Research Article  Open Access
S. A. Ivanov, M. M. Kipnis, V. V. Malygina, "The Stability Cone for a Difference Matrix Equation with Two Delays", International Scholarly Research Notices, vol. 2011, Article ID 910936, 19 pages, 2011. https://doi.org/10.5402/2011/910936
The Stability Cone for a Difference Matrix Equation with Two Delays
Abstract
We provide geometric algorithms for checking the stability of matrix difference equations with two delays such that the matrix is nilpotent. We give examples of how our results can be applied to the study of the stability of neural networks.
1. Introduction
The problem of the stability of the equation with real coefficients is basically solved [1–4].
The stability of matrix (1.1) with special matrices and was studied in [5, 6]. The case , , where is the identity matrix, was studied in [7] without dimension restriction. In the paper [8] the dimension is also not bounded, and the results of [7] are generalized: it assumes that , , . The representation of the solutions of (1.1) with commuting matrices is given in [9] without considering a stability problem.
To the best of the authors' knowledge, the stability of (1.1) with complex coefficients has not been studied yet.
In this paper we provide geometric algorithms for checking the stability of (1.1) with two delays , , for two cases: (1) are complex numbers, (2) are simultaneously triangularizable matrices. The results of this paper are based on the decomposition method (parameter plane method) [10, 11].
Matrices commute in all the above articles, which implies the possibility of simultaneous triangularization [12]. Therefore, our method can be applied to all of the abovementioned cases. In the present paper the case is studied along with other values . The case is very important, so we separately examined it in detail in the paper [13].
The paper is organized as follows. In Section 2, we introduce the curve of decomposition and point out its key property of symmetry. In Section 3, we define the basic ovals and formulate their properties. In Section 4, we define a property of stability, which coincides with usual stability if . Later in that section we solve a problem of geometric checking stability of (1.1) with positive real and complex . In Sections 5 and 6, we give a method of geometric checking the stability of (1.1) with complex coefficients and simultaneously triangularizable matrices, correspondingly. Finally, in Section 7, we employ our results to derive the stability conditions for neural nets.
2. Decomposition Curve for Given
Consider the scalar variant of (1.1). The characteristic polynomial for (1.1) is If , , then the trajectory of (1.1) splits into independent trajectories, and degree of polynomial (2.1) gets smaller after the substitution : Therefore, often (but not always) we will assume that the delays are relatively prime.
Definition 2.1. decomposition curve for given , , is a curve on the complex plane of the variable defined by the equation
Parameter moves along the interval of length , the starting point of which is not fixed. We also call the curve (2.3) hodograph.
In this and the next sections we will consider only real positive values of . Starting from Section 5 we will get rid of this restriction. Obviously, if we assume in (2.1) that and , , then (2.1) will have a root . Hodograph (2.3) splits the complex plane into the connected components. This decomposition is called the decomposition [10]. If we put , , and substitute any two internal points from one of the connected components of decomposition for coefficient in polynomial (2.1), then the polynomials obtained will have equal number of roots inside the circle of radius centred at the origin of the complex plane. In particular, if , , and the substitution of some inner point of a component of decomposition into (1.1) gives a stable equation, then the substitution of any other internal point from that component also gives a stable equation.
Let us point out a key property of symmetry of hodograph (2.3).
Lemma 2.2 (symmetry). If are coprime, then hodograph (2.3) is invariant under the rotation by .
Proof. For coprime there exist such that From (2.3), (2.4) it follows that Lemma 2.2 is proved.
From now and further we will assume that for any complex , while will be assumed as multivalued function, and the equality will mean that one of the values of equals to .
The following lemma asserts that some part of the complex plane is free from points of the hodograph .
Lemma 2.3. Let be coprime, , . Let be the least positive root of the equation , and let . Then, for any , from it follows that .
Proof. The function is periodic, increasing in for any and decreasing in . In addition, for any . Let us assume, in order to get a contradiction, that (2.6) and are true for some . Then there exists a positive integer and such that From (2.3), (2.7) it follows that Since are coprime and , let us find natural numbers such that , . Then (2.8) implies which contradicts (2.6) and the inequality following from (2.7). Lemma 2.3 is proved.
3. Basic Ovals
For hodograph (2.3) the equality takes place. If then let us look at a closed curve on the complex plane, which we call the basic oval. This curve is an image of an interval under the map defined by (2.3). Here is the least positive root of the equation . We are also interested in those parts of hodograph (2.3) that can be obtained by the rotation of the basic oval by the angles , (see Lemma 2.2). We also call them the basic ovals. Here is a formal definition.
