Rational Difference Equations Research Articles

Global behavior of a third order rational difference equation

In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation $$x_{n+1}=\frac {ax_{n}x_{n-1}}{-bx_{n}+ cx_{n-2}},\quad n\in \mathbb {N}_0 $$ where $a$, $b$, $c$ are positive real numbers and the initial conditions $x_{-2}$, $x_{-1}$, $x_0$ are real numbers. We show that every admissible solution of that equation converges to zero if either $a c$ with ${(a-c)}/{b} c$ with ${(a-c)}/{b}>1$, we prove that every admissible solution is unbounded. Finally, when $a=c$, we prove that every admissible solution converges to zero.

Forbidden sets of planar rational systems of difference equations with common denominator

The forbidden sets of systems of first order rational difference equations in the plane in which the denominators are common for all the components of the system is studied. Such forbidden sets are composed of lines which, depending of some spectral properties of an associated matrix, can either be a finite number or lines or an infinity of lines converging to either an invariant line or to a finite number of lines itersecting in a fixed point or else it can be dense in a large subset of R2.

Behavior of an Exponential System of Difference Equations

We study the qualitative behavior of the following exponential system of rational difference equations:xn+1 = αe-yn+βe-yn-1/γ+αxn+βxn-1, yn+1 = α1e-xn+β1e-xn-1/γ1+α1yn+β1yn-1, n = 0,1,…,whereα,β,γ,α1,β1, andγ1and initial conditionsx0, x-1, yo, and y-1are positive real numbers. More precisely, we investigate the boundedness character and persistence, existence and uniqueness of positive equilibrium, local and global behavior, and rate of convergence of positive solutions that converges to unique positive equilibrium point of the system. Some numerical examples are given to verify our theoretical results.

Alternate locations of equilibrium points and poles in complex rational differential equations

We study configurations of simple equilibrium points of first order complex differential equations consisting of the iteration of rational functions. Rational functions which we deal with have the unit circle or the extended real line as Julia sets. Properties of Julia sets and the EulerJacobi formula lead to alternate locations of equilibrium points and poles of the complex differential equations. Mathematics Subject Classification: 37C10, 32A10, 37F10

Global dynamics of some systems of higher-order rational difference equations

Abstract In the present work, we study the qualitative behavior of two systems of higher-order rational difference equations. More precisely, we study the local asymptotic stability, instability, global asymptotic stability of equilibrium points and rate of convergence of positive solutions of these systems. Our results considerably extend and improve some recent results in the literature. Some numerical examples are given to verify our theoretical results. MSC:39A10, 40A05.

The roles of conic sections and elliptic curves in the global dynamics of a class of planar systems of rational difference equations

Consider the class of planar systems of first-order rational difference equations 1´ where , and the parameters are nonnegative and such that both terms in the right-hand side of (1′) are nonlinear. In this paper, we prove the following discretized Poincare-Bendixson theorem for the class of systems (1′). If the map associated to system (1′) is bounded, then the following statements are true: In particular, system (1′) cannot exhibit chaos when its associated map is bounded. Moreover, we show that if both equilibrium curves of (1′) are reducible conics and the map associated to system (1′) is unbounded, then every solution converges to one of up to infinitely many equilibria or to or . MSC:39A05, 39A11.

Global dynamics of an anti-competitive system of rational difference equations in the plane

We investigate global dynamics of the following systems of difference equationswhere the parameters are positive numbers and initial conditions and are arbitrary non-negative numbers. We find all possible dynamical scenarios for this system.

Solution and dynamics of a fourth rational difference equation

In this paper, we get the form of the solutions of the following nonlinear recursive sequences of order four where the initial conditionsare arbitrary real numbers. Also, we study some qualitative behaviors of the solutions as periodicity character of the solutions of these different equations and we obtain the equlibrium points of considered difference equations in order to investigate the local stability of these equlibrium points by using the linearized equation. We presented some numerical examples by giving some numerical values for the initial values and the coefficients of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program Matlab to confirm the obtained results. Key words: Difference equations, recursive sequences, stability, periodic solution, Mathematics subject classification: 39A10.

On a solvable system of rational difference equations

We show that the following system of difference equationswhere , , , and sequences , , and are real, can be solved in closed form. For the case when the sequences , , and are constant and , we apply obtained formulas in the investigation of the asymptotic behaviour of well-defined solutions of the system. We also find domain of undefinable solutions of the system. Our results considerably extend and improve some recent results in the literature.

