International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 910936 | https://doi.org/10.5402/2011/910936

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

Academic Editor: G. Mishuris
Received30 Mar 2011
Accepted25 Apr 2011
Published21 Jun 2011

Abstract

We provide geometric algorithms for checking the stability of matrix difference equations π‘₯𝑛=𝐴π‘₯π‘›βˆ’π‘š+𝐡π‘₯π‘›βˆ’π‘˜ with two delays π‘š,π‘˜ such that the matrix π΄π΅βˆ’π΅π΄ 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 equationπ‘₯𝑛=π‘Žπ‘₯π‘›βˆ’π‘š+𝑏π‘₯π‘›βˆ’π‘˜(1.1) with real coefficients π‘Ž,𝑏 is basically solved [1–4].

The stability of matrix (1.1) with special 2Γ—2 matrices π‘Ž,𝑏 and π‘š=1 was studied in [5, 6]. The case π‘š=1, π‘Ž=𝐼, where 𝐼 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 π‘Ž=𝛼𝐼, π›Όβˆˆβ„, 0⩽𝛼⩽1. The representation of the solutions of (1.1) with commuting matrices π‘Ž,𝑏 is given in [9] without considering a stability problem.

To the best of the authors' knowledge, the stability of (1.1) with complex coefficients π‘Ž,𝑏 has not been studied yet.

In this paper we provide geometric algorithms for checking the stability of (1.1) with two delays π‘š,π‘˜βˆˆβ„€+, π‘˜>π‘šβ©Ύ1, for two cases: (1) π‘Ž,𝑏 are complex numbers, (2) π‘Ž,𝑏 are simultaneously triangularizable matrices. The results of this paper are based on the 𝐷-decomposition method (parameter plane method) [10, 11].

Matrices π‘Ž,𝑏 commute in all the above articles, which implies the possibility of simultaneous triangularization [12]. Therefore, our method can be applied to all of the above-mentioned cases. In the present paper the case π‘š=1 is studied along with other values π‘šβˆˆβ„€+. The case π‘š=1 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 𝐷-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 𝜌-stability, which coincides with usual stability if 𝜌=1. Later in that section we solve a problem of geometric checking 𝜌-stability of (1.1) with positive real π‘Ž and complex 𝑏. 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. 𝐷-Decomposition Curve for Given π‘˜,π‘š,π‘Ž,𝜌

Consider the scalar variant of (1.1). The characteristic polynomial for (1.1) is𝑓(πœ†)=πœ†π‘˜βˆ’π‘Žπœ†π‘˜βˆ’π‘šβˆ’π‘.(2.1) If π‘˜=π‘‘π‘˜1,π‘š=π‘‘π‘š1, 𝑑>1, then the trajectory of (1.1) splits into 𝑑 independent trajectories, and degree of polynomial (2.1) gets smaller after the substitution πœ†π‘‘=πœ‡: 𝑓1(πœ†)=πœ‡π‘˜1βˆ’π‘Žπœ‡π‘˜1βˆ’π‘š1βˆ’π‘.(2.2) Therefore, often (but not always) we will assume that the delays π‘š,π‘˜ are relatively prime.

Definition 2.1. 𝐷-decomposition curve for given π‘˜,π‘šβˆˆβ„€+, π‘Žβˆˆβ„‚, πœŒβˆˆβ„+ is a curve on the complex plane of the variable 𝑏 defined by the equation 𝑏(πœ”)=πœŒπ‘˜exp(π‘–π‘˜πœ”)βˆ’|π‘Ž|πœŒπ‘˜βˆ’π‘šexp(𝑖(π‘˜βˆ’π‘š)πœ”),πœ”βˆˆβ„.(2.3)

Parameter πœ” moves along the interval of length 2πœ‹, 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 π‘Ž. Starting from Section 5 we will get rid of this restriction. Obviously, if we assume in (2.1) that 𝑏=𝑏(πœ”) and π‘Žβˆˆβ„, π‘Žβ‰₯0, then (2.1) will have a root πœ†=𝜌exp(π‘–πœ”). Hodograph (2.3) splits the complex plane into the connected components. This decomposition is called the 𝐷-decomposition [10]. If we put π‘Žβˆˆβ„, π‘Žβ‰₯0, and substitute any two internal points 𝑏1,𝑏2 from one of the connected components of 𝐷-decomposition for coefficient 𝑏 in polynomial (2.1), then the polynomials obtained will have equal number of roots inside the circle of radius 𝜌 centred at the origin of the complex plane. In particular, if π‘Žβˆˆβ„, π‘Žβ©Ύ0, 𝜌=1 and the substitution of some inner point of a component of 𝐷-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 π‘˜,π‘š are coprime, then hodograph (2.3) is invariant under the rotation by 2πœ‹/π‘š.

Proof. For coprime π‘˜,π‘š there exist 𝑠,π‘‘βˆˆβ„€+ such that π‘˜π‘ βˆ’π‘šπ‘‘=1.(2.4) From (2.3), (2.4) it follows that ξ‚€exp𝑖2πœ‹π‘šξ‚ξ‚€π‘(πœ”)=π‘πœ”+2πœ‹π‘ π‘šξ‚.(2.5) Lemma 2.2 is proved.

From now and further we will assume that βˆ’πœ‹<π‘Žπ‘Ÿπ‘”π‘§β©½πœ‹ for any complex 𝑧, while Arg𝑧 will be assumed as multivalued function, and the equality Arg𝑧=𝑣 will mean that one of the values of Arg𝑧 equals to 𝑣.

The following lemma asserts that some part of the complex plane is free from points of the hodograph 𝑏(πœ”).

Lemma 2.3. Let π‘˜,π‘š be coprime, π‘˜>π‘š>1, |π‘Ž|<πœŒπ‘š. Let πœ”1∈[0;πœ‹/π‘š] be the least positive root of the equation arg𝑏(πœ”)=πœ‹/π‘š, and let 0<πœ”2β©½πœ”1. Then, for any πœ”βˆˆβ„, from ξ€·πœ”arg𝑏(πœ”)=arg𝑏2ξ€Έ,(2.6) it follows that |𝑏(πœ”)|β©Ύ|𝑏(πœ”2)|.

