System Of Difference Equations Xn Research Articles

Global asymptotic behavior of a discrete system of difference equations with delays

In the present paper, we mainly investigate the qualitative behavior of the solutions of a discrete system of difference equations xn+1 = ? + Pmi =1 xn?i yn , yn+1 = ? + Pmi =1 yn?i xn , n ? N where ?, ? ? (0,?), m ? Z+, x?i and y?i are non-negative real numbers for i ? {0, 1, ...,m}. Namely, we discuss the boundedness character and the asymptotic stability properties of steady states of the mentioned system. Finally, for this system, we give a rate of convergence result which has an important place in the discrete dynamical systems. Besides, some numerical simulations with graphs are given to emphasize the efficiency of our theoretical results in the article.

Solvable three-dimensional system of higher-order nonlinear difference equations

In this work, we indicate three-dimensional system of difference equations xn = ayn?k + dyn?kxn?k?l/?bxn?k?l+?czn?l, yn = ?zn?k + ?zn?kyn?k?l/??yn?k?l + ??xn?l, zn = exn?k + hxn?kzn?k?l/?fzn?k?l +?g?yn?l , n ? N0, where k and l are positive integers, the parameters a,?b, ?c, d, ?, ??, ??, ?, e, ?f ,?g, h and the initial values x?j, y?j, z?j j = ?1,k+l, are non-zero real numbers, can be solved in closed form. In addition, we obtain explicit formulas for the well-defined solutions of the aforementioned system for the case l = 1. Also, the set of undefinable solutions of the system is found. Finally, an application about a three-dimensional system of difference equations is given.

On a system of difference equations of third order solved in closed form

In this work, we show that the system of difference equationsxn+1=(ayn-2xn-1yn+bxn-1yn-2+cyn-2+d)/(yn-2xn-1yn),yn+1=(axn-2yn-1xn+byn-1xn-2+cxn-2+d)/(xn-2yn-1xn),where n belongs to the set of positive integer numbers, x-2, x-1, x0, y-2, y-1 and y0 are arbitrary nonzero real numbers, and the parameters a, b, c and d are arbitrary real numbers with d nonzero can be solved in a closed form.We will see that when a = b = c = d = 1, the solutions are expressed using the famous Tetranacci numbers. In particular, the results obtained here extend those in our recent work.

Dynamics of positive solutions of a system of difference equations

In this paper, we study the global behavior of positive solutions of the system of difference equations xn+1=A+ynyn−k,yn+1=A+xnxn−k,n=0,1,…with parameter A>0, the initial conditions xi,yi are arbitrary positive numbers for i=−k,−k+1,…,0 and k∈Z+. Moreover, we provide semi-cycle analysis of positive solutions of the above system of difference equations. We also provide numerical examples that support our analytical results.

Periodic Solutions of a System of Nonlinear Difference Equations with Periodic Coefficients

This paper is dealt with the following system of difference equationsxn+1=an/xn+bn/yn,yn+1=cn/xn+dn/yn,wheren∈ℕ0=ℕ∪0, the initial valuesx0 and y0are the positive real numbers, and the sequencesann≥0,bnn≥0,cnn≥0, anddnn≥0are two-periodic and positive. The system is an extension of a system where every positive solution is two-periodic or converges to a two-periodic solution. Here, the long-term behavior of positive solutions of the system is examined by using a new method to solve the system.

On a solvable three-dimensional system of difference equations

In this paper, we show that the following three-dimensional system of difference equations xn = zn-2xn-3/axn-3 + byn-1, yn = xn-2yn-3/cyn-3 + dzn-1, zn = yn-2zn-3/ezn-3+ fxn-1, n ? N0, where the parameters a, b, c, d, e, f and the initial values x-i, y-i, z-i, i ? {1, 2, 3}, are real numbers, can be solved, extending further some results in literature. Also, we determine the asymptotic behavior of solutions and the forbidden set of the initial values by using the obtained formulas.

Periodicities of a System of Difference Equations

We study the periodicities of a system of difference equations xn+1=max1/xn,An/yn-k,yn+1=max1/yn,Bn/xn-k, where initial values (x-k,y-k),…,(x0,y0)∈(0,+∞)×(0,+∞). We show that if An,Bn∈(0,1) are two periodic sequences, then every solution of the above system is eventually periodic with period 2. If k is even, there must be one in xn and yn converges to period two solution.

On the system of difference equations [formula omitted][formula omitted

We show that the following system of difference equationsxn=xn−1yn−2ayn−2+byn−1,yn=yn−1xn−2cxn−2+dxn−1,n∈N0,where parameters a, b, c, d, as well as initial values x−2,x−1,y−2,y−1, are real numbers, is solvable in closed form and by using obtained formulas we give some results on the long term behavior of their solutions. We also give natural explanations and considerably extend some recent results in the literature.

Long-term behavior of positive solutions of a system of max-type difference equations

Some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equationsxn=maxa,yn-1p∏j=2kxn-jqj,yn=maxa,xn-1p∏j=2kyn-jqj,n∈N0,where a,p>0, and qj,j∈{2,…,k}, are nonnegative numbers, are given. The boundedness character of positive solutions of the system is also studied.

