DOI : 10.17577/IJERTV2IS50643
- Open Access
- Total Downloads : 278
- Authors : M. N. Qureshi, A. Q. Khan, Q. Din
- Paper ID : IJERTV2IS50643
- Volume & Issue : Volume 02, Issue 05 (May 2013)
- Published (First Online): 31-05-2013
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Global Behavior Of Third Order System Of Rational Difference Equations
Global behavior of third order system of rational difference equations
M. N. Qureshi A. Q. Khan Q. Din §
Abstract
In this paper, our aim is to study the dynamical behavior of third-order system of rational difference equations
xn+1
x
n2
n2
= , y
+ xnxn1xn2
n+1
1yn2
= , n = 0, 1, · · · . + y y y
= , n = 0, 1, · · · . + y y y
1 1 n n1 n2
where the parameters , , , 1, 1, 1 and initial conditions x0, x1, x2, y0, y1, y2 are positive real numbers. Some numerical examples are given to verify our theoretical results.
Keywords and phrases: System of rational difference equations, stability, global character, rate of convergence
2010 AMS Mathematics subject classifications: 39A10, 40A05.
The theory of difference equations occupies a central position in applicable Analysis. There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. In applications Nonlinear difference equations of order greater than one are of great importance. Such equations also appear naturally as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology,physiology, physics, engineering and economics. It is very interesting to investigate the dynamical behavior of positive solutions for system of higher-order rational difference equations.
C. Cinar [1] investigated the periodicity of the positive solutions of the system of rational difference equations:
xn+1
1
= , y yn
n+1
yn
= .
xn1yn1
S. Stevi´c [2] studied the system of two nonlinear difference equation:
xn+1
un
= , y
1 + vn
n+1
wn
= ,
1 + sn
where un, vn, wn, sn are some sequences xn or yn.
S. Stevi´c [3] studied the system of three nonlinear difference equations:
xn+1
a1xn2
= , y
= , y
b1ynzn1xn2 + c1
n+1
a2yn2
= , z
= , z
b2znxn1yn2 + c2
n+1
a3zn2
= ,
= ,
b3xnyn1zn2 + c3
This work is supported by HEC of Pakistan
Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan, e-mail: nqureshi@ajku.edu.pk
Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan, e- mail:abdulqadeerkhan1@gmail.com
§ Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan, e- mail:qamar.sms@gmail.com
where the parameters ai, bi, ci, i {1, 2, 3} are real numbers.
Ignacio Bajo and Eduardo Liz [4] investigated the global behavior of difference equation:
for all values of real parameters a, b.
xn+1
xn1
= ,
= ,
a + bxn1xn
S. Kalabusi´c, M. R. S. Kulenovi´c and E. Pilav [5] investigated the global dynamics of the following systems of difference equations:
xn+1
1 + 1xn
= , y
A1 + yn
n+1
2yn
= .
A2 + B2xn + yn
A. S. Kurbanli, C. C¸ inar, I. Yal¸cinkaya [7] studied the behavior of positive solutions of the system of rational difference equation:
xn+1
x
n1
n1
= , y
ynxn1 + 1
n+1
yn1
= .
= .
xnyn1 + 1
-
Touafek and E.M. Elsayed [9] studied the periodic nature and got the form of the solutions of the following systems of rational difference equations:
xn+1
x
n3
n3
= , y
±1 ± xn3yn1
n+1
yn3
= .
= .
±1 ± yn3xn1
Similarly, N. Touafek and E.M. Elsayed [10] studied the periodicity nature of the following systems of rational difference equations:
xn+1
yn
= , y
xn1(±1 ± yn)
n+1
xn
= .
yn1(±1 ± xn)
Recently, Q. Zhang, L. Yang, J. Liu [11] studied the dynamics of a system of rational third-order difference equation:
xn+1
x
n2
n2
= , y
B + ynyn1yn2
n+1
yn2
= , n = 0, 1, · · · .A + x x x
= , n = 0, 1, · · · .A + x x x
n n1 n2
Our aim in this paper is to investigate the dynamical behavior of positive solution for third-order rational difference equations:
xn+1
x
n2
n2
= , y
+ xnxn1xn2
n+1
1yn2
= , n = 0, 1, · · · . (1) + y y y
= , n = 0, 1, · · · . (1) + y y y
1 1 n n1 n2
where the parameters , , , 1, 1, 1 and initial conditions x0, x1, x2, y0, y1, y2 are positive real numbers.