Proof. The function |𝑏(πœ”)| is 2πœ‹/π‘š-periodic, increasing in [(2π‘—βˆ’2)πœ‹/π‘š,(2π‘—βˆ’1)πœ‹/π‘š] for any π‘—βˆˆβ„€ and decreasing in [(2π‘—βˆ’1)πœ‹/π‘š,2π‘—πœ‹/π‘š]. In addition, |𝑏(πœ”+πœ‹/π‘š)|=|𝑏(βˆ’πœ”+πœ‹/π‘š)| for any πœ”. Let us assume, in order to get a contradiction, that (2.6) and |𝑏(πœ”)|<|𝑏(πœ”2)| are true for some πœ”>0. Then there exists a positive integer 𝑠<π‘š and π›Ώβˆˆβ„ such that πœ”=2πœ‹π‘ π‘š||𝛿||+𝛿,<πœ”2.(2.7) From (2.3), (2.7) it follows that Arg𝑏(πœ”)=(π‘˜βˆ’π‘š)πœ”+Arg(πœŒπ‘š=exp(π‘–π‘šπœ”)βˆ’|π‘Ž|)2πœ‹π‘˜π‘ π‘š+Arg(πœŒπ‘šexp(π‘–π‘šπ›Ώ)βˆ’|π‘Ž|)+(π‘˜βˆ’π‘š)𝛿=2πœ‹π‘˜π‘ π‘š+Arg𝑏(𝛿).(2.8) Since π‘˜,π‘š are coprime and 𝑠<π‘š, let us find natural numbers 𝑗,π‘ž such that π‘˜π‘ =π‘šπ‘—+π‘ž, π‘š>π‘žβ©Ύ1. Then (2.8) implies Arg𝑏(πœ”)=2πœ‹π‘žπ‘š+Arg𝑏(𝛿),(2.9) which contradicts (2.6) and the inequality |arg𝑏(𝛿)|<πœ‹/π‘š following from (2.7). Lemma 2.3 is proved.

3. Basic Ovals

For hodograph (2.3) the equality π‘ξ…žπœ”(0)=π‘–πœŒπ‘˜βˆ’π‘š(π‘˜πœŒπ‘šβˆ’|π‘Ž|(π‘˜βˆ’π‘š)) takes place. If𝜌|π‘Ž|<π‘šπ‘˜,π‘˜βˆ’π‘š(3.1) 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 [βˆ’πœ”1,πœ”1] under the map 𝑏(πœ”) defined by (2.3). Here πœ”1∈(0,πœ‹/π‘˜] is the least positive root of the equation arg𝑏(πœ”)=πœ‹. 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 2πœ‹π‘—/π‘š, π‘—βˆˆβ„€,0⩽𝑗<π‘š (see Lemma 2.2). We also call them the basic ovals. Here is a formal definition.

Definition 3.1. Let π‘˜,π‘š be coprime, π‘˜>π‘šβ©Ύ1, π‘—βˆˆβ„€, 0⩽𝑗<π‘š, and let (2.4), (3.1) hold. The basic oval 𝐿𝑗 for (1.1) is a closed curve given by (2.3), where the variable πœ” runs from (βˆ’πœ”1+2πœ‹π‘—π‘ /π‘š) to (πœ”1+2πœ‹π‘—π‘ /π‘š), where πœ”1 is the least positive root of the equation arg𝑏(πœ”)=πœ‹.(3.2)

From Lemma 2.2 and formula (2.5) it follows that all π‘š basic ovals can be obtained from 𝐿0 by rotation by the angles 2πœ‹π‘—/π‘š, 𝑗=0,1,…,(π‘šβˆ’1).

Considering Definition 3.1, we get the following. For existence of the basic oval it is necessary that |π‘Ž|<πœŒπ‘šπ‘˜/(π‘˜βˆ’π‘š). If π‘š>1 and |π‘Ž|>πœŒπ‘š, then the complex number 0 is outside any oval, and the intersection of all ovals is empty. If π‘š=1, then for fixed π‘˜, 𝜌, |π‘Ž|∈[0,πœŒπ‘šπ‘˜/(π‘˜βˆ’π‘š)) the basic oval 𝐿0 is unique. That is why the results related to the stability of (1.1) are different for π‘š=1 and π‘š>1.

The basic oval 𝐿𝑗 decreases as |π‘Ž| increases from 0, and it shrinks to the point 𝑏=βˆ’exp(2πœ‹π‘—/π‘š)πœŒπ‘˜/(π‘˜βˆ’1) as |π‘Ž| reaches πœŒπ‘šπ‘˜/(π‘˜βˆ’π‘š) (Figure 1 for π‘š>1 and Figure 2 for π‘š=1).

Lemma 3.2. Let π‘˜,π‘š be coprime, π‘˜>π‘šβ©Ύ1, π‘—βˆˆβ„€, 0⩽𝑗<π‘š, π‘Žβˆˆβ„, and 0β©½π‘Ž<πœŒπ‘šπ‘˜/(π‘˜βˆ’π‘š). If the complex number 𝑏 lies outside the basic oval 𝐿𝑗, then characteristic polynomial (2.1) has a root πœ† such that |πœ†|>𝜌.

Proof. Let us fix π‘˜,π‘š,π‘Ž,𝜌,𝑗, and let the complex number 𝑏 lie outside the basic oval 𝐿𝑗. Having changed 𝜌 to 𝑅>𝜌 in Definition 3.1 let us consider the system of ovals 𝐿𝑗(𝑅). If π‘…β†’βˆž, then the ovals 𝐿𝑗(𝑅) include a circle of an arbitrarily large radius. Therefore, there exists 𝑅0 such that the point 𝑏 is inside the oval 𝐿𝑗(𝑅0). The ovals 𝐿𝑗 and 𝐿𝑗(𝑅0) are homotopic, therefore, there exists 𝑅1∈(𝜌,𝑅0] such that 𝑏 lies on the curve 𝐿𝑗(𝑅1), which means the existence of a root πœ† of characteristic polynomial (2.1) such that |πœ†|=𝑅1>𝜌. Lemma 3.2 is proved.