Definition 3.1. Let be coprime, , , , and let (2.4), (3.1) hold. The basic oval for (1.1) is a closed curve given by (2.3), where the variable runs from to , where is the least positive root of the equation
From Lemma 2.2 and formula (2.5) it follows that all basic ovals can be obtained from by rotation by the angles , .
Considering Definition 3.1, we get the following. For existence of the basic oval it is necessary that . If and , then the complex number 0 is outside any oval, and the intersection of all ovals is empty. If , then for fixed , , the basic oval is unique. That is why the results related to the stability of (1.1) are different for and .
The basic oval decreases as increases from 0, and it shrinks to the point as reaches (Figure 1 for and Figure 2 for ).
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
Lemma 3.2. Let be coprime, , , , , and . If the complex number lies outside the basic oval , then characteristic polynomial (2.1) has a root such that .
Proof. Let us fix , and let the complex number lie outside the basic oval . Having changed to in Definition 3.1 let us consider the system of ovals . If , then the ovals include a circle of an arbitrarily large radius. Therefore, there exists such that the point is inside the oval . The ovals and are homotopic, therefore, there exists such that lies on the curve , which means the existence of a root of characteristic polynomial (2.1) such that . Lemma 3.2 is proved.
4. Localization of Roots of Characteristic Polynomial (2.1) for Real Nonnegative and Complex
For the stability of (1.1) it is required that all the trajectories are bounded. But sometimes one needs to strenghten or weaken the stability requirement. It justifies the following definition.
Definition 4.1. Equation (1.1) is said to be stable if for any of its solutions the sequence is bounded, and asymptotically stable if for any of its solutions one has .
If , then the concept of (asymptotic) stability coincides with the concept of usual (asymptotic) stability. Evidently, (1.1) is stable, if there are no roots of polynomial (2.1) outside the circle of radius centred at the origin, and there are no multiple roots of the polynomial on the boundary of circle. Equation (1.1) is asymptotically stable if and only if all the roots of its characteristic polynomial (2.1) lie inside the circle of radius with the center at 0.
Let us call the equation (asymptotically) unstable if it is not (asymptotically) stable. As we noted in Section 2, if , then the proportional change of both delays in (1.1) has no influence on stability. It is not the case if . It is easy to see that the equation is asymptotically stable if and only if (1.1) is asymptotically stable. It implies the following important observation: proportional increase of both delays with the conservation of the coefficients in (1.1) preserves asymptotic stability if and may not preserve it if .
Definition 4.2. Let be coprime, , , and . The stability domain is defined to be a set of all complex numbers such that for any the number lies inside the basic oval .
Under the same conditions if are not coprime and , let us put .
Theorems 4.3–5.2 will justify the name “stability domain” for a little later. Evidently, for coprime such that the domain has the following properties. If , then is the connected domain on the complex plane, containing 0, whose boundary is the decomposition curve (2.3). If , then the domain degenerates into the point . If , then the domain is empty. If , then the domain is not defined in view of the fact that the basic ovals (Definition 3.1) are not defined.
If , then the domain is a set of points lying inside the oval . If , then includes 0. If , then is nonempty and does not contain 0. If , then degenerates into the point . Finally, if , then the domain is not defined.
The following theorems are based on the localization of roots of polynomial (2.1) with nonnegative and complex with respect to the circle of radius centred at the origin.
Theorem 4.3. Let be coprime, , , . (1)If , then for any (1.1) is unstable.(2)If , then for any complex (1.1) is unstable; for it is stable (nonasymptotically).(3)If , then (1.1) is asymptotically stable if and only if the complex number lies inside the stability domain . (4)If , then (1.1) is stable if and only if the complex number lies either inside or on the boundary of .
Proof. (1) Let . Let us find such that
Taking into account the inequality , let us consider basic ovals , having replaced by in Definition 4.1. Since , the system of ovals has no intersections. Hence, for any complex number there exists such that lies outside the oval . By Lemma 3.2 (1.1) is unstable. Since , it is unstable. Statement 1 is proved.
(2) Let . If , then statement 2 of Theorem 4.3 is obvious. Let . If , then lies outside the oval . If and is even, then lies outside the oval . If and is odd, then lies either outside the oval or outside the oval . In any case (1.1) is unstable by Lemma 3.2. Statement 2 is proved.