Global Attractivity of a Rational Difference Equation

In this paper, we investigate the global attractivity of positive solutions of the nonlinear difference equation xn+1 = ax

Neimark–Sacker bifurcation of a third-order rational difference equation‡

In this paper, we investigate the existence and direction of the Neimark–Sacker bifurcation of a third-order rational difference equation with positive parameters. Firstly, it is found that there exists a Neimark–Sacker bifurcation when the parameter passes a critical value by analysing the characteristic equation. Secondly, the explicit algorithm for determining the direction and stability of the Neimark–Sacker bifurcations is derived by using the normal form theory. Finally, computer simulations are performed to illustrate the analytical results found.

Global dynamics for a higher order rational difference equation

Global dynamics for a higher order rational difference equation

Algebraic Independence of Solutions of First-Order Rational Difference Equations

We prove a theorem on algebraic independence of solutions of first order rational difference equations. By the theorem, we are able to prove algebraic independence of x, the exponential function ex and the Weierstrass function \({\wp(x)}\) over \({\mathbb{C}}\) only by seeing degrees of polynomials associated with their double angle formulas. As a corollary, we obtain a result on unsolvability of a first-order rational difference equation by solutions of other first-order rational difference equations, which implies its irreducibility. Additionally, we introduce some applications to algebraic independence of functions f(x), f(x2), . . . , f(xn).

Non-autonomous homogeneous rational difference equations of degree one: convergence and monotone solutions for second and third order case

Consider the non-autonomous equations: where and also These are some non-autonomous homogeneous rational difference equations of degree one. A reduction in order is consi-dered. Convergence and monoton character of positive solutions are studied. Copyright © 2013 John Wiley & Sons, Ltd.

On the solutions of some nonlinear systems of difference equations

In this paper, we deal with the existence of solutions and the periodicity character of the following systems of rational difference equations with order three: x n + 1 = x n y n − 2 y n − 1 ( ± 1 ± x n y n − 2 ) , y n + 1 = y n x n − 2 x n − 1 ( ± 1 ± y n x n − 2 ) , where the initial conditions x − 2 , x − 1 , x 0 , y − 2 , y − 1 and y 0 are nonzero real numbers.MSC:39A10.

More results on the trichotomy character of a second-order rational difference equation with period-two coefficients

We extend the known results of the non-autonomous difference equation to the situation where (i) the parameters β n , γ n , A n and B n are period-two sequences of non-negative real numbers with γ n not identically zero and A n +B n ≠ 0 and (ii) the initial conditions x − 1 and x 0 are such that and .

Global behaviour of solutions to a class of second-order rational difference equations when prime period-two solutions exist

For non-negative parameters α, β, γ, A, B and C such that A+B+C>0, consider the class of difference equations where either if A = 0, or R = [0,∞)2 if A>0. This class consists of 49 non-trivial second-order rational difference equations. We give a complete qualitative description of the global behaviour of solutions including basins of attraction of equilibria and prime period-two (PP2) solutions for all nonlinear difference equations (E) for which PP2 solutions exist. In particular, we show that every solution of (E) converges to an equilibrium or to a PP2 solution.

A solution form of a class of higher-order rational difference equations

Abstract We obtain in this paper the expressions of solutions of the following class of difference equations x n + 1 = x n - 2 k + 1 ± 1 ± Π i = 1 k x n - 2 i + 1 , n = 0 , 1 , 2 , … with conditions posed on the initial values x−j, j = 0, 1, 2, … , 2k − 1, where k ∈ {1, 2, …}.

Stability analysis for a class of nonlinear discrete-time control systems subject to disturbances and to actuator saturation

This paper addresses the stability characterisation problem for a class of nonlinear discrete-time control systems subject to actuator saturation and energy bounded disturbance inputs. The considered class of systems covers all nonlinear discrete-time systems that can be modelled by rational difference equations. Based on quadratic and piecewise quadratic Lyapunov functions, conditions based on linear matrix inequalities are proposed to analyse the asymptotic stability (internal stability) and the ℓ2 input-to-state stability (external stability) of the closed-loop system. The proposed conditions are then incorporated into convex optimisation problems to either maximise an estimate of the region of attraction or a bound on the admissible ℓ2 disturbances, and also to obtain an estimate of the system ℓ2-gain for an admissible set of disturbances.

On boundedness of solutions of the difference equation $x_{n+1}=p+\frac{x_{n-1}}{x_{n}}$ for p<1

In this paper, we study the difference equation $$x_{n+1}=p+\frac{x_{n-1}}{x_n}, \quad n=0,1,\ldots, $$ where initial values x −1,x 0∈(0,+∞) and 0<p<1, and obtain the set of all initial values (x −1,x 0)∈(0,+∞)×(0,+∞) such that the positive solutions $\{x_{n}\}_{n=-1}^{\infty}$ are bounded. This answers the Open problem 4.8.11 proposed by Kulenovic and Ladas (Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, 2002).