Let us consider six-dimensional discrete dynamical system of the form:
xn+1 = f (xn, xn1, xn2, yn, yn1, yn2), (2)
yn+1 = g(xn, xn1, xn2, yn, yn1, yn2), n = 0, 1, · · · ,
where f : I3 × J 3 I and g : I3 × J 3 J are continuously differentiable functions and I, J are some intervals of real numbers. Furthermore, a solution {(xn, yn)}n=2 of system (2) is uniquely determined by initial conditions (xi, yi) I × J for i {2, 1, 0}. Along with the system (2) we consider the corresponding vector map F = (f, xn, xn1, xn2, g, yn, yn1, yn2). An equilibrium point of (2) is a point (x¯, y¯) that satisfies
x¯ = f (x¯, x¯, x¯, y¯, y¯, y¯)
y¯ = g(x¯, x¯, x¯, y¯ y¯, y¯)
The point (x¯, y¯) is also called a fixed point of the vector map F .
Definition 1. Let (x¯, y¯) be an equilibrium point of the system (2).
-
An equilibrium point (x¯, y¯) is said to be stable if for every > 0 there exists > 0 such that for
,
,
0
every initial condition (xi, yi), i {2, 1, 0} 1 (xi, yi)(x¯, y¯)1 < implies 1(xn, yn)(x¯, y¯)1 <
i=2
for all n > 0, where 1.1 is the usual Euclidian norm in R2.
-
An equilibrium point (x¯, y¯) is said to be unstable if it is not stable.
-
An equilibrium point (x¯, y¯) is said to be asymptotically stable if there exists > 0 such that
,
,
0
1 (xi, yi) (x¯, y¯)1 < and (xn, yn) (x¯, y¯) as n .
i=2
-
An equilibrium point (x¯, y¯) is called global attractor if (xn, yn) (x¯, y¯) as n .
-
An equilibrium point (x¯, y¯) is called asymptotic global attractor if it is a global attractor and
-
stable.
Definition 2. Let (x¯, y¯) be an equilibrium point of the map
F = (f, xn, xn1, xn2, g, yn, yn1, yn2)
where f and g are continuously differentiable functions at (x¯, y¯). The linearized system of (2) about the equilibrium point (x¯, y¯) is
xn
Xn+1 = F (Xn) = FJ Xn,
xn2
xn2
xn1
where Xn = yn
and FJ is Jacobian matrix of the system (2) about the equilibrium point (x¯, y¯).
yn1 yn2
To construct corresponding linearized form of the system (1) we consider the following transfor- mation:
(xn, xn1, xn2, yn, yn1, yn2) 1 (f, f1, f2, g, g1, g2), (3)
where f = xn2 , g = 1yn2 , f1 = xn, f2 = xn1, g1 = yn, g2 = yn1. The Jacobian
+xnxn1 xn2 1+1 ynyn1 yn2
matrix about the fixed point (x¯, y¯) under the transformation (3) is given by
A A B 0 0 o
1 0 0 0 0
FJ (x¯, y¯) =
0 1 0 0 0 0
0 0 o C C D
0 0 0 1 0 0
3
3
0 0 0 0 1 0
x¯
x¯
(+x¯3)2
(+x¯3)2
(1+1 y¯3)2
(1+1 y¯3)2
3
(+x¯3)2
(+x¯3)2
where A =
, B = , C = 1 1 y¯
and D = 11 .
(1+1 y¯3)2
(1+1 y¯3)2
Theorem 1. For the system Xn+1 = F (Xn), n = 0, 1, · · · , of difference equations such that X¯ be a
fixed point of F. If all eigenvalues of the Jacobian matrix JF about X¯ lie inside the open unit disk
|| < 1, then X¯
is unstable.
is locally asymptotically stable. If one of them has a norm greater than one, then X¯
-
Let (x¯, y¯) be an equilibrium point of system (1), then for > and 1 > 1 system (1) has following
two equilibrium points: P
= (0, 0), P = (A, B), where A = ( ) 1 and B = ( 11 ) 1 .
