Difference Equation Xn Research Articles

Eventually periodic solutions of a max-type equation

This paper shows that every well-defined solution of the difference equation xn+1=max{βxn,xn−1}, n∈N0where β∈R is eventually periodic with period two.

Existence, uniqueness and attractivity of prime period two solution for a difference equation of exponential form

In this paper we study the existence, uniqueness and attractivity of prime period two solution of the difference equationxn+1=a+bxn-1e-xnwhere a,b are positive constants and the initial values x-1,x0 are positive numbers.

BEHAVIOR OF SOLUTIONS TO A FUZZY NONLINEAR DIFFERENCE EQUATION

In this paper, we study the existence, asymptotic behavior of the positive solutions of a fuzzy nonlinear difference equation xn+1 = Axn + xn 1 B + xn 1 ; n = 0; 1; ; where (xn) is a sequence of positive fuzzy number, A;B are positive fuzzy numbers and the initial conditions x 1;x0 are positive fuzzy numbers.

On the invariant measure of the random difference equation Xn = AnXn−1 + Bn in the critical case

Nous considérons le modèle autorégressif sur ℝd défini par récurrence par l’équation stochastique Xn = AnXn−1 + Bn, où {(Bn, An)} sont des variables aléatoires à valeurs dans ℝd × ℝ+, indépendantes et de même loi. Le cas critique, c’est-à-dire lorsque $\mathbb{E}[\logA_{1}]=0$, a été étudié par Babillot, Bougerol et Elie, qui ont montré qu’il existe une et une seule mesure de Radon ν invariante pour la chaîne de Markov {Xn}. Dans ce papier nous démontrons que la mesure ν, convenablement dilatée, converge faiblement vers une mesure homogène sur ℝd ∖ {0}.

On a third-order system of difference equations

We show that the system of difference equationsxn+1=a1xn-2b1ynzn-1xn-2+c1,yn+1=a2yn-2b2znxn-1yn-2+c2,zn+1=a3zn-2b3xnyn-1zn-2+c3,n∈N0,where the parameters ai,bi,ci,i∈{1,2,3}, and initial values x-j,y-j,z-j,j∈{0,1,2}, are real numbers, can be solved, developing further the results in the literature.

On stabilization of equilibria using predictive control with and without pulses

Consider a chaotic difference equation xn+1=f(xn). We focus on the problem of control of chaos using a prediction-based control (PBC) method. If f has a unique positive equilibrium, it is proved that global stabilization of this equilibrium can be achieved under mild assumptions on the map f; if f has several positive equilibria, we demonstrate that more than one equilibrium can be stabilized simultaneously. We also show that it is still possible to stabilize an unstable equilibrium using a strategy of control with pulses, that is, the control is only applied after a fixed number of iterations. We illustrate our main results with several examples, mainly from population dynamics.

Some Notes on the Difference Equation xn+1=α+(xn−1/xnk)

We investigate the behavior of the solutions of the recursive sequence, where α ∈ [0, ∞), k ∈ (0, ∞), and the initial conditions x−1, x0 are arbitrary positive numbers. Included are results that considerably improve those in the recently published paper by Hamza and Morsy (2009).

On the Difference Equation xn+1 = xnxn−k/(xn−k+1(a + bxnxn−k))

We show that the difference equation xn+1 = xnxn−k/xn−k+1(a + bxnxn−k), n ∈ ℕ0, where k ∈ ℕ, the parameters a, b and initial values x−i, are real numbers, can be solved in closed form considerably extending the results in the literature. By using obtained formulae, we investigate asymptotic behavior of well‐defined solutions of the equation.

On the Difference Equation xn = anxn−k/(bn + cnxn−1 ⋯ xn−k)

The behavior of well‐defined solutions of the difference equation xn = anxn−k/(bn + cnxn−1 ⋯ xn−k), n ∈ ℕ0, where k ∈ ℕ is fixed, the sequences an, bn and cn are real, (bn, cn)≠(0,0), n ∈ ℕ0, and the initial values x−k, …, x−1 are real numbers, is described.

On the Max‐Type Equation xn+1 = max {1/xn, Anxn−1} with a Period‐Two Parameter

We study the behavior of the well‐defined solutions of the max type difference equation xn+1 = max {1/xn, Anxn−1}, n=0,1,…, where the initial conditions are arbitrary nonzero real numbers and {An} is a period‐two sequence of real numbers with An ∈ [0, ∞).

Local Stability of Period Two Cycles of Second Order Rational Difference Equation

We consider the second order rational difference equation xn+1 = (α + βxn + γxn−1)/(A + Bxn + Cxn−1), n = 0,1,2,…, where the parameters α, β, γ, A, B, C are positive real numbers, and the initial conditions x−1, x0 are nonnegative real numbers. We give a necessary and sufficient condition for the equation to have a prime period‐two solution. We show that the period‐two solution of the equation is locally asymptotically stable. In particular, we solve Conjecture 5.201.2 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.3 in Kulenović and Ladas monograph (2002).