4. Localization of Roots of Characteristic Polynomial (2.1) for Real Nonnegative π‘Ž and Complex 𝑏

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 𝜌-stable if for any of its solutions (π‘₯𝑛) the sequence (|π‘₯𝑛|/πœŒπ‘›) is bounded, and asymptotically 𝜌-stable if for any of its solutions (π‘₯𝑛) one has limπ‘›β†’βˆž|π‘₯𝑛|/πœŒπ‘›=0.

If 𝜌=1, then the concept of (asymptotic) 𝜌-stability coincides with the concept of usual (asymptotic) stability. Evidently, (1.1) is 𝜌-stable, if there are no roots of polynomial (2.1) outside the circle of radius 𝜌 centred at the origin, and there are no multiple roots of the polynomial on the boundary of circle. Equation (1.1) is asymptotically 𝜌-stable if and only if all the roots of its characteristic polynomial (2.1) lie inside the circle of radius 𝜌 with the center at 0.

Let us call the equation (asymptotically) 𝜌-unstable if it is not (asymptotically) 𝜌-stable. As we noted in Section 2, if 𝜌=1, then the proportional change of both delays π‘˜,π‘š in (1.1) has no influence on 𝜌-stability. It is not the case if πœŒβ‰ 1. It is easy to see that the equationπ‘₯𝑛=π‘Žπ‘₯π‘›βˆ’π‘šπ‘‘+𝑏π‘₯π‘›βˆ’π‘˜π‘‘(4.1) is asymptotically 𝜌-stable if and only if (1.1) is asymptotically πœŒπ‘‘-stable. It implies the following important observation: proportional increase of both delays π‘˜,π‘š with the conservation of the coefficients π‘Ž,𝑏 in (1.1) preserves asymptotic 𝜌-stability if 𝜌>1 and may not preserve it if 𝜌<1.

Definition 4.2. Let π‘˜,π‘š be coprime, π‘˜>π‘šβ©Ύ1, π‘Žβˆˆβ„‚, and 0β©½|π‘Ž|<πœŒπ‘šπ‘˜/(π‘˜βˆ’π‘š). The stability domain 𝐷(π‘˜,π‘š,π‘Ž,𝜌) is defined to be a set of all complex numbers 𝑏 such that for any 𝑗(0⩽𝑗<π‘š) the number 𝑏 lies inside the basic oval 𝐿𝑗.
Under the same conditions if π‘˜,π‘š are not coprime and 𝑑=𝑔𝑐𝑑(π‘˜,π‘š), let us put 𝐷(π‘˜,π‘š,π‘Ž,𝜌)=𝐷(π‘˜/𝑑,π‘š/𝑑,π‘Ž,πœŒπ‘‘).

Theorems 4.3–5.2 will justify the name β€œstability domain” for 𝐷(π‘˜,π‘š,π‘Ž,𝜌) a little later. Evidently, for coprime π‘˜,π‘š such that π‘˜>π‘š>1 the domain 𝐷(π‘˜,π‘š,π‘Ž,𝜌) has the following properties. If 0β©½|π‘Ž|<πœŒπ‘š, then 𝐷(π‘˜,π‘š,π‘Ž,𝜌) is the connected domain on the complex plane, containing 0, whose boundary is the 𝐷-decomposition curve (2.3). If |π‘Ž|=πœŒπ‘š, then the domain 𝐷(π‘˜,π‘š,π‘Ž,𝜌) degenerates into the point 𝑏=0. If πœŒπ‘š<|π‘Ž|<πœŒπ‘šπ‘˜/(π‘˜βˆ’π‘š), then the domain 𝐷(π‘˜,π‘š,π‘Ž,𝜌) is empty. If |π‘Ž|β©ΎπœŒπ‘šπ‘˜/(π‘˜βˆ’π‘š), then the domain 𝐷(π‘˜,π‘š,π‘Ž,𝜌) is not defined in view of the fact that the basic ovals (Definition 3.1) are not defined.

If π‘š=1, then the domain 𝐷(π‘˜,1,π‘Ž,𝜌) is a set of points lying inside the oval 𝐿0. If 0β©½|π‘Ž|<𝜌, then 𝐷(π‘˜,1,π‘Ž,𝜌) includes 0. If 𝜌⩽|π‘Ž|<πœŒπ‘˜/(π‘˜βˆ’1), then 𝐷(π‘˜,1,π‘Ž,𝜌) is nonempty and does not contain 0. If |π‘Ž|=πœŒπ‘˜/(π‘˜βˆ’1), then 𝐷(π‘˜,1,π‘Ž,𝜌) degenerates into the point 𝑏=βˆ’πœŒπ‘˜/(π‘˜βˆ’1). Finally, if |π‘Ž|>πœŒπ‘˜/(π‘˜βˆ’1), then the domain 𝐷(π‘˜,1,π‘Ž,𝜌) is not defined.

The following theorems are based on the localization of roots of polynomial (2.1) with nonnegative π‘Ž and complex 𝑏 with respect to the circle of radius 𝜌 centred at the origin.

Theorem 4.3. Let π‘˜,π‘š be coprime, π‘˜>π‘š>1, π‘Žβˆˆβ„+, 𝜌>0. (1)If π‘Ž>πœŒπ‘š, then for any 𝑏 (1.1) is 𝜌-unstable.(2)If π‘Ž=πœŒπ‘š, then for any complex 𝑏≠0 (1.1) is 𝜌-unstable; for 𝑏=0 it is 𝜌-stable (nonasymptotically).(3)If 0β©½π‘Ž<πœŒπ‘š, then (1.1) is asymptotically 𝜌-stable if and only if the complex number 𝑏 lies inside the stability domain 𝐷(π‘˜,π‘š,a,𝜌). (4)If 0β©½π‘Ž<πœŒπ‘š, then (1.1) is 𝜌-stable if and only if the complex number 𝑏 lies either inside or on the boundary of 𝐷(π‘˜,π‘š,π‘Ž,𝜌).