1
1
0 1 3 3
Theorem 2. Let (xn, yn) be positive solution of system (1), then for every m 0 the following results
hold:
0 xn (
)m+1×2, if n = 3m + 1,
0 xn
m+1
( ) x1
, if n = 3m + 2,
0 x
m+1
n
n
0
0
( ) x , if n = 3m + 3,
0 yn
0 yn
1 m+1
1
1
( ) y2
1 m+1
1
1
( ) y1
, if n = 3m + 1,
, if n = 3m + 2,
0 y
1 m+1
( ) y , if n = 3m + 3,
n 1 0
Proof. The results are obviously true for m = 0. Suppose that results are true for m = k 1, i.e.,
0 xn
0 xn
k+1
( ) x2
k+1
( ) x1
, if n = 3k + 1,
, if n = 3k + 2,
0 x
k+1
n
n
0
0
( ) x , if n = 3k + 3,
0 yn
0 yn
1 k+1
1
1
( ) y2
1 k+1
1
1
( ) y1
, if n = 3k + 1,
, if n = 3k + 2,
0 y
1 k+1
( ) y , if n = 3k + 3,
n 1 0
Now, for m = k + 1 using (1) one has
0 x
x3k+1 x3k+1 k+2
= ( ) x ,
3k+4
+ x3k+3x3k+2x3k+1
2
0 x
x3k+2 x3k+2 k+2
= ( ) x ,
3k+5
+ x3k+4x3k+3x3k+2
1
0 x
x3k+3 x3k+3 k+2
= ( ) x ,
3k+6
+ x3k+5y3k+4y3k+3 0
0 y
1y3k+1 1y3k+1 1 k+2
= ( ) y ,
3k+4
1 + 1y3k+3y3k+2x3k+1 1
1 2
0 y
1y3k+2 1y3k+2 1 k+2
= ( ) y ,
3k+5
1 + 1y3k+4y3k+3y3k+2 1
1 1
0 y
1y3k+3 1y4k+3 1 k+2
= ( ) y ,
3k+6
+ 1 + + 1y3k+5y3k+4y3k+3 1
1 0
Theorem 3. For the equilibrium point P0 of Equation (1) the following results hold:
-
If < and 1 < 1, then equilibrium point P0 of the system (1) is locally asymptotically stable.
-
If > or 1 > 1, then equilibrium point P0 is unstable.
Proof. (i) The linearized system of (1) about the equilibrium point P0 is given by:
Xn+1 = FJ (P0)Xn,
xn
0 0 0 0 o
xn1
x
x
1 0 0 0 0 0
1
1
0
0
0
0
0
(P
0
0
0)
0
0
is gi
0
0
ven
1
0
by
0
1
0
0
0
0
0
(P
0
0
0)
0
0
is gi
0
0
ven
1
0
by
0
1
0 1 0 0 0 0
where Xn =
n2
yn yn1
and FJ (0, 0) =
1 .
0
0
0
0
yn2
The characteristic polynomial of FJ
0
yn2
The characteristic polynomial of FJ
0
6 3
6 3
1 1
P () = ( + ) + .. (4)
1 1
1
1
1
1
1
1
1
1
The roots of P () are ± and ± repeated roots. Since | | < 1 and | | < 1, whenever < and
1 < 1. Thus, by Theorem 1 P0 is locally asymptotically stable.
(ii) It is easy to see that if > or 1 > 1, then there exists at least one root of Equation (4) such that || > 1. Hence, by Theorem 1 if > or 1 > 1, then (0, 0) is unstable.
Theorem 4. If < or 1 < 1, then positive equilibrium point P1 of Equation (1) is unstable. Proof. The linearized system of (1) about the equilibrium point P1 is given by:
Xn+1 = FJ (P1)Xn,
xn
xn1
n
n
1 + 1 +
J
1
J
1
0 0 0
=
=
yn
yn
where X
xn2 xn3
and F
(P ) =
1 0 0 0 0 0
1 1
1 1
0 1 0 0 0 0
1
0 0 0 1 + 1
1
1 + 1
1
0 0 0 1 + 1
1
1 + 1
,
yn1 yn2 yn3
0 0 0 1 0 0
0 0 0 0 1 0
1
1
One of the roots of characteristic polynomial of FJ (P1) is given by 1 . Hence, by Theorem 1 if
1 > 1 then P1 is unstable.
Theorem 5. Let < and 1 < 1, then the equilibrium point P0 of Equation (1) is globally asymptotically stable.
Proof. For < and 1 < 1, from Theorem 3 P0 is locally asymptotically stable. From Theorem 2, it is easy to see that every positive solution (xn, yn) is bounded, i.e., 0 xn µ and 0 yn for all n = 0, 1, 2, · · · , where µ = max{x2, x1, x0} and = max{y2, y1, y0}. Now, it is sufficient to prove that (xn, yn) is decreasing. From system (1) one has
xn+1
xn2
= ,
= ,
+ xnxn1xn2
n2
n2
x
< x
n2.
This implies that x3n+1 < x3n2 and x3n+4 < y3n+1. Also
yn+1
yn2
= ,
= ,
+ ynyn1yn2
n2
n2
y
n2
n2
< y .