(3) Let . Let the number be inside the domain . Then for any the number lies inside the oval . By Lemma 2.3 the beam drawn on the complex plane from 0 to does not intersect curve (2.3). Therefore, polynomial (2.1) has the same number of roots inside the circle of radius for given and for . However if , then all the roots of (2.1) lie inside the circle of radius centred at 0. Therefore, (1.1) is asymptotically stable for given .
If lies on the boundary of the domain or outside it, then lies either on the boundary of one of the basic ovals or outside one of them, and by Lemma 3.2 (1.1) is asymptotically unstable.
(4) If lies outside the domain , then the conclusion of statement 4 of Theorem 4.3 is a straightforward consequence of Lemma 3.2. If lies inside , then the conclusion of statement 4 of Theorem 4.3 is a straightforward consequence of statement 3 of Theorem 4.3. Let lie on the boundary of . Then for any root of polynomial (2.1) either or . In the latter case in view of the inequality we have
hence the root such that is simple. Theorem 4.3 is proved.
If in (1.1) the least delay is equal to 1, then the situation is essentially different from the case .
Theorem 4.4. Let , , . (1)If , then for all complex numbers (1.1) is unstable.(2)If , then (1.1) is asymptotically stable if and only if the complex number lies inside the domain .(3)If , then (1.1) is stable if and only if the complex number lies inside .
Proof. (1) Let , and let be a given complex number. Let us find such that and the point is located outside the oval obtained from Definition 3.1 by substituting for . By Lemma 3.2 there exists a complex root of polynomial (2.1) such that , so instability of (1.1) is proved. Let . Then the previous arguments also prove instability provided that . However, if , then under the assumption that the number is a multiple root of polynomial (2.1), and consequently, (1.1) is also unstable. Statement 1 of Theorem 4.4 is proved.
(2) Let . Since is the domain of inner points of the oval , it is connected. The function (see (2.3)) increases as moves either from 0 to or from 0 to . Therefore, there are no points of hodograph (2.3) inside . To complete the proof of asymptotical stability of (1.1) at any point of it is sufficient to prove that there exists at least one point inside the oval such that the equation is asymptotically stable for .
CASE 1. Let . Then the point lies inside . If , then polynomial (2.1) has the multiple root and the simple root . This gives the asymptotic stability, in view of .
CASE 2. Let . Let us consider the point at the boundary of and consider characteristic polynomial (2.1) with given :
The equation transforms into
One of roots of (4.5) is equal to , while others lie inside the circle of radius centred at the origin in view of the inequality [14].
Let us return to (2.1), and let us figure out in what direction the root moves as the coefficient moves from the point toward the interior of so that , . From (2.1) it follows that for we have
and in view of we get . Therefore, implies . Consequently, there exist values of inside providing asymptotic stability of (1.1), therefore, for any value inside (1.1) is asymptotically stable.
(3) The proof of Statement 3 of Theorem 4.4 is analogous to the proof of Statement 4 of Theorem 4.3. Theorem 4.4 is proved.
5. Stability of (1.1) with Complex Coefficients
Let us change the variables in (1.1) so that it has no influence on (asymptotic) stability: Equation (1.1) changes to where The characteristic polynomial for (5.2) has the form It is related to (2.1) by the change , that saves the absolute values of roots of the equation. It is important for us that new (5.2) has a real nonnegative coefficient at , in view of (5.3). This allows us to apply the results of the previous section. Therefore, from Theorems 4.3 and 4.4 we immediately derive the following theorems providing an answer to the question on the stability of (1.1) with complex coefficients , .
Theorem 5.1. Let be coprime, , , .
(1)If , then for any complex (1.1) is unstable.(2)If , then for any (1.1) is unstable; for it is stable (nonasymptotically).(3)If , then (1.1) is asymptotically stable if and only if the complex number lies inside the domain .(4)If , then (1.1) is stable if and only if the complex number lies either inside or on its boundary.
Theorem 5.2. Let , , .
(1)If , then for any complex (1.1) is unstable.(2)If , then (1.1) is asymptotically stable if and only if the complex number lies inside the domain .(3)If , then (1.1) is stable if and only if the complex number lies either inside or on its boundary.
