Strictly convex metric spaces with round balls and fixed points

Eventually periodic solutions of single neuron model

In this paper, we consider a nonautonomous piecewise linear difference equation that describes a discrete version of a single neuron model with a periodic (period two and period three) internal decay rate. We investigated the periodic behavior of solutions relative to the periodic internal decay rate in our previous papers. Our goal is to prove that this model contains a large quantity of initial conditions that generate eventually periodic solutions. We will show that only periodic solutions and eventually periodic solutions exist in several cases.

Periodic Solutions of the Second Order Quadratic Rational Difference Equation $$x_{n+1}=\frac{\alpha }{(1+x_n)x_{n-1}} $$ x n + 1 = α ( 1 + x n ) x n - 1

The aim of this article is to investigate the periodic nature of solutions of a rational difference equation $$x_{n+1}=\frac{\alpha }{(1+x_n)x_{n-1}}. {(*)} $$ We explore Open Problem 3.3 given in Amleh et al. (Int J Differ Equ 3(1):1–35, 2008, [2]) that requires to determine all periodic solutions of the equation (*). We conclude that for the equation (*) there are no periodic solution with prime period 3 and 4. Period 7 is first period for which exists nonnegative parameter \(\alpha \) and nonnegative initial conditions.

Periodic orbits of a neuron model with periodic internal decay rate

In this paper we will study a non-autonomous piecewise linear difference equation which describes a discrete version of a single neuron model with a periodic internal decay rate. We will investigate the periodic behavior of solutions relative to the periodic internal decay rate. Furthermore, we will show that only periodic orbits of even periods can exist and show their stability character.

Arrow-Hahn economic models with weakened conditions of continuity

On Uncertain Discontinuous Functions and Quasi-equilibrium in Some Economic Models

In the paper is studied some properties of uncertain discontinuous mappings, the so-called w-discontinuous mappings. Based on them, the existence of a quasi-equilibrium for a new economic model is proved.

On the stability of the Bohl — Brouwer — Schauder Theorem

Periodic orbits of single neuron models with internal decay rate 0 < β ≤ 1

In this paper we consider a discrete dynamical system x n+1=βx n – g(x n ), n=0,1,..., arising as a discrete-time network of a single neuron, where 0 &lt; β ≤ 1 is an internal decay rate, g is a signal function. A great deal of work has been done when the signal function is a sigmoid function. However, a signal function of McCulloch-Pitts nonlinearity described with a piecewise constant function is also useful in the modelling of neural networks. We investigate a more complicated step signal function (function that is similar to the sigmoid function) and we will prove some results about the periodicity of solutions of the considered difference equation. These results show the complexity of …

Periodic and Chaotic Orbits of a Neuron Model

In this paper we study a class of difference equations which describes a discrete version of a single neuron model. We consider a generalization of the original McCulloch-Pitts model that has two thresholds. Periodic orbits are investigated accordingly to the different range of parameters. For some parameters sufficient conditions for periodic orbits of arbitrary periods have been obtained. We conclude that there exist values of parameters such that the function in the model has chaotic orbits. Models with chaotic orbits are not predictable in long-term.

Chaotic mappings in symbol space

Der Rigaer Deutsch-Baltische Mathematiker Piers Bohl (1865–1921)

Vor kurzem jahrte sich zum 125. Male der Geburtstag und zum 70. Male der Todestag eines der bedeutendsten, vielleicht sogar des bedeutendsten Mathematikers Lettlands, Piers Bohls. Er lebte unter wechselnden politischen Regimen, aber verstand es stets, sich seiner Arbeit zu widmen. Piers Bohl wurde am 23. Oktober 1865 als Spros einer deutschen Kaufmannsfamilie im Stadtchen Walk (an der Grenze Lettlands und Estlands) geboren. Uber seine fruhe Kindheit scheint nichts bekannt zu sein. Ersten Unterricht erhielt er durch Privatlehrer, er besuchte dann die stadtische Elementarschule zu Walk sowie das livlandische ritterschaftliche Landesgymnasium in Fellin, dem heutigen estnischen Wiland. Fast gle…

On the population model with a sine function

In the interval [0,1] function sr(x) = r sin πx behaves similar to logistic function h μ (x) = μx(1‐ x). We prove that for every r &gt; there exists subset ? ⊂ [0,1] such that sr : ? → ? is a chaotic function. Since the logistic function is chaotic in another subset of [0,1] but both functions have similar graphs in [0,1] we conclude that it can lead to errors in practice. First Published Online: 14 Oct 2010

Construction of chaotic dynamical system

The first‐order difference equation xn+ 1 = f(xn ), n = 0,1,…, where f: R → R, is referred as an one‐dimensional discrete dynamical system. If function f is a chaotic mapping, then we talk about chaotic dynamical system. Models with chaotic mappings are not predictable in long‐term. In this paper we consider family of chaotic mappings in symbol space S 2. We use the idea of topological semi‐conjugacy and so we can construct a family of mappings in the unit segment such that it is chaotic. First published online: 09 Jun 2011