- Research Article
- Open Access

# Reducibility and Stability Results for Linear System of Difference Equations

- Aydin Tiryaki
^{1}and - Adil Misir
^{2}Email author

**2008**:867635

https://doi.org/10.1155/2008/867635

© A. Tiryaki and A. Misir. 2008

**Received:**8 August 2008**Accepted:**29 October 2008**Published:**10 November 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.

## Keywords

- Difference Equation
- Unit Disk
- Asymptotic Stability
- Matrix Function
- Stability Result

## 1. Introduction

where is a nonsingular matrix with real entries and

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.

where

This problem is equivalent to solving (2.1). □

Corollary 2.2.

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)

where is a real constant and , are arbitrary real constants such that

Corollary 2.4.

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)
- (ii)

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

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

for which the eigenvalues are

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

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].

## Declarations

### Acknowledgment

The authors would like to thank to Professor Ağacık Zafer for his valuable contributions to Section 3.

## Authors’ Affiliations

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