Rational Difference Equations Research Articles

A Generalised Difference Equation and Its Dynamics and Solutions

Rational difference equations have a wide range of applications in various fields of science. To illustrate, the equation xn+1=a+bxnc+dxn,n=0,1,..., known as the Riccati difference equation, has been applied in the field of optics. In this study, the global asymptotic stability of the difference equation xn+1=Axn−2k+j+1B+Cxn−(k+j)xn−2k+j+1,n=0,1,..., is proved. The solutions of this difference equation are obtained by applying the standard iteration method, and the periodicity of these solutions is determined. Furthermore, this difference equation represents a generalisation of the results obtained in previous studies.

Periodically and Stability Properties of a Higher Order Rational Difference Equation

The aim of this research is to study the local, global, and boundedness of the difference equationTη+1 = r + p1Tη−l1 / Tη−m1 + p2Tη−l1 / Tη−m2 + ... + psTη−l1 / Tη−ms,where l1, m1, m2, ..., ms, s, are positive real numbers. It also studies periodic solutions of special case of this equation. Finally, numerical examples are given to confirm results.

Neimark–Sacker bifurcation of two second-order rational difference equations

We investigate the Neimark–Sacker bifurcation of the equilibrium of two special cases of the difference Equation x n + 1 = β x n x n − 1 + γ x n − 1 2 + δ x n B x n x n − 1 + C x n − 1 2 + D x n where the parameters β, γ, δ, B, C, D are non-negative numbers which satisfy B + C + D>0 and the initial conditions x − 1 and x 0 are arbitrary nonnegative numbers such that B x n x n − 1 + C x n − 1 2 + D x n > 0 for all n ≥ 0 . More precisely, we consider special cases where either γ = D = 0 or β = D = 0 . As we will show both equations exhibit Neimark–Sacker bifurcation, where one of equations ( γ = D = 0 ) probably exhibits Chenciner bifurcation, with two invariant curves, while another Equation ( β = D = 0 ) exhibits simple Neimark–Sacker bifurcation with one invariant curve. We will also obtain some global asymptotic stability result for each equation in the subset of the parametric region of local stability.

On the Laplacian index of tadpole graphs

Abstract In this article, we study the Laplacian index of tadpole graphs, which are unicyclic graphs formed by adding an edge between a cycle C k {C}_{k} and a path P n {P}_{n} . Using two different approaches, we show that their Laplacian index converges to Δ 2 Δ − 1 = 9 2 \frac{{\Delta }^{2}}{\Delta -1}=\frac{9}{2} as n , k → ∞ n,k\to \infty , where Δ = 3 \Delta =3 is the maximum degree of the graph. This limit is known as the Hoffman’s limit for the Laplacian matrix. The first technique is a linear time algorithm presented in [R. Braga, V. Rodrigues, and R. Silva, Locating eigenvalues of a symmetric matrix whose graph is unicyclic, Trends in Comput. Appl. Math. 22 (2021), no. 4, 659–674] that diagonalizes the matrix preserving its inertia. Here, we adapt this algorithm to the Laplacian index of a tadpole graph. The second method is to apply a formula presented in [V. Trevisan and E. R. Oliveira, Applications of rational difference equations to spectra graph theory, J. Difference Equ. Appl. 27 (2021), 1024–1051] for solving rational difference equations that appear when applying the diagonalization algorithm in some cases.

On a Solvable Systems of Third Order Rational Difference Equations

On a Solvable Systems of Third Order Rational Difference Equations

Qualitative Behavior of Sixth Order of Rational Difference Equation

The field of difference equations (DEs)has gained considerable prominence in applied analysis. The primary aim of this research is to conduct a comprehensive analysis on the periodicity of solutions, local asymptotic stability, and global behavior of DEsUn+1=sUn−2+tUn−5+hUn−2+rUn−5/cUn−5−e, n=0,1,2, . . . .

An Example of a Globally Asymptotically Stable Anti-monotonic System of Rational Difference Equations in the Plane

An Example of a Globally Asymptotically Stable Anti-monotonic System of Rational Difference Equations in the Plane

Analysis and qualitative behaviour of a tenth-order rational difference equation

In this article, we examine the qualitative behavior of the solutionsof the following di¤erence equationzn+1 = aZn-4 +bZn-4/cZn-4-dzn4; n = 0,1,....where the initial conditions Z_9; Z_8; Z_7; Z_6; Z_5; Z_4; Z_3; Z_2; Z_1;Z0 are arbitrary non-zero real numbers and a, b, c, d are positive constants.

Dynamics and general form of the solutions of rational difference equations

Dynamics and general form of the solutions of rational difference equations

A qualitative investigation of some rational difference equations

In this paper, we study some qualitative properties of the solutions for the following difference equation% \[ y_{n+1}=\frac{\alpha +\alpha _{0} y_{n}^{r}+\alpha _{1} y_{n-1}^{r}+\cdots+\alpha _{k} y_{n-k}^{r}}{\beta +\beta _{0} y_{n}^{r}+\beta _{1} y_{n-1}^{r}+\cdots+\beta _{k} y_{n-k}^{r}}% ,~~~~n\geq 0,\tag{I}\label{i} \] where \(r, \alpha , \alpha _{0}, \alpha _{1},\ldots, \alpha _{k}, \beta , \beta _{0}, \beta _{1},\ldots, \beta _{k}\in (0,\infty )\) and \(k \) is a non-negative integer number. We find the equilibrium points for the considered equation. Then classify these points in terms of local stability or not. We investigate the boundedness and the global stability of the solutions for the considered equation. Also, we study the existence of periodic solutions of Eq. (I).

