Dynamics of a Second-Order System of Nonlinear Difference Equations

Numerical Examples ◽

Unbounded Solutions ◽

In this paper, we investigate the equilibrium points, stability of two equilibrium points, convergences of negative equilibrium point, periodic solutions, and existence of bounded or unbounded solutions of a system of nonlinear difference equations xn+1 =xn-1yn - 1, yn+1 = yn-1xn - 1 n = 0,1,..., where the initial values are real numbers. Additionally we present some numerical examples to verify our theoretical results.

QUALITATIVE BEHAVIOURS OF A SYSTEM OF NONLINEAR DIFFERENCE EQUATIONS

Journal of Science and Arts ◽

10.46939/j.sci.arts-21.1-a05 ◽

2021 ◽

Vol 21 (1) ◽

pp. 39-56

Author(s):

ERKAN TAŞDEMİR ◽

YÜKSEL SOYKAN

Keyword(s):

Real Number ◽

Difference Equations ◽

Equilibrium Points ◽

Bounded Solutions ◽

Related System ◽

Numerical Examples ◽

Existence Of Periodic Solutions ◽

The Stability ◽

The paper aims to study the dynamics of a system of nonlinear difference equations x_(n+1)=x_(n-1) y_n+A,y_(n+1)=y_(n-1) x_n+A where A is real number. We especially investigate the stability of equilibrium points, convergence of equilibrium points, existence of periodic solutions, and existence of bounded solutions of related system. Moreover, we present some numerical examples to verify the theoretical results.

On Stability Analysis of Higher-Order Rational Difference Equation

Discrete Dynamics in Nature and Society ◽

10.1155/2020/3094185 ◽

2020 ◽

Vol 2020 ◽

pp. 1-10

Author(s):

Abdul Khaliq ◽

H. S. Alayachi ◽

M. S. M. Noorani ◽

A. Q. Khan

Keyword(s):

Stability Analysis ◽

Equilibrium Points ◽

Global Behavior ◽

Real Numbers ◽

Positive Real ◽

Numerical Examples ◽

Local Asymptotic Stability ◽

Recursive Sequence ◽

Rational Difference Equation ◽

In this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, global behavior of equilibrium points, boundedness and periodicity of the rational recursive sequence wn+1=wn−pα+βwn/γwn+δwn−r, where γwn≠−δwn−r for r∈0,∞, α, β, γ, δ∈0,∞, and r>p≥0. With initial values w−p,w−p+1,…,w−r,w−r+1,…,w−1, and w0 are positive real numbers. Some numerical examples are given to verify our theoretical results.

On the Solutions of a System of Third-Order Rational Difference Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2018/1743540 ◽

2018 ◽

Vol 2018 ◽

pp. 1-11 ◽

Cited By ~ 3

Author(s):

A. M. Alotaibi ◽

M. S. M. Noorani ◽

M. A. El-Moneam

Keyword(s):

Difference Equations ◽

Initial Conditions ◽

Theoretical Discussion ◽

Real Numbers ◽

Positive Real ◽

Third Order ◽

Numerical Examples ◽

Rational Difference Equations

The structure of the solutions for the system nonlinear difference equations xn+1=ynyn-2/(xn-1+yn-2), yn+1=xnxn-2/(±yn-1±xn-2), n=0,1,…, is clarified in which the initial conditions x-2, x-1, x0, y-2, y-1, y0 are considered as arbitrary positive real numbers. To exemplify the theoretical discussion, some numerical examples are presented.

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

Journal of Mathematics ◽

10.1155/2020/6636105 ◽

2020 ◽

Vol 2020 ◽

pp. 1-7

Author(s):

Durhasan Turgut Tollu

Keyword(s):

Periodic Solution ◽

Positive Solution ◽

Periodic Solutions ◽

Difference Equations ◽

Real Numbers ◽

System Of Difference Equations ◽

Positive Real ◽

Long Term Behavior

This paper is dealt with the following system of difference equations x n + 1 = a n / x n + b n / y n , y n + 1 = c n / x n + d n / y n , where n ∈ ℕ 0 = ℕ ∪ 0 , the initial values x 0 and y 0 are the positive real numbers, and the sequences a n n ≥ 0 , b n n ≥ 0 , c n n ≥ 0 , and d n n ≥ 0 are 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.

