Reducibility and Stability Results for Linear System of Difference Equations
Advances in Difference Equations volume 2008, Article number: 867635 (2008)
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.
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 .
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
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:
This problem is equivalent to solving (2.1). □
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
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 .
Consider the system
are real-valued functions defined for such that for all
We also assume that for all ,
It is easy to verify that if we take
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
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 . Indeed, this allows using the Floquet theory for linear periodic system (1.7).
Let be the fundamental matrix of (1.1) with
The zero solution of (1.1) is
stable if and only if there exists a positive constant M such that(3.1)
asymptotically stable if and only if(3.2)
where is a norm in.
Consider system (1.1) with a constant regular matrix. Then its zero solution is
stable if and only if and the eigenvalues of unit modulus are semisimple;
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
The zero solution of (3.3) is
stable if and only if the Floquet exponents have modulus less than or equal to one; those with modulus of one are semisimple;
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).
The zero solution of (1.1) is stable if and only if there exists a regular periodic matrix satisfying (2.8) such that
the Floquet exponents of have modulus less than or equal to one; those with modulus of one are semisimple;
; those eigenvalues of of unit modulus are semisimple.
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
all the Floquet exponents of lie inside the unit disk and ; those eigenvalues of of unit modulus are semisimple; or
the Floquet exponents of have modulus less than or equal to one; those with modulus of one are semisimple; and
Let be periodic with period N. The Floquet exponents mentioned in Theorem 3.3 are the eigenvalues of where
Consider the system
Note that the conditions of Example 2.3 are all satisfied. It follows that
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
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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