Stability conditions a class of linear delay difference systems

where A is a 2 × 2 constant matrix, k is a nonnegative integer and a is a real number. Difference equations and their stability are the appropriate mathematical representations for discrete processes, which have special importance in areas such as population models. Difference equations are mentioned below which usually happen as a result of linearization the population models. Recently, stability of the difference equations as the system (1) has been investigated by many researchers. For instance Levin and May (1976) obtained necessary and sufficient conditions for the asymptotic stability of the delay difference equation. (1) xn+1 − axn − Axn−k = 0 *Corresponding author: Serbun Ufuk Değer, Institute of Sciences, Kastamonu University, Kastamonu, Turkey E-mail: sudeger@kastamonu.edu.tr


Introduction
In this paper we study the asymptotic stability of the solutions of linear delay difference systems; where A is a 2 × 2 constant matrix, k is a nonnegative integer and a is a real number. Difference equations and their stability are the appropriate mathematical representations for discrete processes, which have special importance in areas such as population models. Difference equations are mentioned below which usually happen as a result of linearization the population models. Recently, stability of the difference equations as the system (1) has been investigated by many researchers. For instance Levin and May (1976) obtained necessary and sufficient conditions for the asymptotic stability of the delay difference equation. (1) x n+1 − ax n − Ax n−k = 0

ABOUT THE AUTHORS
My key research activities include: Delay Differential Equations, Delay Difference Equations, Neutral Differential Equations, Neutral Difference Equations and stability of these equations. More generally, we can say Differential Equations, Difference Equations and stability of these equations.
The research reported in this paper relates to the stability of the systems that cannot be examined by differential equations such as populations and economics. In order to investigate the stability of these systems, difference equations should be used. Our research represents a generalized method to define the stability of these systems which involve populations and economics.

PUBLIC INTEREST STATEMENT
In this paper, stability analysis of a class of Linear Delay Difference Systems was investigated. This system is a generalized version of the difference equation of a population model using a matrix A instead of scaler a. In stability analysis of a class of Linear Delay Difference Systems, we researched the asymptotic stability of these systems. It is known that these systems are asymptotically stable if and only if all the roots of characteristic equation of the system are inside the unit disk. We applied a root analysis method using a qualitative approach in order to prove the asymptotic stability of the systems. We have created new necessary and sufficient conditions for the systems which are asymptotically stable; in case of matrix, A has complex coefficients.
where b is a real number and k is a nonnegative integer. As a result, Levin and May (1976) obtained zero solution of (2) which is asymptotically stable iff Later, Clark (1976) studied the delay difference equation where a, b are arbitrary real numbers and k is a positive integer. Clark indicated if then (3) is asymptotically stable, although his work brings significant innovations which only give sufficient condition for the asymptotic stability of (3). After that, Kuruklis (1994) demonstrated that the zero solution of (3) is asymptotically stable iff |a| < k+1 k , and where is the solution of sin k Figure 1). Matsunaga and Hara (1999) considered difference system and they obtained necessary and sufficient conditions for the asymptotic stability of the zero solution of the system, where A is a 2 × 2 constant matrix and k is a nonnegative integer. Matsunaga (2004) showed new stability conditions of generalized linear delay difference system (2) (3) if k is even, x n+1 − ax n + Bx n−k = 0, which was an extension of Matsunaga (1999). Kipnis and Malygina (2011) and Cermák and Jánsky (2014) have also investigated a similar problem and the authors should clearly demonstrate the novelty and originality of their results like ours where B is a 2 × 2 constant matrix, k is a nonnegative integer and a is a real number. The purpose of this paper is to obtain new results for the asymptotic stability of zero solution of system (1) when A is a constant matrix. Thus, we need to show that zero solution of system (1) is asymptotically stable iff all the roots of its characteristic equation are inside the unit disk (Elaydi, 2005). Now we will give some basic information that we use the lemmas.

Preliminary
We consider system (1). Characteristic equation of system (1): where I is a 2 × 2 identity matrix. If we write x n = Py n for a regular matrix P, then we get following system: Thus, A can be given one of the following two matrices in Jordan form (Elaydi, 2005): Now, (I) and (II) will be investigated, respectively ( Figure ref2); (I) We consider (4) where A is given by (I), If the results of Kuruklis are applied to (5), then Theorem 1 is obtained.
Theorem 1 Assume that a ≠ 0. Then, the zero solution of system (1) is asymptotically stable if and only if |a| < k+1 k , and for j = 1, 2 where is the solution of Thus, it is just enough to consider the case Δ ( ) = 0 under the condition 2 < < to investigate the roots of (7). We know that system (1) is asymptotically stable iff all the roots of (6) with 2 < < are inside the unit disk. Moreover, Δ ( ) has no real root when b ≠ 0. For a ≠ 0 and 2 < < , (6) can be written equivalently: where = a and q = b a k+1 . Thus, all the roots of (6) are inside the unit disk iff all the roots of (8) are inside the disk | | < 1 |a| . Now, some auxiliary lemmas can be given.