Proof. (1) Let π‘Ž>πœŒπ‘š. Let us find π‘…βˆˆβ„ such that πœŒπ‘š<π‘…π‘šπ‘…<π‘Ž<π‘šπ‘˜.π‘˜βˆ’π‘š(4.2) Taking into account the inequality π‘Ž<π‘…π‘šπ‘˜/(π‘˜βˆ’π‘š), let us consider π‘š basic ovals 𝐿𝑗(𝑅), 𝑗=0,1,…,π‘šβˆ’1 having 𝜌 replaced by 𝑅 in Definition 4.1. Since π‘…π‘š<π‘Ž, the system of ovals 𝐿𝑗(𝑅) has no intersections. Hence, for any complex number 𝑏 there exists π‘—βˆˆβ„€,0⩽𝑗<π‘š such that 𝑏 lies outside the oval 𝐿𝑗(𝑅). By Lemma 3.2 (1.1) is 𝑅-unstable. Since 𝑅>𝜌, it is 𝜌-unstable. Statement 1 is proved.
(2) Let π‘Ž=πœŒπ‘š. If 𝑏=0, then statement 2 of Theorem 4.3 is obvious. Let 𝑏≠0. If Re𝑏⩾0, then 𝑏 lies outside the oval 𝐿0. If Re𝑏⩽0 and π‘š is even, then 𝑏 lies outside the oval πΏπ‘š/2. If Re𝑏⩽0 and π‘š is odd, then 𝑏 lies either outside the oval 𝐿(π‘šβˆ’1)/2 or outside the oval 𝐿(π‘š+1)/2. In any case (1.1) is 𝜌-unstable by Lemma 3.2. Statement 2 is proved.
(3) Let 0β©½π‘Ž<πœŒπ‘š. Let the number 𝑏 be inside the domain 𝐷(π‘˜,π‘š,π‘Ž,𝜌). Then for any 𝑗(0⩽𝑗<π‘š) the number 𝑏 lies inside the oval 𝐿𝑗. By Lemma 2.3 the beam drawn on the complex plane from 0 to 𝑏 does not intersect curve (2.3). Therefore, polynomial (2.1) has the same number of roots inside the circle of radius 𝜌 for given 𝑏 and for 𝑏=0. However if 𝑏=0, then all the roots of (2.1) lie inside the circle of radius 𝜌 centred at 0. Therefore, (1.1) is asymptotically 𝜌-stable for given 𝑏.
If 𝑏 lies on the boundary of the domain 𝐷(π‘˜,π‘š,π‘Ž,𝜌) or outside it, then 𝑏 lies either on the boundary of one of the basic ovals 𝐿𝑗 or outside one of them, and by Lemma 3.2 (1.1) is asymptotically 𝜌-unstable.
(4) If 𝑏 lies outside the domain 𝐷(π‘˜,π‘š,π‘Ž,𝜌), then the conclusion of statement 4 of Theorem 4.3 is a straightforward consequence of Lemma 3.2. If 𝑏 lies inside 𝐷(π‘˜,π‘š,π‘Ž,𝜌), then the conclusion of statement 4 of Theorem 4.3 is a straightforward consequence of statement 3 of Theorem 4.3. Let 𝑏 lie on the boundary of 𝐷(π‘˜,π‘š,π‘Ž,𝜌). Then for any root πœ† of polynomial (2.1) either |πœ†|<𝜌 or |πœ†|=𝜌. In the latter case in view of the inequality 0β©½π‘Ž<πœŒπ‘š<πœŒπ‘šπ‘˜/π‘˜βˆ’π‘š we have π‘‘π‘“π‘‘πœ†=πœ†π‘˜βˆ’π‘šβˆ’1(π‘˜πœ†π‘šβˆ’π‘Ž(π‘˜βˆ’π‘š))β‰ 0,(4.3) hence the root πœ† such that |πœ†|=𝜌 is simple. Theorem 4.3 is proved.

If in (1.1) the least delay π‘š is equal to 1, then the situation is essentially different from the case π‘š>1.

Theorem 4.4. Let π‘˜>π‘š=1, π‘Žβˆˆβ„+, 𝜌>0. (1)If π‘Žβ©ΎπœŒπ‘˜/(π‘˜βˆ’1), then for all complex numbers 𝑏 (1.1) is 𝜌-unstable.(2)If 0β©½π‘Ž<πœŒπ‘˜/(π‘˜βˆ’1), then (1.1) is asymptotically 𝜌-stable if and only if the complex number 𝑏 lies inside the domain 𝐷(π‘˜,1,π‘Ž,𝜌).(3)If 0β©½π‘Ž<πœŒπ‘˜/(π‘˜βˆ’1), then (1.1) is 𝜌-stable if and only if the complex number 𝑏 lies inside 𝐷(π‘˜,1,π‘Ž,𝜌).