Example 5.3. Let , , in (1.1). Let . For every given value let us find all values of the complex coefficient for which (1.1) is stable. The answer is demonstrated by Figure 4. Let us give some comments. First calculate , . If , then to find stability domain one does not need to use Theorems 4.3–5.2. The domain is a circle given in Figure 4(a). Since are coprime, then for , by Theorem 5.1, (1.1) stable if and only if . The corresponding “curved hexagon” is shown in Figure 4(a). Similarly for the condition is necessary and sufficient for asymptotic stability of (1.1). The corresponding “curved hexagon” is shown in Figure 4(b). For the stability criterion is the condition . The corresponding “curved triangle” is shown in Figure 4(a). Similarly the “digon” for is shown in Figure 4(a), and the “curved triangle” for is shown in Figure 4(b). For , according to Theorem 5.2, the stability criterion for (1.1) is (Figure 4(b)). The corresponding “stability oval” is shown in Figure 4(a).
6. Stability Cones for Matrix Equation (1.1) with Simultaneously Triangularizable Matrices
Let us consider a matrix equation; . The characteristic equation for (6.1) is
Definition 6.1. Matrix equation (6.1) is called stable if for every solution the sequence is bounded. Equation (6.1) is called asymptotically stable if holds for every solution .
Obviously, matrix equation (6.1) is asymptotically stable if and only if all the roots of characteristic polynomial (6.2) lie inside the circle of radius with the center at 0. We also observe that if at least one root of (6.2) lies outside the circle of radius with the center at 0, then (6.1) is unstable.
In this paper we consider (6.1) only with triangularizable matrices . It is known [12] that if the matrix is nilpotent, then can be simultaneously triangularized.
Definition 6.2. If , then the stability cone for given is a set of points such that and the intersection of the set with any plane is the stability domain . If , then the stability cone for given is a set of points such that and the intersection of the set with the plane is the domain .
Let us define a stability cone as the stability cone for .
Returning to Figure 3, we can interpret the figures in Figure 3(a) as sections of the stability cone for at different heights , and the ones in Figure 3(b) as sections of the stability cone for , and so on.
(a)
(b)
(c)
(d)
(a)
(b)
The stability cones for are the intersections of conical surfaces formed by the basic ovals as the parameter changes from 0 to (Figure 5).
Let us consider the simple case of a diagonal system with complex entries , . Let us construct the points in in the following way:
It follows from the definition of the stability cone and from Theorems 5.1–5.2 that (6.3) is asymptotically stable if and only if all the points lie inside the stability cone for given . All the points with , are considered as inner points of the stability cone.
The natural extension of the the class of diagonal systems is that of systems with simultaneously triangularizable matrices. The following theorem is our main result.
Theorem 6.3. Let , let the numbers be coprime, and . Let , and , and , where and are lower triangle matrices with elements . Let one construct the points , by the formulas (cf. (6.4))
Then (6.1) is asymptotically stable if and only if all the points lie inside the stability cone for the given .
If some point lies outside the stability cone, then (6.1) is unstable.
Proof. Let us make the change . Then (6.1) transforms to the following one: The characteristic polynomial for (6.6) has the form It coincides with the characteristic polynomial of diagonal system (6.3). Therefore, from statement 3 of Theorem 5.1 (for ) and from statement 2 of Theorem 5.2 (for ) we obtain asymptotic stability if all the points lie inside the stability cone. Similarly from statement 4 of Theorem 5.1 (for ) and statement 3 of Theorem 5.2 (for ) we obtain instability of (6.1) if some point lies outside the cone. Theorem 6.3 is proved.
7. Applications to Neural Networks
Let us apply the results of the previous sections to the problem of the stability of discrete neural networks similar to continuous networks studied in [15, 16]. Let us consider a ring configuration of neurons (Figure 6) interchanging signals with the neighboring neurons.
Let be a signal of the th neuron at the th moment of time. Let us suppose that the neuron reaction on its state, as well as on that of the previous neuron, is units delayed, and reaction on the next neuron is units delayed. The neuron chain is closed, and the first neuron is next to the th one. Let us assume that the neurons interchange the signals according to the equations where are sufficiently smooth realvalued functions of a real variable. Let us assume that there is a real number such that the stationary sequences form a solution of (7.1). Let us introduce the variables and the vector , and let us linearize system (7.1) in new variables about zero. We get (6.1) with the circulant [17] matrices Here , , . Let us introduce a matrix (a lines permutation operator): Then , , therefore, diagonalization generates simultaneous diagonalization of . The eigenvalues of are 1, , , where . Therefore, Granting (7.4), by Theorem 6.3, we can build points in for system (6.1), (7.2): We get the following consequence of Theorem 6.3.
Corollary 7.1. If for every the point defined by formulas (7.5) lies inside the stability cone for given , then system (6.1), (7.2) is asymptotically stable. If at least one point lies outside the stability cone for given , then system (6.1), (7.2) is unstable.
