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Reducibility and Stability Results for Linear System of Difference Equations
Advances in Difference Equations volume 2008, Article number: 867635 (2008)
Abstract
We first give a theorem on the reducibility of linear system of difference equations of the form . Next, by the means of Floquet theory, we obtain some stability results. Moreover, some examples are given to illustrate the importance of the results.
1. Introduction
Consider the homogeneous linear system of difference equations

where is a
nonsingular matrix with real entries and
If for some

is specified, then (1.1) is called an initial value problem (IVP). The solution of this IVP is given by

where is the fundamental matrix defined by

However, (1.1) is called reducible to equation

if there is a nonsingular matrix with real entries such that

Let be a
matrix function whose entries are real-valued functions defined for
. Consider the system

Let be a fundamental matrix of (1.7) satisfying
. This
can be used to transform (1.1) into (1.5).
Stability properties of (1.1) can be deduced by considering the reduced form (1.5) under some additional conditions. In this study, we first give a theorem on the reducibility of (1.1) into the form of (1.5) and then obtain asymptotic stability of the zero solution of (1.1).
2. Reducible Systems
In this section, we give a theorem on the structure of the matrix , and provide an example for illustration. The results in this section are discrete analogues of the ones given in [1].
Theorem 2.1.
The homogeneous linear difference system (1.1) is reducible to (1.5) under the transformation (1.6) if and only if there exists a regular real matrix
such that

hold.
Proof.
Let and
be defined as above. Under the transformation (1.6), (1.1) becomes

and after reorganizing, we get

Thus, (1.1) is reducible to (1.5) with

Clearly, is the unique solution of the IVP:

where
This problem is equivalent to solving (2.1). â–¡
Corollary 2.2.
The homogeneous linear system of difference equation (1.1) is reducible to

with a constant matrix under transformation (1.6) if and only if there exists a
regular real matrix
defined for
such that


hold.
Below, we give an example for Corollary 2.2 in the special case . To obtain the matrix
, we choose a suitable form of the matrix
.
Example 2.3.
Consider the system

where
-
(i)
are real-valued functions defined for
such that
for all
-
(ii)
for all
-
(iii)
We also assume that for all ,

It is easy to verify that if we take

where



then (2.7) holds. Moreover, from (2.8) we have

In case for every
, that is,

the relations (2.10), (2.12), and (2.13) take the form

where is a real constant and
,
are arbitrary real constants such that
Corollary 2.4.
If there exists a regular constant matrix
such that

then (1.1) reduces to (2.6) with
It should be noted that in case the constant matrices and
commute, that is,
, then
must be a constant matrix as well.
3. Stability of Linear Systems
It turns out that to obtain a stability result, one needs take , a periodic matrix [2]. Indeed, this allows using the Floquet theory for linear periodic system (1.7).
We need the following three well-known theorems [3–5].
Theorem 3.1.
Let be the fundamental matrix of (1.1) with
The zero solution of (1.1) is
-
(i)
stable if and only if there exists a positive constant M such that
(3.1) -
(ii)
asymptotically stable if and only if
(3.2)where
is a norm in
.
Theorem 3.2.
Consider system (1.1) with a constant regular matrix. Then its zero solution is
-
(i)
stable if and only if
and the eigenvalues of unit modulus are semisimple;
-
(ii)
asymptotically stable if and only if
, where
is an eigenvalue of
is the spectral radius of
Consider the linear periodic system

where ,
for some positive integer N.
From the literature, we know that if with
is a fundamental matrix of (3.3), then there exists a constant
matrix, whose eigenvalues are called the Floquet exponents, and periodic matrix
with period N such that
Theorem 3.3.
The zero solution of (3.3) is
-
(i)
stable if and only if the Floquet exponents have modulus less than or equal to one; those with modulus of one are semisimple;
-
(ii)
asymptotically stable if and only if all the Floquet exponents lie inside the unit disk.
In view of Theorems 3.1, 3.2, and 3.3, we obtain from Corollary 2.2 the following new stability criteria for (1.1).
Theorem 3.4.
The zero solution of (1.1) is stable if and only if there exists a regular periodic matrix
satisfying (2.8) such that
-
(i)
the Floquet exponents of
have modulus less than or equal to one; those with modulus of one are semisimple;
-
(ii)
; those eigenvalues of
of unit modulus are semisimple.
Theorem 3.5.
The zero solution of (1.1) is asymptotically stable if and only if there exists a regular periodic matrix
satisfying (2.8) such that either
-
(i)
all the Floquet exponents of
lie inside the unit disk and
; those eigenvalues of
of unit modulus are semisimple; or
-
(ii)
the Floquet exponents of
have modulus less than or equal to one; those with modulus of one are semisimple; and
Remark 3.6.
Let be periodic with period N. The Floquet exponents mentioned in Theorem 3.3 are the eigenvalues of
where
Example 3.7.
Consider the system

Note that the conditions of Example 2.3 are all satisfied. It follows that

Now,

for which the eigenvalues are
On the other hand, for

if
and
if
.
Applying Theorems 3.4 and 3.5, we see that the zero solution of (3.4) is asymptotically stable if and is stable if
In fact, the unique solution of (3.4) satisfying is

where ,
,
,
, and
.
It is easy to see that if
and
is bounded if
Remark 3.8.
In the computation of ,
is calculated by using Example 2.3, and
is obtained by the method given in [6, 7].
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Acknowledgment
The authors would like to thank to Professor Ağacık Zafer for his valuable contributions to Section 3.
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Tiryaki, A., Misir, A. Reducibility and Stability Results for Linear System of Difference Equations. Adv Differ Equ 2008, 867635 (2008). https://doi.org/10.1155/2008/867635
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DOI: https://doi.org/10.1155/2008/867635