Proof. (1) Let π‘Ž>πœŒπ‘˜/(π‘˜βˆ’π‘š), and let 𝑏 be a given complex number. Let us find 𝑅>𝜌 such that πœŒπ‘˜/(π‘˜βˆ’π‘š)<π‘Ž<π‘…π‘˜/(π‘˜βˆ’π‘š) and the point 𝑏 is located outside the oval 𝐿0(𝑅) obtained from Definition 3.1 by substituting 𝑅 for 𝜌. By Lemma 3.2 there exists a complex root πœ† of polynomial (2.1) such that |πœ†|>𝑅>𝜌, so 𝜌-instability of (1.1) is proved. Let π‘Ž=πœŒπ‘˜/(π‘˜βˆ’1). Then the previous arguments also prove 𝜌-instability provided that π‘β‰ βˆ’πœŒπ‘˜/(π‘˜βˆ’1). However, if 𝑏=βˆ’πœŒπ‘˜/(π‘˜βˆ’1), then under the assumption that π‘Ž=πœŒπ‘˜/(π‘˜βˆ’1) the number πœ†=𝜌 is a multiple root of polynomial (2.1), and consequently, (1.1) is also 𝜌-unstable. Statement 1 of Theorem 4.4 is proved.
(2) Let 0β©½π‘Ž<πœŒπ‘˜/(π‘˜βˆ’1). Since 𝐷(π‘˜,1,π‘Ž,𝜌) is the domain of inner points of the oval 𝐿0, it is connected. The function |𝑏(πœ”)| (see (2.3)) increases as πœ” moves either from 0 to πœ‹ or from 0 to (βˆ’πœ‹). Therefore, there are no points of hodograph (2.3) inside 𝐿0. To complete the proof of asymptotical 𝜌-stability of (1.1) at any point of 𝐿0 it is sufficient to prove that there exists at least one point 𝑏0 inside the oval 𝐿0 such that the equation is asymptotically 𝜌-stable for 𝑏=𝑏0.
CASE 1. Let 0β©½π‘Ž<𝜌. Then the point 𝑏=0 lies inside 𝐿0. If 𝑏=0, then polynomial (2.1) has the (π‘˜βˆ’1)-multiple root πœ†=0 and the simple root πœ†=π‘Ž. This gives the asymptotic 𝜌-stability, in view of π‘Ž<𝜌.
CASE 2. Let πœŒβ©½π‘Ž<πœŒπ‘˜/(π‘˜βˆ’1). Let us consider the point 𝑏=πœŒπ‘˜βˆ’π‘ŽπœŒπ‘˜βˆ’1 at the boundary of 𝐿0 and consider characteristic polynomial (2.1) with given 𝑏: 𝑓2(πœ†)=πœ†π‘˜βˆ’π‘Žπœ†π‘˜βˆ’1βˆ’πœŒπ‘˜+π‘ŽπœŒπ‘˜βˆ’1.(4.4) The equation 𝑓2(πœ†)=0 transforms into πœ†ξ‚΅ξ‚΅πœŒξ‚Άξ‚Άξƒ©ξ‚΅πœ†βˆ’1πœŒξ‚Άπ‘˜βˆ’1βˆ’π‘Žξ‚΅ξ‚΅πœŒξ‚Άξ‚Άβˆ’1π‘˜βˆ’2𝑗=0ξ‚΅πœ†πœŒξ‚Άπ‘˜βˆ’π‘—βˆ’2ξƒͺ=0.(4.5) One of roots of (4.5) is equal to 𝜌, while others lie inside the circle of radius 𝜌 centred at the origin in view of the inequality 0β©½π‘Žβˆ’πœŒ<𝜌/(π‘˜βˆ’1) [14].
Let us return to (2.1), and let us figure out in what direction the root πœ†=𝜌 moves as the coefficient 𝑏 moves from the point 𝑏=πœŒπ‘˜βˆ’π‘ŽπœŒπ‘˜βˆ’1 toward the interior of 𝐿0 so that dπ‘βˆˆβ„, d𝑏<0. From (2.1) it follows that for πœ†=𝜌 we have dπœ†=𝜌dπ‘βˆ’π‘˜+2,π‘˜πœŒβˆ’π‘Ž(π‘˜βˆ’1)(4.6) and in view of π‘Ž<πœŒπ‘˜/(π‘˜βˆ’1) we get dπœ†/d𝑏>0. Therefore, d𝑏<0 implies πœ†<𝜌. Consequently, there exist values of 𝑏 inside 𝐿0 providing asymptotic 𝜌-stability of (1.1), therefore, for any value 𝑏 inside 𝐿0 (1.1) is asymptotically 𝜌-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 π‘Ž,𝑏

Let us change the variables in (1.1) so that it has no influence on (asymptotic) 𝜌-stability:π‘₯𝑛=𝑦𝑛𝑖𝑛expπ‘šξ‚.argπ‘Ž(5.1) Equation (1.1) changes to𝑦𝑛=π›Όπ‘¦π‘›βˆ’π‘š+π›½π‘¦π‘›βˆ’π‘˜,(5.2) whereξ‚€π‘˜π›Ό=|π‘Ž|,𝛽=𝑏expβˆ’π‘–π‘šξ‚.argπ‘Ž(5.3) The characteristic polynomial for (5.2) has the formπœ“(πœ‡)=πœ‡π‘˜βˆ’π›Όπœ‡π‘˜βˆ’π‘šβˆ’π›½.(5.4) It is related to (2.1) by the change πœ‡=πœ†exp(βˆ’π‘–(1/π‘š)argπ‘Ž), 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 π‘¦π‘›βˆ’π‘š, 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 π‘Ž, 𝑏.

Theorem 5.1. Let π‘˜,π‘š be coprime, π‘˜>π‘š>1, π‘Žβˆˆβ„‚, 𝜌>0.
(1)If |π‘Ž|>πœŒπ‘š, then for any complex 𝑏 (1.1) is 𝜌-unstable.(2)If |π‘Ž|=πœŒπ‘š, then for any 𝑏≠0 (1.1) is 𝜌-unstable; for 𝑏=0 it is 𝜌-stable (nonasymptotically).(3)If |π‘Ž|<πœŒπ‘š, then (1.1) is asymptotically 𝜌-stable if and only if the complex number 𝛽=𝑏exp(βˆ’π‘–(π‘˜/π‘š)argπ‘Ž) lies inside the domain 𝐷(π‘˜,π‘š,π‘Ž,𝜌).(4)If |π‘Ž|<πœŒπ‘š, then (1.1) is 𝜌-stable if and only if the complex number 𝛽=𝑏exp(βˆ’π‘–(π‘˜/π‘š)argπ‘Ž) lies either inside 𝐷(π‘˜,π‘š,π‘Ž,𝜌) or on its boundary.

