconvergence and stability in essentially strongly order-preserving semiflows. Finally, some applications are made to quasimonotone systems of delay differential equations and reaction–diffusion equations with delay, and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption.

Without assuming boundedness and differentiability of the activation functions and any symmetry of interconnections, we employ Lyapunov functions to establish some sufficient conditions ensuring existence, uniqueness, global asymptotic stability, and even global exponential stability of equilibria for the Cohen–Grossberg neural networks with and without delays. Our results are not only presented in terms of system parameters and can be easily verified and also less restrictive than previously known criteria and can be applied to neural networks, including Hopeld neural networks, bidirectional association memory neural networks, and cellular neural networks.

In this paper, we study the effect of synaptic delay of signal transmission on the pattern formation and some properties of non-linear waves in a ring of identical neurons. First, linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. Regarding the delay as a bifurcation parameter, we obtained the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Second, global continuation conditions for Hopf bifurcating periodic orbits are derived by using the equivariant degree theory developed by Geba et al. and independently by Ize & Vignoli. Third, we show that the coincidence of these periodic solutions is completely determined either by a scalar delay differential equation if the number of neurons is odd, or by a system of two coupled delay differential equations if the number of neurons is even. Fourth, we summarize some important results about the properties of Hopf bifurcating periodic orbits, including the direction of Hopf bifurcation, stability of the Hopf bifurcating periodic orbits, and so on. Fifth, in an excitatory ring network, solutions of most initial conditions tend to stable equilibria, the boundary separating the basin of attraction of these stable equilibria contains all of periodic orbits and homoclinic orbits. Finally, we discuss a trineuron network to illustrate the theoretical results obtained in this paper and conclude that these theoretical results are important to complement the experimental and numerical observations made in living neurons systems and articial neural networks, in order to understand the mechanisms underlying the system dynamics better.

Considered is a system of delay differential equations modeling a time-delayed connecting network of three neurons without selffeedback. We investigate the linear stability of the trivial solution and Hopf bifurcation of this system. The general formula for the direction, the estimation formula of period and stability of Hopf bifurcating periodic solution are also given.

In this paper, we consider a class of pseudo monotone semiflows, which only enjoy some weak monotonicity properties and are defined on product-ordered topological spaces. Under certain conditions, several convergence principles are established for each precompact orbit of such a class of semiflows to tend to an equilibrium, which improve and extend some corresponding results already known. Some applications to delay differential equations are presented.

In this paper, we consider a ring of neurons with self-feedback and delays. As a result of our approach based on global bifurcation theorems of delay differential equations coupled with representation theory of Lie groups, the coexistence of its asynchronous periodic solutions (i.e. mirror-reflecting waves, standing waves and discrete waves), bifurcated simultaneously from the trivial solution at some critical values of the delay, will be established for delay not only near to but also far away from the critical values. Therefore, we can obtain wave solutions of large amplitudes. In addition, we consider the coincidence of these periodic solutions.

We consider a delayed network of two neurons with both self-feedback and interaction described by an all-or-none threshold function. The model describes a combination of analog and digital signal processing in the network and takes the form of a system of delay differential equations with discontinuous nonlinearity. We show that the dynamics of the network can be understood in terms of the iteration of a one-dimensional map, and we obtain simple criteria for the convergence of solutions, the existence, multiplicity and attractivity of periodic solutions.

Employing Brouwer’s xed point theorem, matrix theory, a continuation theorem of the coincidence degree and inequality analysis, the authors make a further investigation of a class of cellular neural networks with delays (DCNNs) in this Letter. A family of sufcient conditions are given for checking global exponential stability and the existence of periodic solutions of DCNNs. These results have important leading significance in the design and applications of globally stable DCNNs and periodic oscillatory DCNNs. Our results extend and improve some earlier publications.

In this paper, the convergence of a class of systems of delay differential equations is considered. We show that every bounded solution of such systems tends to an equilibrium under certain hypotheses. Our results extend some corresponding ones already known.

Lyapunov functionals and Lyapunov functions coupled with the Razumikhin technique are still the most popular tools in studying the stability of large-scale retarded nonlinear systems. However, it is generally difficult to construct Lyapunov functionals or functions that satisfy the strong conditions required in the classical stability theory. In this paper, we show that for some delay differential systems such as additive neural networks with delays, we can weaken the condition that the Lyapunov functional or function is positive definite, by using the equivalence between the state stability and the output stability. We apply our general theory to obtain some new stability conditions for cellular neural network models. It is represented that it is easy to construct Lyapunov functionals or functions satisfied conditions of our theorems.