Reducibility and Stability Results for Linear System of Difference Equations
© A. Tiryaki and A. Misir. 2008
Received: 8 August 2008
Accepted: 29 October 2008
Published: 10 November 2008
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 .
This problem is equivalent to solving (2.1). □
Consider the system
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).
The zero solution of (1.1) is
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).
Consider the system
The authors would like to thank to Professor Ağacık Zafer for his valuable contributions to Section 3.
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