0000000000827715
AUTHOR
Aija Anisimova
On the Second Order Rational Difference Equation $$x_{n+1}=\beta +\frac{1}{x_n x_{n-1}}$$ x n + 1 = β + 1 x n x n - 1
The author investigates the local and global stability character, the periodic nature, and the boundedness of solutions of the second-order rational difference equation $$x_{n+1}=\beta +\frac{1}{x_{n}x_{n-1}}, \quad n=0,1,\ldots ,$$ with parameter \(\beta \) and with arbitrary initial conditions such that the denominator is always positive. The main goal of the paper is to confirm Conjecture 8.1 and to solve Open Problem 8.2 stated by A.M. Amleh, E. Camouzis and G. Ladas in On the Dynamics of a Rational Difference Equations I (International Journal of Difference Equations, Volume 3, Number 1, 2008, pp.1–35).
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 < 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.