Theorem 5.2. Let π‘˜>π‘š=1, π‘Žβˆˆβ„‚, 𝜌>0.
(1)If |π‘Ž|β©ΎπœŒπ‘˜/(π‘˜βˆ’1), then for any complex 𝑏 (1.1) is 𝜌-unstable.(2)If |π‘Ž|<πœŒπ‘˜/(π‘˜βˆ’1), then (1.1) is asymptotically 𝜌-stable if and only if the complex number 𝛽=𝑏exp(βˆ’π‘–π‘˜argπ‘Ž) lies inside the domain 𝐷(π‘˜,1,π‘Ž,𝜌).(3)If |π‘Ž|<πœŒπ‘˜/(π‘˜βˆ’1), then (1.1) is 𝜌-stable if and only if the complex number 𝛽=𝑏exp(βˆ’π‘–π‘˜argπ‘Ž) lies either inside 𝐷(π‘˜,1,a,𝜌) or on its boundary.

Example 5.3. Let π‘š=6, π‘Ž=0.8+0.9𝑖, 𝜌=1.15 in (1.1). Let 6β©½π‘˜β©½12. For every given value π‘˜ let us find all values of the complex coefficient 𝑏 for which (1.1) is 𝜌-stable. The answer is demonstrated by Figure 4. Let us give some comments. First calculate |π‘Ž|=1.204, argπ‘Žβ‰ƒ0.844. If π‘˜=6, then to find 𝜌-stability domain one does not need to use Theorems 4.3–5.2. The domain is a circle given in Figure 4(a). Since 6,7 are coprime, then for π‘˜=7, by Theorem 5.1, (1.1) 𝜌-stable if and only if π‘βˆˆexp(𝑖(7/6)argπ‘Ž)𝐷(7,6,π‘Ž,𝜌). The corresponding β€œcurved hexagon” is shown in Figure 4(a). Similarly for π‘˜=11 the condition π‘βˆˆexp(𝑖(11/6)argπ‘Ž)𝐷(11,6,π‘Ž,𝜌) is necessary and sufficient for asymptotic stability of (1.1). The corresponding β€œcurved hexagon” is shown in Figure 4(b). For π‘˜=8 the stability criterion is the condition π‘βˆˆexp(𝑖(4/3)argπ‘Ž)𝐷(4,3,π‘Ž,𝜌2). The corresponding β€œcurved triangle” is shown in Figure 4(a). Similarly the β€œdigon” exp(𝑖(3/2)argπ‘Ž)𝐷(3,2,π‘Ž,𝜌3) for π‘˜=9 is shown in Figure 4(a), and the β€œcurved triangle” exp(𝑖(5/3)argπ‘Ž)𝐷(5,3,π‘Ž,𝜌2) for π‘˜=10 is shown in Figure 4(b). For π‘˜=12, according to Theorem 5.2, the stability criterion for (1.1) is π‘βˆˆexp(𝑖⋅2argπ‘Ž)𝐷(2,1,π‘Ž,𝜌6) (Figure 4(b)). The corresponding β€œstability oval” is shown in Figure 4(a).

6. Stability Cones for Matrix Equation (1.1) with Simultaneously Triangularizable Matrices

Let us consider a matrix equationπ‘₯𝑛=𝐴π‘₯π‘›βˆ’π‘š+𝐡π‘₯π‘›βˆ’π‘˜,(6.1)π‘₯βˆΆβ„€+→ℂ𝑙; 𝐴,π΅βˆˆβ„‚π‘™Γ—π‘™. The characteristic equation for (6.1) isπœ“ξ€·(πœ†)=detπΌπœ†π‘˜βˆ’π΄πœ†π‘˜βˆ’π‘šξ€Έ.βˆ’π΅(6.2)

Definition 6.1. Matrix equation (6.1) is called 𝜌-stable if for every solution (π‘₯𝑛) the sequence (|π‘₯𝑛|/πœŒπ‘›) is bounded. Equation (6.1) is called asymptotically 𝜌-stable if limπ‘›β†’βˆž|π‘₯𝑛|/πœŒπ‘›=0 holds for every solution (π‘₯𝑛).

Obviously, matrix equation (6.1) is asymptotically 𝜌-stable if and only if all the roots of characteristic polynomial (6.2) lie inside the circle of radius 𝜌 with the center at 0. We also observe that if at least one root of (6.2) lies outside the circle of radius 𝜌 with the center at 0, then (6.1) is 𝜌-unstable.

In this paper we consider (6.1) only with triangularizable matrices 𝐴,𝐡. It is known [12] that if the matrix π΄π΅βˆ’π΅π΄ is nilpotent, then 𝐴,𝐡 can be simultaneously triangularized.

Definition 6.2. If π‘˜>π‘š>1, then the 𝜌-stability cone for given π‘˜,π‘š,𝜌 is a set of points 𝑀=(𝑒1,𝑒2,𝑒3)βˆˆβ„3 such that 0⩽𝑒3β©½1 and the intersection of the set with any plane 𝑒3=π‘Ž(0β©½π‘Žβ©½1) is the stability domain 𝐷(π‘˜,π‘š,π‘Ž,𝜌). If π‘˜>π‘š=1, then the 𝜌-stability cone for given π‘˜,𝜌 is a set of points 𝑀=(𝑒1,𝑒2,𝑒3)βˆˆβ„3 such that 0⩽𝑒3β©½π‘˜/(π‘˜βˆ’1) and the intersection of the set with the plane 𝑒3=π‘Ž(0β©½π‘Žβ©½π‘˜/(π‘˜βˆ’1)) is the domain 𝐷(π‘˜,1,π‘Ž,𝜌).

Let us define a stability cone as the 𝜌-stability cone for 𝜌=1.

Returning to Figure 3, we can interpret the figures in Figure 3(a) as sections of the stability cone for π‘˜=5,π‘š=1 at different heights 𝑒3=π‘Ž, and the ones in Figure 3(b) as sections of the stability cone for π‘˜=5,π‘š=2, and so on.