Let us proceed to the problem of stability of a neural network with a large number of neurons. The points defined by (7.5) lie on the closed curve
If , then the points are dense in the curve (7.6). We get the following consequence of Theorem 6.3.
Corollary 7.2. Let one consider system (6.1), (7.2) with matrices . If any point of the curve (7.6) lies inside the stability cone for given , then system (6.1), (7.2) is asymptotically stable for any . If at least one point of the curve (7.6) lies outside the stability cone for given , then there exists such that system (6.1), (7.2) is unstable for any .
Example 7.3. Let us consider the ring of neurons shown in Figure 6. Put , , , , . Let us pose a question: what are the values of for which the system of two neurons described by (6.1), (7.2) is stable?
For applications of Corollaries 7.1 and 7.2 we construct the curves (7.6) for six values of (Figures 7(a) and 7(b)). Assuming , we construct the points , , in each of the six curves. Then we construct the stability cone for given (Figure 7(b)). It is the stability cone for . In Figure 7(b) we see that two curves (7.6) corresponding to the values are hidden inside the cone. The point corresponding to the values is on the surface of the cone, while all other points of the curve (7.6) for lie inside the cone. Therefore, according to Corollaries 7.1 and 7.2, if , then system (6.1), (7.2) is asymptotically stable for any . In our interpretation it means that the neuron configuration in Figure 6 is stable for any number of neurons. If , then all the curves (7.6) lie entirely or partially outside the cone. In view of Corollaries 7.1 and 7.2 system (6.1), (7.2) is unstable. In our interpretation it means that even the configuration of two neurons is unstable.
(a)
(b)
Now let us consider Example 7.3 without the condition . Under assumptions of Example 7.3, for any value there exists a number such that system (6.1), (7.2) is asymptotically stable if and unstable if . Corollaries 7.1 and 7.2 allow us to find by means of construction of different stability cones. Table 1 shows how depends on . The value is a Lyapunov exponent [18] for system (6.1), (7.2).

Example 7.4. Let us consider the neuron chain shown in Figure 6. Let us fix the parameters: , , , , . In this example we demonstrate how the change of the parameter in system (6.1), (7.2) changes the mutual location of the curve (7.6) and the stability cone. In Figure 8(a) we show the curves (7.6) corresponding to the values , , , and the points , , for . The upper part of the stability cone corresponding to is removed. In Figure 8(b), one third of the lateral surface of the stability cone is removed too. Figure 8 demonstrates that the curves (7.6) lie inside the cone if . Therefore, if , then system (6.1), (7.2) is asymptotically stable for any . If , then the point defined by (7.5) lies on the cone surface. If , then the point lies outside the cone, and this shows the instability of system (6.1), (7.2). In our interpretation, for , the neuron chain is stable no matter how many neurons are in the chain, and for it is unstable even if it consists only of two neurons.
(a)
(b)
8. Conclusion
The condition is sufficient for the asymptotic stability of matrix (6.1) [19], and it does not require simultaneous triangularization of the matrices . There are sufficient conditions for stability of nonautonomous scalar difference equations in [20–22].
Stability cones for differential matrix equations with one delay are introduced in the paper [23] and for some integrodifferential equations in the paper [24].
There are images of the stability domains in the space of parameters of scalar differential equations with delays [25, 26] and scalar difference equations with delays [25]. The results of the papers [25, 26] imply that there is no simple complete description of the stability domains for these equations.
Acknowledgments
The authors are indebted to K. Chudinov, A. Makarov, and D. Scheglov for the very useful comments.
