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In this note, we study a discontinuous three-term recurrence relation which arises from seeking the steady states of a cellular neural network with step control function. Several collections of per...

Simple dynamic systems representing time varying states of interconnected neurons may exhibit extremely complex behaviors when bifurcation parameters are switched from one set of values to another. In this paper, motivated by simulation results, we examine the steady states of one such system with bang-bang control and two real parameters. We found that nonnegative and negative periodic states are of special interests since these states are solutions of linear nonhomogeneous three-term recurrence relations. Although the standard approach to analyse such recurrence relations is the method of finding the general solutions by means of variation of parameters, we find novel alternate geometric methods that offer the tracking of solution trajectories in the plane. By means of this geometric approach, we are then able, without much tedious computation, to completely characterize the nonnegative and negative periodic solutions in terms of the bifurcation parameters.

It is mainly proved: Let F be a family of meromorphic function in D, a(z)(6= 0) and b(z)( 6≡ 0) be two holomorphic functions on D. Suppose that admits the zeros of multiplicity at least 3 for each function f ∈ F. For each f ∈ F, if f = a(z) ⇔ f ′ = b(z) , then F is normal in D. Some example shows that the multiplicity of zeros of f is best in some sense. And the result of paper improve and supplement the result of Lei, Yang and Fang [J. Math. Anal. App. 364 (2010), 143–150].

In 1973, Kopell and Howard introduced a λ–ω reaction–diffusion system and found an explicit family of periodic travelling wave solutions lying on circles with radius less than 1. Since λ–ω systems represent universal models for studying chemical processes, and onset of turbulent behaviour, etc., explicit solutions of λ–ω systems with delays or discrete λ–ω systems can be of further help when the only method for obtaining other solutions is through numerical computation. There are now much investigations of various λ–ω systems. However, it is of interest to note that none attempts to find explicit travelling wave solutions. In this paper, we investigate the existence of explicit solutions for the simplest Euler scheme of a λ–ω system with delays or advancements which is described as a coupled pair of partial difference equations. We are able to provide necessary as well as sufficient conditions for the existence of numerical periodic travelling wave solutions. Additionally, we also provide some examples to...

Periodic travelling wave solutions of reaction–diffusion equations were studied by many authors. The λ – ω type reaction–diffusion system is a notable special model that admits explicit periodic travelling wave solutions and was introduced by Kopell and Howard in 1973. There are now similar systems which are investigated by means of autonomous dynamics. In contrast, there are few papers which are concerned with non-autonomous cases. For this reason, we apply Mawhin’s continuation theorem to derive the existence of periodic travelling wave solutions for non-autonomous λ – ω systems, and we describe the ‘disappearance’ of periodic travelling wave solutions under special situations. Our main result is also illustrated by examples.

Discontinuous dynamical systems arise in major engineering problems involving discontinuous control functions such as the Heaviside function H. Yet complete analyses of these models are extremely difficult and to our knowledge, even the prototype mathematical model defined over the set of integers has not been studied. In this paper, we give a complete analysis of its periodic behaviours. In particular, we show that any solution is periodic and can be determined by two of its consecutive terms, that there are solutions with arbitrary large periods and that there are exactly four types of periodic solutions. As applications, we can then give exact steady-state solutions to neural networks with the above equation as its steady-state equation, and we can also provide the exact periodic behaviours of the complex dynamical system . It is hoped that the techniques used in this paper can be applied to many other discontinuous dynamical systems in the future.

In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation where η > 0, p, and g are continuous functions on [0,∞) such that p(t) ≥ 0, g(t) ≤ t, g′(t) ≥ α > 0, and lim t→∞ g(t) =∞. It is important to note that the condition g′(t) ≥ α > 0 is required. In the paper, we remove this restriction under the superlinear assumption η > 1. In fact, we can do even better by considering impulsive differential equations with delay and obtain necessary and sufficient conditions for the existence of nonoscillatory solutions and also a comparison theorem that enables us to apply known oscillation results for impulsive equations without forcing terms to get oscillation criteria for the analyzed equations. У 1997 роцi, А. Х. Наср отримав необхiднi та достатнi осциляцiйнi умови для рiвняння де η>0, p та g — неперервнi функцiї на [0,∞) такi, що p(t)≥0,g(t)≤t,g′(t)≥α>0 та limt→∞g(t)=∞. Слiд зауважити, що необхiдною тут є умова g′(t)≥α>0. У данiй статтi ми усуваємо це обмеження при суперлiнiйному припущеннi η>0. Насправдi, можна отримати навiть кращий результат, розглядаючи iмпульснi диференцiальнi рiвняння з запiзненням, i встановити необхiднi та достатнi умови iснування неосциляцiйних розв’язкiв, а також теорему порiвняння, яка дає змогу застосувати вiдомi осциляцiйнi результати для iмпульсних рiвнянь без збурюючих членiв, щоб отримати осциляцiйнi критерiї для наших рiвнянь.

Monotonicity of solutions is an important property in the investigation of oscillatory behaviors of differential equations. A number of papers provide some existence criteria for eventually positive increasing solutions. However, relatively little attention is paid to eventually positive solutions that are also eventually decreasing solutions. For this reason, we establish several new and sharp oscillatory criteria for impulsive functional differential equations from this viewpoint.

Second order periodic differential equations such as the Hill’s equation are used to model damped oscillators, vibrating elliptical drumheads, rotating electric dipoles, etc. and existence of periodic solutions are important. In this paper we study second order equations subject to ‘adjustable’ feedback control functions with delays. Based on the fixed point theorem for an ordered Banach space, the existence, multiplicity, and the nonexistence of periodic solutions for our equation are obtained. The uniqueness of periodic solutions and the dependence of periodic solutions on the adjustable scale factor are also studied.