The stability cones for π‘š>1 are the intersections of π‘š conical surfaces formed by the basic ovals as the parameter π‘Ž changes from 0 to π‘˜/(π‘˜βˆ’π‘š) (Figure 5).

Let us consider the simple case of a diagonal systemπ‘¦π‘›ξ€·π‘Žβˆ’diag11,…,π‘Žπ‘™π‘™ξ€Έπ‘¦π‘›βˆ’π‘šξ€·π‘βˆ’diag11,…,π‘π‘™π‘™ξ€Έπ‘¦π‘›βˆ’π‘˜=0(6.3) with complex entries π‘Žπ‘—π‘—,𝑏𝑗𝑗, 1⩽𝑗⩽𝑙. Let us construct the points 𝑀𝑗=(𝑒1𝑗,𝑒2𝑗,𝑒3𝑗)   (1⩽𝑗⩽𝑙) in ℝ3 in the following way:𝑒1𝑗𝑏=Reπ‘—π‘—ξ‚€π‘˜expβˆ’π‘–π‘šargπ‘Žπ‘—π‘—ξ‚ξ‚,𝑒2𝑗𝑏=Imπ‘—π‘—ξ‚€π‘˜expβˆ’π‘–π‘šargπ‘Žπ‘—π‘—ξ‚ξ‚,𝑒3𝑗=||π‘Žπ‘—π‘—||.(6.4)

It follows from the definition of the 𝜌-stability cone and from Theorems 5.1–5.2 that (6.3) is asymptotically 𝜌-stable if and only if all the points 𝑀𝑗(1⩽𝑗⩽𝑙) lie inside the 𝜌-stability cone for given π‘˜,π‘š. All the points 𝑀𝑗 with 𝑒3𝑗=0, 𝑒21𝑗+𝑒22𝑗<πœŒπ‘˜ are considered as inner points of the 𝜌-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 π‘˜>π‘šβ©Ύ1, let the numbers π‘˜,π‘š be coprime, and 𝜌>0. Let 𝐴,𝐡,π‘†βˆˆβ„π‘™Γ—π‘™, and π‘†βˆ’1𝐴𝑆=𝐴𝑇, and π‘†βˆ’1𝐡𝑆=𝐡𝑇, where 𝐴𝑇 and 𝐡𝑇 are lower triangle matrices with elements π‘Žπ‘—π‘ ,𝑏𝑗𝑠(1⩽𝑗,𝑠⩽𝑙). Let one construct the points 𝑀𝑗=(𝑒1𝑗,𝑒2𝑗,𝑒3𝑗), (1⩽𝑗⩽𝑙) by the formulas (cf. (6.4)) 𝑒1𝑗=||𝑏𝑗𝑗||ξ‚€cosargπ‘π‘—π‘—βˆ’π‘˜π‘šargπ‘Žπ‘—π‘—ξ‚,𝑒2𝑗=||𝑏𝑗𝑗||ξ‚€sinargπ‘π‘—π‘—βˆ’π‘˜π‘šargπ‘Žπ‘—π‘—ξ‚,𝑒3𝑗=||π‘Žπ‘—π‘—||.(6.5) Then (6.1) is 𝜌-asymptotically stable if and only if all the points 𝑀𝑗(1⩽𝑗⩽𝑙) lie inside the 𝜌-stability cone for the given π‘˜,π‘š,𝜌.
If some point 𝑀𝑗 lies outside the 𝜌-stability cone, then (6.1) is 𝜌-unstable.

Proof. Let us make the change 𝑦𝑛=𝑆π‘₯𝑛. Then (6.1) transforms to the following one: 𝑦𝑛=π΄π‘‡π‘¦π‘›βˆ’π‘š+π΅π‘‡π‘¦π‘›βˆ’π‘˜.(6.6) The characteristic polynomial for (6.6) has the form πœ“(πœ†)=𝑙𝑗=1ξ€·πœ†π‘˜βˆ’π‘Žπ‘—π‘—πœ†π‘˜βˆ’π‘šβˆ’π‘π‘—π‘—ξ€Έ.(6.7) It coincides with the characteristic polynomial of diagonal system (6.3). Therefore, from statement 3 of Theorem 5.1 (for π‘š>1) and from statement 2 of Theorem 5.2 (for π‘š=1) we obtain asymptotic 𝜌-stability if all the points 𝑀𝑗 lie inside the 𝜌-stability cone. Similarly from statement 4 of Theorem 5.1 (for π‘š>1) and statement 3 of Theorem 5.2 (for π‘š=1) we obtain 𝜌-instability of (6.1) if some point 𝑀𝑗 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 𝑙 neurons (Figure 6) interchanging signals with the neighboring neurons.