References
 Yu. Nikolaev, “The geometry of Ddecomposition of a twodimensional plane of arbitrary coefficients of the characteristic polynomial of a discrete system,” Automation and Remote Control, vol. 65, no. 12, pp. 1904–1914, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 F. M. Dannan, “The asymptotic stability of $x(n+k)+ax(n)+bx(nl)=0$,” Journal of Difference Equations and Applications, vol. 10, no. 6, pp. 589–599, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. M. Kipnis and R. M. Nigmatulin, “Stability of trinomial linear difference equations with two delays,” Automation and Remote Control, vol. 65, no. 11, pp. 1710–1723, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. S. Cheng and S. Y. Huang, “Alternate derivations of the stability region of a difference equation with two delays,” Applied Mathematics ENotes, vol. 9, pp. 225–253, 2009. View at: Google Scholar  Zentralblatt MATH
 E. Kaslik and St. Balint, “Bifurcation analysis for a twodimensional delayed discretetime Hopfield neural network,” Chaos, Solitons & Fractals, vol. 34, no. 4, pp. 1245–1253, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H. Matsunaga, “Stability regions for a class of delay difference systems,” in Differences and Differential Equations, vol. 42 of Fields Institute Communications, pp. 273–283, American Mathematical Society, Providence, RI, USA, 2004. View at: Google Scholar  Zentralblatt MATH
 I. S. Levitskaya, “A note on the stability oval for ${x}_{n+1}={x}_{n}+A{x}_{nk}$,” Journal of Difference Equations and Applications, vol. 11, no. 8, pp. 701–705, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 E. Kaslik, “Stability results for a class of difference systems with delay,” Advances in Difference Equations, vol. 2009, Article ID 938492, 13 pages, 2009. View at: Google Scholar  Zentralblatt MATH
 J. Diblik and D. Ya. Khusainov, “Representation of solutions of discrete delayed system $x(k+1)=Ax(k)+Bx(km)+f(k)$ with commutative matrices,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 63–76, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 E. N. Gryazina and B. T. Polyak, “Stability regions in the parameter space: Ddecomposition revisited,” Automatica, vol. 42, no. 1, pp. 13–26, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 D. Šiljak, “Parameter space methods for robust control design: a guided tour,” IEEE Transactions on Automatic Control, vol. 34, no. 7, pp. 674–688, 1989. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 R. Horn and C. Johnson, Matrix Theory, Cambridge University Press, Cambridge, UK, 1986.
 M. Kipnis and V. Malygina, “The stability cone for a matrix delay difference equation,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 860326, 15 pages, 2011. View at: Google Scholar
 A. Cohn, “Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise,” Mathematische Zeitschrift, vol. 14, no. 1, pp. 110–148, 1922. View at: Publisher Site  Google Scholar  MathSciNet
 S. A. Campbell, Y. Yuan, and S. Bungay, “Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling,” Nonlinearity, vol. 18, no. 6, pp. 2827–2847, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 Y. Yuan and S. A. Campbell, “Stability and synchronization of a ring of identical cells with delayed coupling,” Journal of Dynamics and Differential Equations, vol. 16, no. 3, pp. 709–744, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 P. J. Davis, Circulant Matrices, AMS Chelsea, New York, NY, USA, 1994.
 J. Kaplan and J. Yorke, “Chaotic behavior of multidimensional difference equations,” in Functional Differential Equations and Approximation of Fixed Points, H. O. Peitgen and H. O. Walther, Eds., vol. 730 of Lecture Notes in Mathematics, pp. 204–227, Springer, Berlin, Germany, 2006. View at: Google Scholar
 M. M. Kipnis and D. A. Komissarova, “Stability of delay difference system,” Advances in Difference Equations, vol. 2006, Article ID 314109, 9 pages, 2006. View at: Google Scholar
 A. Yu. Kulikov, “Stability of a linear nonautonomous difference equation with bounded delays,” Russian Mathematics, vol. 54, no. 11, pp. 18–26, 2010. View at: Publisher Site  Google Scholar
 A. Yu. Kulikov and V. V. Malygina, “On the stability of nonautonomous difference equations with several delays,” Russian Mathematics, vol. 52, no. 3, pp. 15–23, 2008. View at: Publisher Site  Google Scholar
 V. V. Malygina and A. Yu. Kulikov, “On the stability of semiautonomous difference equations,” Russian Mathematics, vol. 55, no. 5, pp. 19–27, 2011. View at: Google Scholar
 T. N. Khokhlova, M. M. Kipnis, and V. V. Malygina, “The stability cone for a delay differential matrix equation,” Applied Mathematics Letters, vol. 24, no. 5, pp. 742–745, 2011. View at: Publisher Site  Google Scholar
 T. L. Sabatulina and V. V. Malygina, “Several stability tests for linear autonomous differential equations with distributed delay,” Russian Mathematics, vol. 51, no. 6, pp. 52–60, 2007. View at: Publisher Site  Google Scholar
 M. M. Kipnis and I. S. Levitskaya, “Stability of delay difference and differential equations: similarities and distinctions,” in Difference Equations, Special functions and Orthogonal Polynomials, pp. 315–324, World Scientific, Hackensack, NJ, USA, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 I. S. Levitskaya, “Stability domain of a linear differential equation with two delays,” Computers & Mathematics with Applications, vol. 51, no. 1, pp. 153–159, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
Copyright
Copyright © 2011 S. A. Ivanov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.