This implies that y3n+1 < x3n2 and y3n+4 < x3n+1. Hence, the subsequences {x3n+1}, {x3n+2}, {x3n+3}
and {y3n+1}, {y3n+2}, {y3n+3} are decreasing.Therefore the sequences {xn} and {yn} are decreasing.
Hence, lim
n
xn = lim
n
yn = 0.
-
-
In this section we will determine the rate of convergence of a solution that converges to the unique positive equilibrium point of the system (1). The following result gives the rate of convergence of solution of a system of difference equations
Xn+1 = (A + B(n)) Xn, (5)
where Xn is an m-dimensional vector, A Cm×m is a constant matrix, and B : Z+ Cm×m is a matrix function satisfying
1B(n)1 0 (6)
j
j
as n ,where 1 · 1 denotes any matrix norm which is associated with the vector norm
1(x, y)1 = x2 + y2
Proposition 1. (Perrons Theorem)[?] Suppose that condition (6) holds. If Xn is a solution of (5) , then either Xn = 0 for large n or
= lim (1X 1) , (7)
= lim (1X 1) , (7)
1/n
n
n
or
= lim
n
1Xn+11
1Xn1
(8)
exists and is equal to the norm of one the eigenvalues of the matrix A.
giv
giv
Assume that lim xn = x¯ and lim
yn = y¯. First we will find a system of limiting equations for
n
the map F . The error terms are
nenas
2 2 2 2
xn+1 x¯ = , Ai (xni x¯) + , Bi (yni y¯) , yn+1 y¯ = , Ci (xni x¯) + , Di (yni y¯) .
i=0
i=0
i=0
i=0
i=0
i=0
i=0
i=0
i=0
i=0
i=0
i=0
i=0
i=0
i=0
i=0
Set e1 = xn x¯ and e2 = yn y¯, one has
2
2
2
2
n n
2
2
2
2
n+1
n+1
ni
ni
ni
ni
e1 = , Aie1 + , Bie2
i=0
i=0
i=0
i=0
, e2
= , Cie1
i=0
i=0
+ , Die2 .
i=0
i=0
i=0
i=0
i=0
i=0
i=0
i=0
i=0
i=0
2
n+1
n+1
ni
ni
ni
ni
2
x¯ xni
=
=
where A
= i=1 , A
2
2
x¯2 xn2
2
2
, A = ,
2
2
0
+
2
2
i=0
1
xn i(+x¯3)
+
2
2
i=0
2
xn i(+x¯3)
2
+
2
2
i=0
xn i(+x¯3)
=
=
1 1 y¯ yni
B = 0 for i {0, 2}, C
= 0 for i {0, 2}, D
= i=1 , D
2
2
1 1 y¯2 yn2 ,
2
2
i i
D =
D =
1 1
1 1
,
,
1 1
2
2
2
0
1 1
1 1
+
2
2
i=0
1
1 1
1 1
yn i( + y¯3)
+
2
1 1
1 1
1 1
1 1
2
i=0
yn i( + y¯3)
1 1
1 1
+
2
2
i=0
yn i( + y¯3)
Taking the limits, we obtain lim
Ai =
x¯3
3 2 for i {0, 1}, lim
A2 =
3 2 ,
n
( + x¯ )
n
1 1
1 1
11y¯3
( + x¯ )
lim
n
Bi = 0 for i 0, 2 , lim
{ }
{ }
n
Ci = 0 for i 0, 2 , lim
{ }
{ }
n
Di =
( + y¯3)2 for i {0, 1},
lim D2
n
11
= , So, the limiting system of error terms can be written as E
(1 + 1y¯3)2
n+1
= FJ
(0, 0)En,
e
e
1
n
e1
where where En =
where where En =
. Using proposition (1), one has following result.
. Using proposition (1), one has following result.
e
e
2
2
e
e
n1
n2 2
en
n2 2
en
e1
n 1
2
n2
{ }
{ }
Theorem 6. Assume that (xn, yn) be a positive solution of the system (1) such that lim
n
xn = x¯,
and lim yn = y¯, where (x¯, y¯) = (0, 0). Then, the error vector En of every solution of (1) satisfies
n
both of the following asymptotic relations
1
1
lim ( En
n
1
1) n
= |FJ
|
|
(x¯, y¯) , lim
n
1En+11 = F
J
J
|
|
1En1
(x¯, y¯)|,
where FJ (x¯, y¯) are the characteristic roots of the Jacobian matrix FJ (x¯, y¯) about (0, 0).
In order to verify our theoretical results we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to the system of nonlinear difference equations (1). All plots in this section are drawn with mathematica.