Doubly periodic travelling waves can be used to describe dynamic patterns of signals that govern movements of animals. In this paper, we study the existence of such waves in cellular networks involving the discontinuous Heaviside step function. This is done by finding ω-periodic solutions of an accompanying recurrence relation with a priori unknown parameters and the Heaviside function. Since analytic tools cannot be used to handle discontinuous models such as ours, existence of periodic solutions is investigated by means of symmetry, combinatorial techniques and accompanying linear systems. By such means, we are able to obtain all periodic solutions with least periods 1 through 6. Our techniques are new and good for other periodic solutions with relatively small periods.

In this paper, we study a very simple three term recurrence relation involving the discontinuous Heaviside step function. One reason for studying such an relation is that solutions of our recurrence relation are steady state distributions in some basic neural network models. Since analytic tools cannot be used to handle discontinuous models such as ours, existence of periodic solutions is investigated by combining combinatorial elimination technique as well as existence arguments for linear systems. By such means, we are able to obtain all periodic solutions with least periods 1 through 8. Some periodic solutions with periods 12, 20 and 36 can also be found, but exhaustive results are not yet available.

Positive and negative steady state solutions of a neural network are sought as critical points of functionals. We approach our problems in an elementary manner and our results are easy to use. A number of illustrative examples are also provided.

This paper is concerned with upper bounds for the distances between adjacent zeros of solutions of first-order linear delay differential equations with variable coefficients of the form x ′ ( t ) + p ( t ) x ( t - τ ) = 0 , t ⩾ t ∗ , where τ > 0 and p ( t ) ∈ C ([ t ∗ , ∞), R ). By introducing a sequence of testing functions, we are able to derive upper bounds that are better than those in the literature.

In a study (Szekely, 1965) [1] of the locomotion of salamanders, it is observed that a 'doubly periodic traveling wave solution' of a logical neural network can be used to explain a dynamic pattern of movements. We show here that nonlinear and nonlogical artificial neural network can also be built by means of reaction diffusion principles and existence or nonexistence of doubly periodic traveling waves can be guaranteed by adjusting parameters built into these networks. Our derivations for existence are based on implicit function theorems and the invariance properties of our model; while nonexistence is based on boundedness properties of the polynomial reaction term. We also give illustrative examples as well as comments on the differences between present results with those obtained for linear models studied earlier in Cheng and Lin (2009) [2,3].

We show that in an artificial dynamic neural network that depends on a real parameter μ, steady states do not exist for μ ≤ -2, and positive and negative steady states exist for μ > -2. We hope that such a bifurcation phenomenon in our network model may explain some of the real observations in nature.

In this paper, the simultaneous existence of positive, negative and sign-changing periodic solutions for a class of integral equations of the form @f(x)[email protected]!"["x","x"+"@w"]"@?"@WK(x,y)f(y,@f([email protected](y)))dy,[email protected][email protected], is considered, where @W is a closed subset of R^N with a periodic structure. Our main result is different from most existing results since they provide three constant sign periodic solutions only.

The model of hematopoiesis proposed by Mackey and Glass (1977) [1] has been studied by many authors and sufficient conditions are obtained for the global attractivity of the positive equilibrium. Here we obtain another sufficient condition which is relatively simple.

We consider a difference equation involving three parameters and a piecewise constant control function with an additional positive threshold . Treating the threshold as a bifurcation parameter that varies between 0 and , we work out a complete asymptotic and bifurcation analysis. Among other things, we show that all solutions either tend to a limit 1-cycle or to a limit 2-cycle and, we find the exact regions of attraction for these cycles depending on the size of the threshold. In particular, we show that when the threshold is either small or large, there is only one corresponding limit 1-cycle which is globally attractive. It is hoped that the results obtained here will be useful in understanding interacting network models involving piecewise constant control functions.

Biological systems are able to switch their neural systems into inhibitory states and it is therefore important to build mathematical models that can explain such phenomena. If we interpret such inhibitory modes as `positive' or `negative' steady states of neural networks, then we will need to find the corresponding fixed points. This paper shows positive fixed point theorems for a particular class of cellular neural networks whose neuron units are placed at the vertices of a regular polygon. The derivation is based on elementary analysis. However, it is hoped that our easy fixed point theorems have potential applications in exploring stationary states of similar biological network models.

In a study (Szekely, Acta Physiol. Hung. 27 (1965), pp. 285–289) of the locomotion of salamanders, it is observed that a ‘doubly periodic travelling wave solution’ of a logical neural network can be used to explain a dynamic pattern of movements. We show here that a relatively simple (nonlogical) artificial neural network can also be built and necessary and sufficient conditions for the existence of doubly periodic travelling wave solutions can be found. It is hoped that our investigation will set some foundation in the future design of other artificial neural networks that also allow periodic travelling wave solutions.

A multi-point boundary value problem involving the one dimensional $p$-Laplacian and depending on a parameter is studied in this paper and existence of positive solutions is established by means of a fixed point theorem for operators defined on Banach spaces with cones.

In this paper, by means of frequency measure (also called asymptotic density or natural density) defined for integer subsets, we obtain some new explicit sufficient and necessary conditions for real sequences to be frequently convergent. These conditions involve frequent Cauchy sequences and statistical pre-Cauchy sequences as well as statistical Cauchy sequences.