Stability Analysis of a System of Exponential Difference Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2014/375890 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11 ◽

Cited By ~ 9

Author(s):

Q. Din ◽

K. A. Khan ◽

A. Nosheen

Keyword(s):

Difference Equations ◽

Existence And Uniqueness ◽

Initial Conditions ◽

Positive Equilibrium ◽

Global Behavior ◽

Real Numbers ◽

Positive Real ◽

Numerical Examples ◽

Boundedness Character ◽

We study the boundedness character and persistence, existence and uniqueness of positive equilibrium, local and global behavior, and rate of convergence of positive solutions of the following system of exponential difference equations:xn+1=(α1+β1e-xn+γ1e-xn-1)/(a1+b1yn+c1yn-1),yn+1=(α2+β2e-yn+γ2e-yn-1)/(a2+b2xn+c2xn-1), where the parametersαi, βi, γi, ai, bi, andcifori∈{1,2}and initial conditionsx0, x-1, y0, andy-1are positive real numbers. Furthermore, by constructing a discrete Lyapunov function, we obtain the global asymptotic stability of the positive equilibrium. Some numerical examples are given to verify our theoretical results.

Stability analysis of a discrete biological model

International Journal of Biomathematics ◽

10.1142/s1793524516500212 ◽

2016 ◽

Vol 09 (02) ◽

pp. 1650021 ◽

Cited By ~ 1

Author(s):

A. Q. Khan ◽

M. N. Qureshi

Keyword(s):

Equilibrium Point ◽

Initial Conditions ◽

Biological Model ◽

Positive Equilibrium ◽

Global Behavior ◽

Real Numbers ◽

Positive Real ◽

Numerical Examples ◽

Positive Equilibrium Point ◽

In this paper, we investigate the equilibrium point, local and global behavior of the unique positive equilibrium point, and rate of convergence of positive solutions of following discrete biological model: [Formula: see text] where parameters [Formula: see text] and the initial conditions [Formula: see text] are positive real numbers. Some numerical examples are given to verify theoretical results.

On a System of k-Difference Equations of Order Three

Mathematical Problems in Engineering ◽

10.1155/2020/6638700 ◽

2020 ◽

Vol 2020 ◽

pp. 1-11

Author(s):

Ibrahim Yalçınkaya ◽

Hijaz Ahmad ◽

Durhasan Turgut Tollu ◽

Yong-Min Li

Keyword(s):

Positive Solutions ◽

Difference Equations ◽

Initial Conditions ◽

Global Behavior ◽

Real Numbers ◽

Positive Real ◽

Numerical Examples ◽

In this paper, we deal with the global behavior of the positive solutions of the system of k -difference equations u n + 1 1 = α 1 u n − 1 1 / β 1 + α 1 u n − 2 2 r 1 , u n + 1 2 = α 2 u n − 1 2 / β 2 + α 2 u n − 2 3 r 2 , … , u n + 1 k = α k u n − 1 k / β k + α k u n − 2 1 r k , n ∈ ℕ 0 , where the initial conditions u − l i l = 0,1,2 are nonnegative real numbers and the parameters α i , β i , γ i , and r i are positive real numbers for i = 1,2 , … , k , by extending some results in the literature. By the end of the paper, we give three numerical examples to support our theoretical results related to the system with some restrictions on the parameters.

Dynamic Behaviors of a Class of High-Order Fuzzy Difference Equations

Journal of Mathematics ◽

10.1155/2020/1737983 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Lili Jia

Keyword(s):

Asymptotic Stability ◽

Difference Equation ◽

Difference Equations ◽

Existence And Uniqueness ◽

Fuzzy Numbers ◽

Initial Conditions ◽

Equilibrium Points ◽

High Order ◽

Numerical Examples ◽

The purpose of this paper is to give the conditions for the existence and uniqueness of positive solutions and the asymptotic stability of equilibrium points for the following high-order fuzzy difference equation: xn+1=Axn−1xn−2/B+∑i=3kCixn−i n=0,1,2,…, where xn is the sequence of positive fuzzy numbers and the parameters A,B,C3,C4,…,Ck and initial conditions x0,x−1,x−2,x−ii=3,4,…,k are positive fuzzy numbers. Besides, some numerical examples describing the fuzzy difference equation are given to illustrate the theoretical results.