Global attractivity of a rational difference equation with higher order and its applications

<p>We study in this paper the global attractivity for a higher order rational difference equation. As application, our results not only include and generalize many known ones, but also formulate some new results for several conjectures presented by Camouzis and Ladas, et al.</p>

Global stability and co-balancing numbers in a system of rational difference equations

<abstract><p>This paper investigates both the local and global stability of a system of rational difference equations and its connection to co-balancing numbers. The study delves into the intricate dynamics of mathematical models and their stability properties, emphasizing the broader implications of global stability. Additionally, the investigation extends to the role of co-balancing numbers, elucidating their significance in achieving equilibrium within the solutions of the rational difference equations. The interplay between global stability and co-balancing numbers forms a foundational aspect of the analysis. The findings contribute to a deeper understanding of the mathematical structures underlying dynamic systems and offer insights into the factors influencing their stability and equilibrium. This article serves as a valuable resource for mathematicians, researchers, and scholars interested in the intersection of global stability and co-balancing sequences in the realm of rational difference equations. Moreover, the presented examples and figures consistently demonstrate the global asymptotic stability of the equilibrium point throughout the paper.</p></abstract>

On the solutions of some systems of rational difference equations

<p>In this paper, we considered some systems of rational difference equations of higher order as follows</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} u_{n+1} & = &\frac{v_{n-6}}{1\pm v_{n}u_{n-1}v_{n-2}u_{n-3}v_{n-4}u_{n-5}v_{n-6}}, \\ v_{n+1} & = &\frac{u_{n-6}}{1\pm u_{n}v_{n-1}u_{n-2}v_{n-3}u_{n-4}v_{n-5}u_{n-6}}, \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p>where the initial conditions $ u_{0, } $ $ u_{-1}, $ $ u_{-2}, $ $ u_{-3}, $ $ u_{-4}, $ $ u_{-5}, $ $ u_{-6}, $ $ v_{0, } $ $ v_{-1}, $ $ v_{-2}, $ $ v_{-3}, $ $ v_{-4}, $ $ v_{-5} $ and $ v_{-6} $ were arbitrary real numbers. We obtained a closed form of the solutions for each considered system and also some periodic solutions of some systems were found. We presented some numerical examples to explain the obtained theoretical results.</p>

Experimenting with discrete dynamical systems

We demonstrate the power of Experimental Mathematics and Symbolic Computation to study intriguing problems on rational difference equations, studied extensively by Difference Equations giants, Saber Elaydi and Gerry Ladas (and their students and collaborators). In particular we rigorously prove some fascinating conjectures made by Amal Amleh and Gerry Ladas back in 2000. For other conjectures we are content with semi-rigorous proofs. We also extend the work of Emilie Purvine (formerly Hogan) and Doron Zeilberger for rigorously and semi-rigorously proving global asymptotic stability of arbitrary rational difference equations (with positive coefficients), and more generally rational transformations of the positive orthant of R k into itself.

ON PERIODIC SOLUTIONS OF SYSTEM OF GENERALIZED RATIONAL DIFFERENCE EQUATIONS

In this article, the authors establish periodic solutions for system of generalized rational difference equations (GRDE) with nonzero initial conditions. Additionally, we discussed the sensitivity of the GRDE’s generic kl variables.

On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations

The principle goal of this paper is to look at some of the qualitative behavior of the critical point of the rational difference equation
 
 Ψ_{n+1}=αΨ_{n-2}+((βΨ_{n-2}Ψ_{n-3})/(γΨ_{n-3}+δΨ_{n-6})), n=0,1,2,...,
 
 where α,β,γ and δ are arbitrary positive real numbers. We also used the proposed equation to get the general solution for particular cases and provided numerical examples to demonstrate our results.

Form of the solutions of difference equations via Lie symmetry analysis and Fibonacci numbers

In this paper, we give the form of the solutions of the following rational difference equation where an and bn are sequences of real numbers, by using Lie symmetry analysis and associated with Fibonacci numbers, respectively. We find an interesting relation between the exact solution of Equation (1) and the classical Fibonacci sequence.

Behavior and formula of the solutions of rational difference equations of order six

This paper is devoted to find the form of the solution of the following rational difference equations : xn+1 = xn−3xn−5 xn−1(±1 ± xn−3xn−5) , where the initial conditions x−5, x−4, x−3, x−2, x−1, x0 are arbitrary non zero real numbers. Also, we study the behavior of the solutions.

Sparse Triangular Decomposition for Computing Equilibria of Biological Dynamic Systems Based on Chordal Graphs.

Many biological systems are modeled mathematically as dynamic systems in the form of polynomial or rational differential equations. In this paper we apply sparse triangular decomposition to compute the equilibria of biological dynamic systems by exploiting the inherent sparsity of parameter-free systems via the chordal graph and by constructing suitable elimination orderings for parametric systems using the newly introduced block chordal graph. Our experiments with parameter-free systems provide practical information on suitable algorithms for chordal completion and verify the performance gains of sparse triangular decomposition against the ordinary one in the settings of computation of the equilibria. Then we establish full characterizations of block chordal graphs and propose algorithms for testing block chordality and constructing minimal block chordal completions. Based on these results, which are of their own merits in graph theory, we present a new algorithm of sparse triangular decomposition for parametric systems and apply it to detect the equilibria of parametric biological dynamic systems, with remarkable speedups against ordinary triangular decomposition verified by the experiments.

Analytical Solution to a Third-Order Rational Difference Equation.

Inspired by some open conjectures in a rational dynamical system by G. Ladas and Palladino, in this paper, we consider the problem of solving a third-order difference equation. We comment the conjecture by Ladas. A third-order rational difference equation is solved analytically. The solution is compared with the solution to the linearized equation. We show that the solution to the linearized equation is not good, in general. The methods employed here may be used to solve other rational difference equations. The period of the solution is calculated. We illustrate the accuracy of the obtained solutions in concrete examples.