On a nonlinear second order system of difference equations

We prove that all positive solutions of the next system of difference equationsxn+1=A+ynpxn-1q,yn+1=A+xnpyn-1q,n∈N0,where parameters A,p and q are positive numbers, are bounded if one of the following conditions is satisfied p2<4q, or 2q⩽p<1+q,q∈(0,1), and that the system has positive unbounded solutions when p2⩾4q>4, or p⩾1+q, q⩽1, completely describing the boundedness character of the system in this case.

On a system of difference equations of odd order solvable in closed form

It is shown that the next system of difference equationsxn=cnyn-(2k-1)an+bnyn-(2k-1)∏i=1k-1yn-(2i-1)xn-2i,yn=γnxn-(2k-1)αn+βnxn-(2k-1)∏i=1k-1xn-(2i-1)yn-2in∈N0, where an,bn,cn,αn,βn,γn, n∈N0, and initial values x-i,y-i, i∈{1,2,…,2k-1} are real numbers, is solved in closed form, and the behavior of its well-defined solutions when all the coefficients in the system are constant is described.

On a solvable system of difference equations of fourth order

The system of difference equationsxn=cnxn-4an+bnyn-1xn-2yn-3xn-4,yn=γnyn-4αn+βnxn-1yn-2xn-3yn-4,n∈N0, where the sequences an,bn,cn,αn,βn,γn,n∈N0, and initial values x-i,y-i,i∈{1,2,3,4} are real numbers, such that cn≠0,γn≠0,n∈N0, is solved in closed form, and based on the obtained formulae is studied the asymptotic behaviour of well-defined solutions of the system.

On the system of difference equations [formula omitted], [formula omitted

We show that the system of difference equationsxn=cnyn-3an+bnyn-1xn-2yn-3,yn=γnxn-3αn+βnxn-1yn-2xn-3,n∈N0,where the sequences an, bn, cn, αn, βn, γn, n∈N0, and initial values x-i, y-i, i∈{1,2,3} are real numbers, such that cn≠0, γn≠0, n∈N0, can be solved in closed form, and for the case when all the sequences an, bn, cn, αn, βn and γn are constant we describe the asymptotic behavior of well-defined solutions of the system.

On the system [formula omitted], [formula omitted

It is shown that the system of difference equationsxn+1=ynxn-kyn-k+1(an+bnynxn-k),yn+1=xnyn-kxn-k+1(cn+dnxnyn-k),n∈N0,where k∈N, the sequences an,bn,cn,dn, n∈N0, and the initial values x-i,y-i, i=0,k¯ are real numbers can be solved in closed form. Asymptotic behavior of well-defined solutions of the system for the case when all the sequences an,bn,cn and dn are constant is studied in detail.

Basins of attraction of an anti-competitive discrete rational system

C Dedicated to Professor Mustafa Kulenovic on the occasion of his 60th birthday Abstract. We investigate the global asymptotic behavior of solutions of the following anti-competitive system of difference equations xn+1 = 1yn A1 + xn ; yn +1 = 2xn + 2yn yn ; n = 0; 1;:::; where the parameters 1; 2; 2;A1 are positive numbers and the initial conditions x0 0;y0 > 0. We find the basins of attraction of all at- tractors of the system, which are the equilibrium point and period-two solutions.

On a solvable rational system of difference equations

We show that the system of difference equationsxn+1=xnyn-kyn-k+1(an+bnxnyn-k),yn+1=ynxn-kxn-k+1(cn+dnynxn-k),n∈N0,where k∈N, the sequences an,bn,cn,dn, n∈N0, and initial values x-i,y-i,i=0,k¯ are real numbers can be solved in closed form. By using obtained formulae we investigate asymptotic behavior of well-defined solutions of the system for the case when all the sequences an, bn,cn and dn are constant.

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 the Solutions of the System of Difference Equationsxn+1=max{A/xn,yn/xn},yn+1=max{A/yn,xn/yn}

We study the behavior of the solutions of the following system of difference equationsxn+1=max⁡{A/xn,yn/xn},yn+1=max⁡{A/yn,xn/yn}where the constantAand the initial conditions are positive real numbers.

ON THE SOLUTIONS OF A CLASS OF DIFFERENCE EQUATIONS SYSTEMS

In this paper we discuses the periodic solutions of particular cases of the following general system of difference equations xn+1 = a1 + a2yn

Asymptotic behavior of a competitive system of linear fractional difference equations

We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = (a+x n )/(b+y n ), yn+1 = (d+y n )/(e+x n ), n = 0,1,..., where the parameters a, b, d, and e are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. In certain range of parameters, we prove the existence of the global stable manifold of the unique positive equilibrium of this system which is the graph of an increasing curve. We show that the stable manifold of this system separates the positive quadrant of initial conditions into basins of attraction of two types of asymptotic behavior. In the case where a = d and b = e, we find an explicit equation for the stable manifold to be y = x.