System Of Rational Difference Equations Research Articles

Global dynamics of some systems of rational difference equations

In this paper, we study the qualitative behavior of some systems of second-order rational difference equations. More precisely, we study the equilibrium points, local asymptotic stability of equilibrium point, unstability of equilibrium points, global character of equilibrium point, periodicity behavior of positive solutions and rate of convergence of positive solutions of these systems. Some numerical examples are given to verify our theoretical results.

On a second order rational systems of difference equations

In this paper we study the periodicity and the form of the solutions of the following systems of difference equations of order two xn+1 = \frac{y_{n}x_{n-1}}{\pm x_{n-1}\pm y_{n}}, yn+1 = \frac{x_{n}y_{n-1}}{\pm x_{n}\pm y_{n-1}}, n ∈ ℕ0, with nonzero real numbers initial conditions.

Global Attractor of Solutions of a Rational System in the Plane

We consider a two-dimensional autonomous system of rational difference equations with three positive parameters. It was conjectured that every positive solution of this system converges to a finite limit. Here we confirm this conjecture, subject to an additional assumption.

Global dynamics of a competitive system of rational difference equations

In this paper, we study the qualitative behavior of a competitive system of second‐order rational difference equations. More precisely, we investigate the boundedness character, existence and uniqueness of positive equilibrium point, local asymptotic stability and global stability of the unique positive equilibrium point and rate of convergence of positive solutions of the system. Some numerical examples are given to verify our theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.

Dynamics of a two-dimensional competitive system of rational difference equations with quadratic terms

Abstract We investigate global dynamics of the following systems of difference equations: { x n + 1 = b 1 x n 2 A 1 + y n 2 , y n + 1 = a 2 + c 2 y n 2 x n 2 , n = 0 , 1 , 2 , … , where the parameters b 1 , a 2 , A 1 , c 2 are positive numbers and the initial condition y 0 is an arbitrary nonnegative number and x 0 is a positive number. We show that this system has rich dynamics which depends on the part of a parametric space. We find precisely the basins of attraction of all attractors including the points at ∞. MSC:39A10, 39A30, 37E99, 37D10.

On the solutions and periodic nature of some systems of difference equations

In this paper, we consider the solution of the following rational systems of difference equations: [Formula: see text] where initial conditions are nonzero real numbers.

Folding, cycles and chaos in planar systems

A typical planar system of difference equations can be folded or transformed into a scalar difference equation of order two plus a passive (non-dynamic) equation. We discuss this method and its application to identify and prove the existence or nonexistence of cycles and chaos in systems of rational difference equations with variable coefficients. These include some systems that converge to autonomous systems and some that do not, e.g. systems with periodic coefficients.

On Positive Solutions for the Rational Difference Equation Systems x n+1 = A/x n y n (2), and y n+1 = By n /x n-1 y n-1.

Our aim in this paper is to investigate the behavior of positive solutions for the following systems of rational difference equations: x n+1 = A/x n y n 2, and y n+1 = By n/x n−1 y n−1, n = 0,1,…, where x −1, x 0, y −1, and y 0 are positive real numbers and A and B are positive constants.

On A System of Rational Difference Equation

Abstract In this paper, we study local asymptotic stability, global character and periodic nature of solutions of the system of rational difference equations given by xn+1= , yn= , n=0, 1,…, where the parameters a; b; c; d; e; f ∊ (0; ∞), and with initial conditions x0; y0 ∊ (0; ∞). Some numerical examples are given to illustrate our results.

On a Third Order Rational Systems of Difference Equations

Abstract In this paper we investigate the form of the solutions of the following systems of difference equations of order three with a nonzero real numbers initial conditions.

Invariant quadrics and orbits for a family of rational systems of difference equations

We study the existence of invariant quadrics for a class of systems of difference equations in Rn defined by linear fractionals sharing denominator. Such systems can be described in terms of some square matrix A and we prove that there is a correspondence between non-degenerate invariant quadrics and solutions to a certain matrix equation involving A. We show that if A is semisimple and the corresponding system admits non-degenerate quadrics, then every orbit of the dynamical system is contained either in an invariant affine variety or in an invariant quadric.

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.

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.

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.

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.

English

There has been a great interest in studying difference equations and systems. One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models describing real life situations in population biology, geometry, economics, probability theory, genetics, physics etc. In this paper, we investigate the solutions of the system of rational difference equations where the initial values QUOTE QUOTE are real numbers and the initial values are non-zero real numbers such that QUOTE QUOTE and QUOTE QUOTE are not equal to 1. We give general solutions of the system. Also, we obtain necessary conditions for every solution of the system to be limited or unlimited. Key words: Difference equation, system of difference equations, solutions.

On the Symmetrical System of Rational Difference Equation x<sub>n+1</sub>=<i>A</i>+y<sub>n-k</sub>/y<sub>n</sub>, y<sub>n+1</sub>=<i>A</i>+x<sub>n-k</sub>/x<sub>n</sub>

In this paper, we study the behavior of the symmetrical system of rational difference equation: where A > o and xi, yi ∈(0, ∞), for i= -k,-k+1,…,0.

Solutions Form for Some Rational Systems of Difference Equations

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Also we investigate some properties of the obtained solutions and present some numerical examples.