The Form of the Solutions and Periodicity of Some Systems of Difference Equations

This paper is devoted to get the form of the solutions and the periodic nature of the following systems of rational difference equations xn+1 = xn−5/(−1 + xn−5yn−2), yn+1 = yn−5/(±1 ± yn−5xn−2), where the initial conditions are real numbers.

On the Solutions of a General System of Difference Equations

We deal with the solutions of the systems of the difference equations xn+1 = 1/xn−pyn−p, yn+1 = xn−pyn−p/xn−qyn−q, and xn+1 = 1/xn−pyn−pzn−p, yn+1 = xn−pyn−pzn−p/xn−qyn−qzn−q, zn+1 = xn−qyn−qzn−q/xn−ryn−rzn−r, with a nonzero real numbers initial conditions. Also, the periodicity of the general system of k variables will be considered.

On the solutions of systems of rational difference equations

In this paper we deal with the periodic nature and the form of the solutions of the following systems of rational difference equations xn+1=xn−3±1±xn−3yn−1,yn+1=yn−3±1±yn−3xn−1 with a nonzero real number’s initial conditions.

On the difference equation xn = xn−2 /( bn + cnxn−1 xn−2 )

We prove that the difference equation x n = x n - 2 b n + c n x n - 1 x n - 2 , n ∈ N 0 , where ( b n ) n ∈ N 0 and ( c n ) n ∈ N 0 are sequences periodic with period two and the initial values x −2, x −1 are real numbers, can be solved explicitly. Some applications of obtained formulae are given.

Dynamic behavior of a nonlinear rational difference equation and generalization

This paper is concerned about the dynamic behavior for the following high order nonlinear difference equation xn =( xn-k + xn-m + xn-l)/(xn-kxn-m + xn-mxn-l +1) with the initial data {x−l,x−l+1, ... ,x−1 }∈ R l and 1 ≤ k ≤ m ≤ l. The convergence of solution to this equation is investigated by introducing a new sequence, which extends and includes corresponding results obtained in the references (Li in J Math Anal Appl 312:103-111, 2005; Berenhaut et al. Appl. Math. Lett. 20:54-58, 2007; Papaschinopoulos and Schinas J Math Anal Appl 294:614-620, 2004) to a large extent. In addition, some propositions for generalized equations are reported.

Dynamics of a two-dimensional system of rational difference equations of Leslie--Gower type

Abstract We investigate global dynamics of the following systems of difference equations x n + 1 = α 1 + β 1 x n A 1 + y n y n + 1 = γ 2 y n A 2 + B 2 x n + y n , n = 0 , 1 , 2 , … where the parameters α 1, β 1, A 1, γ 2, A 2, B 2 are positive numbers, and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. We show that this system has rich dynamics which depends on the region of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or non-hyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or non-hyperbolic equilibrium points. We give examples of a globally attractive non-hyperbolic equilibrium point and a semi-stable non-hyperbolic equilibrium point. We also give an example of two local attractors with precisely determined basins of attraction. Finally, in some regions of parameters, we give an explicit formula for the global stable manifold. Mathematics Subject Classification (2000) Primary: 39A10, 39A11 Secondary: 37E99, 37D10

Dynamics of a Higher Order Rational Difference Equation

In this paper, we address the difference equation xn=pxn-s+xn-t/q+xn-t n=0,1,... with positive initial conditions where s, t are distinct nonnegative integers, p, q > 0. Our results not only include some previously known results, but apply to some difference equations that have not been investigated so far.

Discrete delays and piecewise constant argument in neuronics

Sufficient conditions are established for the convergence of solutions of difference equation xn+1 = a tanh[xn - bn-1 - c], n = 0,, 1, 2, 3, ., discritized from the mother version of continuous case dx(t)/dt = -x(t) + a tanh[x(t) - bx(t - τ) - c], t > 0, with a, b, c and τ are all positive, that models the dynamics of a single effective neuron with dynamical threshold with delay. A similar approach can also be applied to establish the stability conditions for a differential equation with piecewise constant arguments as in the following dx(t)/dt = -ax(t) + tanh[bx(t) + cx([t])], t > 0, t ≠ 0, 1, 2, .... The stability analysis of the generalized time delay model dx(t)/dt = -ax(t) + tanh{bx(t) +∑j=0k cjx([t - j])}, t > 0, t ≠ 0, 1, 2, ..., is also discussed using the lemma due to Cooke and Huang 1991.

Unbounded Solutions of the Difference Equation xn = xn−lxn−k − 1

The following difference equation xn = xn−lxn−k − 1, n ∈ ℕ0, where k, l ∈ ℕ, k < l, gcd(k, l) = 1, and the initial values x−l, …, x−2, x−1 are real numbers, has been investigated so far only for some particular values of k and l. To get any general result on the equation is turned out as a not so easy problem. In this paper, we give the first result on the behaviour of solutions of the difference equation of general character, by describing the long‐term behavior of the solutions of the equation for all values of parameters k and l, where the initial values satisfy the following condition min{x−l, …, x−2, x−1}.