Example 1. Consider the system (1) with initial conditions x2 = 7.9, x1 = 0.19, x0 = 1.2, y2 = 3.6, y1 = 2.3, y0 = 9.1. Moreover, choosing the parameters = 970, = 990, = 110, 1 = 770, 1 = 790, 1 = 90. Then, the system (1) can be written as:
xn+1
970x
n2
n2
= , y
990 + 110xnxn1xn2
n+1
770yn2
= , n = 0, 1, · · · , (9)790 + 90y y y
= , n = 0, 1, · · · , (9)790 + 90y y y
n n1 n2
n = 0, 1, · · · and with initial conditions x2 = 7.9, x1 = 0.19, x0 = 1.2, y2 = 3.6, y1 = 2.3, y0 =
9.1 .The plot of system (9) is shown in Figure (1) and its global attractor is shown in Figure(2).
Example 2. Consider the system (1) with initial conditions x2 = 2.9, x1 = 0.19, x0 = 1.2, y2 = 3.6, y1 = 5.3, y0 = 1.1. Moreover, choosing the parameters = 197, = 199, = 210, 1 = 177, 1 = 179, 1 = 190. Then, the system (1) can be written as:
xn+1
197x
n2
n2
= , y
199 + 210xnxn1xn2
n+1
177yn2
= , n = 0, 1, · · · , (10)179 + 190y y y
= , n = 0, 1, · · · , (10)179 + 190y y y
n n1 n2
n = 0, 1, · · · and with initial conditions x2 = 2.9, x1 = 0.19, x0 = 1.2, y2 = 3.6, y1 = 5.3, y0 =
-
.The plot of system (10) is shown in Figure (3) and its global attractor is shown in Figure (4).
Figure 1: Plot of system (9)
Figure 2: An attractor of system (9)
Figure 3: Plot of system (10)
Figure 4: An attractor of system (10)
Conclusion
In the paper, we investigate some dynamics of a six-dimensional discrete system. The system has two positive equilibrium points. The linearization method is used to show that equilibrium point (0, 0) is locally asymptotically stable. The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state. An approach to this problem consists of determining the possible global behaviors of the system and determining which initial conditions lead to these long-term behaviors. In case of higher-order dynamical systems, it is very difficult to discuss global behavior of the system. Some powerful tools such as semiconjugacy and weak contraction cannot be used to analyze global behavior of system (1). In the paper,using simple techniques to prove the global asymptotic stability of equilibrium point (0, 0). Some numerical examples are provided to support our theoretical results.
-
C. Cinar, On the positive solutions of the difference equation system xn+1 = 1 ; yn+1 = yn ,
Applied Mathematics and Computation, 158, 303305(2004).
yn xn1 yn1
-
S. Stevi´c, On some solvable systems of difference equations, App. Math. and Comput., 218, 50105018(2012).
-
S. Stevi´c, On a third-order system of difference equations, App. Math. and Comput., 218, 7649 7654(2012).
-
I. Bajo, E. Liz, Global behaviour of a second-order nonlinear difference equation, J. Differ. Eq. Appl., 17(10), 14711486(2011).
-
S. Kalabusi´c, M. R. S. Kulenovi´c, E. Pilav, Dynamics of a two-dimensional system of ra- tional difference equations of LeslieGower type, Advances in Difference Equations, (2011), doi:10.1186/1687-1847-2011-29.
-
S. Kalabusi´c, M. R. S. Kulenovi´c, E. Pilav, Global Dynamics of a Competitive System of Rational Difference Equations in the Plane, Advances in Difference Equations, (2009), Article ID 132802, 30 pages.
-
A. S. Kurbali, C. C¸ inar, I. Yal¸cinkaya, On the behavior of positive solutions of the system of rational difference equations xn+1 = xn1 , yn+1 = yn1 , Mathematical and Computer
Modelling, 53, 12611267(2011).
ynxn1+1
xnyn1+1
-
A. S. Kurbanli, On the behavaior of positive solutions of the system of rational difference equa- tions xn+1 = xn1 , yn+1 = yn1 , zn+1 = 1 , Advances in Difference Equations, 2011,
2011:40.
ynxn1 1
xnyn1 1
ynzn
-
N. Touafek, E.M. Elsayed, On the solutions of systems of rational difference equations, Mathe- matical and Computer Modelling, 55, 19871997(2012).
-
N. Touafek, E.M. Elsayed, On the periodicity of some systems of nonlinear difference equations,
Bull. Math. Soc. Sci. Math. Roumanie, 2, 217224(2012).
-
Q. Zhang, L. Yang, J. Liu, Dynamics of a system of rational third order difference equation,
-
Advances in Difference Equations, doi:10.1186/1687-1847-2012-136, 8 pages.