Some auxiliary lemmas and main theorems
Lemma 1 (Matsunaga, 2004) Let a ≠ 0 and q = b a k+1 . Then, the zero solution of system (1) is asymptotically stable iff all the roots of (8) are inside the disk | | < 1 |a| .
Now our aim is to calculate the locations according to movement of the roots via root analysis of the system (1) as q varies. Note that, for q = 0, the roots of (8) are 0 multiplicity k and 1(simple). Furthermore, (8) has no real root when q ≠ 0. Firstly, the existence region for arguments of complex roots of (8) will be found.
Lemma 2 Suppose that q > 0 and 2 < < . Let re i with r > 0 be a complex root of (8). Then, the following are provided: where ‖ ⋅ ‖ denotes greatest integer function.
Proof Let = re i with r > 0. From (6), we have Thus, we get From the real part and the imaginary part of (9) and are obtained. From (10), the following equality can be derived From (10) and (12), we can write Consider q > 0 and − 2 < 2 − < 2 . Using (12) and (13), we derive cos k + 1 − > 0 and cos k − > 0. Thus, we can write and From the above inequalities, the following result is obtained: Hence, the proof is completed. For 2 < 2 − < 3 2 , the proof is similar. ✷ Lemma 3 Suppose that q < 0 and 2 < < . Let re i with r > 0 be a complex root of (8). Then, the following holds: Proof For q < 0, the proof is analogous to the above. ✷ Since Lemmas 2 and 3, the following notation can be used as whole existence region for arguments of complex roots of (8).
The following lemma calculates arguments of complex roots of (8) on the unit circle.

Lemma 5
We suppose that 2 < < . For n = 0, where = 0 is argument of complex roots of (8) on the unit circle.
Proof The proof is obvious. ✷ Now, the movement of the roots of (8) will be investigated as q varies on the complex plane.
Lemma 6 Suppose that 2 < < . When q increases to 0, simple root = 1 of (8) is inside the unit circle.
Case (b) 2 < 2 − < 3 2 . It will be similar to case (a), from the increasing property of I( ) andI . This way the claim of this lemma is provided with (17).  (17), cos k − < 0 and the increasing property of Thus, we show that roots of (8) move continuously away from zero, while |q| increases except for the root ̃ from Lemmas 6 and 7.
Proof We should note that takes minimum value at n = 0. Also squaring both of (10) and (11), adding side by side to them, we get |q| = 2 sin 2 , and for n = 0, we have ✷ Theorem 2 Assume that 2 < | | < , a = 1 and = n (n = 0, ±1, ±2, … , ±k.) are arguments of complex roots of (8) on the unit circle while b < 0. Then system (1) is asymptotically stable iff where k is a nonnegative integer. Proof Assume that system (1) is asymptotically stable with 2 < | | < as a = 1. It is known that the minimum value of q is − cos k+ 1 2 − 2k+1 when = 1 from Lemma 8. Thus, we use the continuity q with respect to ; it is written for every | | < 1. Now, assume that above inequality is provided. It is known that the minimum value of q is − cos k+ 1 2 − 2k+1 when = 1 from Lemma 8. Also via Lemma 6 and continuity q with respect to , we can write every root of is inside unit disk as q is in the neighborhood of 0. So system (1) is asymptotically stable. ✷ If the matrix A is chosen n-dimensionally, that is, A is a n × n constant complex matrix, then the following theorem is obtained: Theorem 3 Assume that 2 < | | < anda = 1. Let q j e i j j = 1, 2, … , n be the eigenvalues of A. Then, the system (1) is asymptotically stable iff where q j , j are real numbers and 2 Proof Let b j e i j j = 1, 2, … , n be the eigenvalues of A; the characteristic equation of the system (1) is given by Thus, Theorem 3 can be seen as a result of Theorem 2. ✷ Now the values |q| will be found for a root of (8) on the circle = 1 |a| . The following lemma provided these values of |q| in terms of |a| and the complex root arguments .
Lemma 9 Let re i be a root of (8) (18), the value of |q| increases with respect to | | and, then the minimum value of |q| is equivalent to the minimum value of | | which provides r = 1 |a| . If it is used the notation − cos then the minimum value of S( ) becomes equivalent to the minimum value of |q|. Hence, by means of Lemmas 2, 3 and 7, it is obvious that S( ) is strictly increasing with respect to | |, and S( ) only has a local minimum at + . Now necessary and sufficient conditions for the roots of (8) to be inside the disk | | = 1 |a| are provided. For q > 0, a graphic of S( ) with k = 1 and = 3 is presented.

Proof
Case (i) 1 |a| ≥ 1, i.e. 0 < |a| ≤ 1. From Figure 3, there is at least one root 1 of S( ) in − 3 2 k+1 , 0 . Therefore, from Lemmas 6-9, trace of branch | | = 1 shows that this trace is inside the disk | | = 1 |a| while it is increasing from 0 to Q 1 (a) |a| k+1 , it is on the circle | | = 1 |a| when it is at Q 1 (a) |a| k+1 and it is outside the disk | | = 1 |a| while it is increasing from Q 1 (a) |a| k+1 . Hence, our claim is the following: all the roots of (8) are This shows that there is a root of (8) belonging to the branch | | = 1 as | | ≥ 1 |a| . ✷ Now for the following lemma for q < 0, a graphic of S( ) with k = 1 and = 3 ispresented.
Proof Firstly, assume that −Q 4 (a) < −Q 3 (a) < −Q 2 (a). It is known that the value of satisfying r = S( ) = 1 inside H + 0 is 0 . Now, the locations of the roots of (8) are investigated. In case of q = 0, the roots of (8) are 0 and 1. A branch of | | = 1 is in the region H + 0 inside while it is increasing to 0 from q. (at a very small neighborhood of q = 0) There are three cases: Figure 4, there is at least one root 2 of S( ) in 0 , − 2 k+1 . So, from Lemmas 6-9, trace of branch | | = 1 shows that this trace is inside the disk | | = 1 |a| while q is decreasing from 0 to −Q 2 (a) |a| k+1 , it is on the circle | | = 1 |a| when it is at −Q 2 (a) |a| k+1 and it is outside the disk | | = 1 |a| while it is increasing from −Q 2 (a) |a| k+1 . Here our claim is: All the roots of (8) are inside the disk | | = 1 |a| iff −Q 2 (a) |a| k+1 < q < 0.

Then, system (1) is asymptotically stable provided
where k is a nonnegative integer.