In this paper, we investigate the dynamical behaviors of the rational difference equation $x_n=\left(a{x}_{n-1}\right)/\left(b+c{x}_n{x}_{n-1}\right)$ with arbitrary initial conditions, where a, b, and c are real numbers. A general solution is obtained. The asymptotic stability of the equilibrium points is investigated, using a nonlinear stability criterion combined with basin of attraction analysis and simulation to determine the stability regions of the equilibrium points. The existence of the periodic solutions is discussed. We investigate the codim-1 bifurcations of the equation. We show that the equation exhibits a Neimark–Sacker bifurcation. For this bifurcation, the topological normal form is computed. To confirm our theoretical results, we performed a numerical simulation as well as numerical bifurcation analysis by using the Matlab package MatContM.

In this paper, we consider the explicit solution of the following system of nonlinear rational difference equations: $x_{n+1}=x_{n-1}/\left(x_{n-1}+r\right)$ , $y_{n+1}=x_{n-1}{y}_n/\left(x_{n-1}{y}_n+r\right)$ , with initial conditions $x_{-1}$ , $x_0$ and $y_0$ , which are arbitrary positive real numbers. By doing this, we encounter the hypergeometric function. We also investigate global dynamics of this system. The global dynamics of this system consists of two kind of bifurcations.

In this paper, we discuss and mathematically compute the eigenvalues and the characteristic polynomials of special even square matrices of orders 4x4 and 8x8. Also, we introduce two 8th order compound magic squares. The computed values are verified using Maple software. First, for the 4th order square matrix, the characteristic polynomial was derived to be: λ(λ-2s)( λ²+4Θ) with the eigenvalues: 0,2 s, and two other conjugates. In further analysis, we performed numerical classification of the squares for the matrices of order 4. Second, for the 8th order magic square, the characteristic polynomial was obtained in the form: λ3(λ-4s)(λ4+Ωλ2+θ) where Ω,Θ are constants; the eigenvalues are 0,4 s, ∓√λ1, ∓√λ2; where λ1, λ2 are the roots of the quadratic equation: λ2+Ωλ+Θ=0. Third, for the franklin square, we obtained the eigenvalues 0,4 s, and the roots of the equation: λ2+aλ+b. Finally, we suggested a hybrid image encryption technique based on Franklin magic square matrices and improved substitution technique. The proposed a grayscale image encryption/ decryption algorithm uses circular rotation of bits and Franklin magic squares’ properties in conjunction with substitution techniques to obtain a very secure algorithm against attacks.

American scientific publishing group

This paper, we study systems of difference equations numerically and theoretically. These systems have been examined by many researchers. We focus on their general forms and limits. We consider different orders of difference systems. In certain cases, we use a computer in order to verify the summation laws that we encountered. We consider a system with two parameters. For certain values of the parameters, we determine the explicit form of the solution and show that the limit of this sequence tends to zero. We show that for the values of a parameter between 0 and 1, the limit of the sequence is a nonzero value, while for the values greater than 1, the limit of the sequence becomes zero.