Let 𝑦𝑛(𝑗) be a signal of the 𝑗-th neuron at the 𝑛-th moment of time. Let us suppose that the neuron reaction on its state, as well as on that of the previous neuron, is m-units delayed, and reaction on the next neuron is π‘˜-units delayed. The neuron chain is closed, and the first neuron is next to the 𝑙-th one. Let us assume that the neurons interchange the signals according to the equations𝑦𝑛(1)𝑦=𝑓(1)π‘›βˆ’π‘šξ‚ξ‚€π‘¦+𝑔(𝑙)π‘›βˆ’π‘šξ‚ξ‚€π‘¦+β„Ž(2)π‘›βˆ’π‘˜ξ‚,𝑦𝑛(2)𝑦=𝑓(2)π‘›βˆ’π‘šξ‚ξ‚€π‘¦+𝑔(1)π‘›βˆ’π‘šξ‚ξ‚€π‘¦+β„Ž(3)π‘›βˆ’π‘˜ξ‚,⋯𝑦𝑛(𝑙)𝑦=𝑓(𝑙)π‘›βˆ’π‘šξ‚ξ‚€π‘¦+𝑔(π‘™βˆ’1)π‘›βˆ’π‘šξ‚ξ‚€π‘¦+β„Ž(1)π‘›βˆ’π‘˜ξ‚,(7.1) where 𝑓,𝑔,β„Ž are sufficiently smooth real-valued functions of a real variable. Let us assume that there is a real number π‘¦βˆ— such that the stationary sequences 𝑦1𝑛=π‘¦βˆ—,…,𝑦𝑙𝑛=π‘¦βˆ— form a solution of (7.1). Let us introduce the variables π‘₯𝑛(𝑗)=𝑦𝑛(𝑗)βˆ’π‘¦βˆ— and the vector π‘₯𝑛=(π‘₯𝑛(1),…,π‘₯𝑛(𝑙))𝑇, and let us linearize system (7.1) in new variables about zero. We get (6.1) with the circulant [17] matricesβŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ π΄=𝛼00⋯𝛽𝛽𝛼0β‹―00𝛽𝛼⋯0β‹―β‹―β‹―β‹―β‹―000⋯𝛼,𝐡=0𝛾0β‹―000𝛾⋯0000β‹―0⋯⋯⋯⋯⋯𝛾00β‹―0.(7.2) Here 𝛼=𝑑𝑓(π‘¦βˆ—)/𝑑𝑦, 𝛽=𝑑𝑔(π‘¦βˆ—)/𝑑𝑦, 𝛾=π‘‘β„Ž(π‘¦βˆ—)/𝑑𝑦. Let us introduce a matrix 𝑃 (a lines permutation operator):βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ π‘ƒ=010β‹―0001β‹―0000β‹―0β‹―β‹―β‹―β‹―β‹―100β‹―0.(7.3) Then 𝐡=𝛾𝑃, 𝐴=𝛼𝐼+π›½π‘ƒπ‘™βˆ’1, therefore, diagonalization 𝑃 generates simultaneous diagonalization of 𝐴,𝐡. The eigenvalues of 𝑃 are 1, πœ€, πœ€2,…,πœ€π‘™βˆ’1, where πœ€=exp(𝑖2πœ‹/𝑙). Therefore,𝐴𝑇=𝛼𝐼+𝛽diag1,πœ€π‘™βˆ’1,πœ€π‘™βˆ’2ξ€Έ,…,πœ€,𝐡𝑇=𝛾diag1,πœ€,πœ€2,…,πœ€π‘™βˆ’1ξ€Έ.(7.4) Granting (7.4), by Theorem 6.3, we can build points 𝑀𝑗=(𝑒1𝑗,𝑒2𝑗,𝑒3𝑗) in ℝ3 for system (6.1), (7.2):𝑒1𝑗=𝛾cos2πœ‹(π‘—βˆ’1)π‘™βˆ’ξ‚€π‘˜π‘šξ‚ξ‚΅ξ‚΅βˆ’arg𝛼+𝛽exp2πœ‹π‘–(π‘—βˆ’1)𝑙,𝑒2jξ‚΅=𝛾sin2πœ‹(π‘—βˆ’1)π‘™βˆ’ξ‚€π‘˜π‘šξ‚ξ‚΅ξ‚΅βˆ’arg𝛼+𝛽exp2πœ‹π‘–(π‘—βˆ’1)𝑙,𝑒3𝑗=||||ξ‚΅βˆ’π›Ό+𝛽exp2πœ‹π‘–(π‘—βˆ’1)𝑙||||.(7.5) We get the following consequence of Theorem 6.3.

Corollary 7.1. If for every 𝑗(1⩽𝑗⩽𝑙) the point 𝑀𝑗=(𝑒1𝑗,𝑒2𝑗,𝑒3𝑗) defined by formulas (7.5) lies inside the 𝜌-stability cone for given π‘˜,π‘š,𝜌, then system (6.1), (7.2) is asymptotically 𝜌-stable. If at least one point 𝑀𝑗 lies outside the 𝜌-stability cone for given π‘˜,π‘š,𝜌, then system (6.1), (7.2) is 𝜌-unstable.

Let us proceed to the problem of stability of a neural network with a large number of neurons. The points 𝑀𝑗=(𝑒1𝑗,𝑒2𝑗,𝑒3𝑗) defined by (7.5) lie on the closed curve𝑒1ξ‚€ξ‚€π‘˜(𝑑)=𝛾cosπ‘‘βˆ’π‘šξ‚ξ‚,𝑒arg(𝛼+𝛽exp(βˆ’π‘–π‘‘))2ξ‚€ξ‚€π‘˜(𝑑)=𝛾sinπ‘‘βˆ’π‘šξ‚ξ‚,𝑒arg(𝛼+𝛽exp(βˆ’π‘–π‘‘))3||||(𝑑)=𝛼+𝛽exp(βˆ’π‘–π‘‘),0⩽𝑑⩽2πœ‹.(7.6)

If π‘™β†’βˆž, then the points 𝑀𝑗 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 𝑙×𝑙 matrices 𝐴,𝐡. If any point of the curve (7.6) lies inside the 𝜌-stability cone for given π‘˜,π‘š,𝜌, then system (6.1), (7.2) is asymptotically 𝜌-stable for any 𝑙. If at least one point of the curve (7.6) lies outside the 𝜌-stability cone for given π‘˜,π‘š,𝜌, then there exists 𝑙0 such that system (6.1), (7.2) is 𝜌-unstable for any 𝑙>𝑙0.

Example 7.3. Let us consider the ring of neurons shown in Figure 6. Put π‘˜=4, π‘š=3, 𝜌=1, 𝛽=0.1, 𝛾=0.4. Let us pose a question: what are the values of 𝛼>0 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 𝛼 (Figures 7(a) and 7(b)). Assuming 𝑙=3, we construct the points 𝑀1, 𝑀2, 𝑀3 in each of the six curves. Then we construct the stability cone for given π‘˜,π‘š,𝜌 (Figure 7(b)). It is the 𝜌-stability cone for 𝜌=1. In Figure 7(b) we see that two curves (7.6) corresponding to the values 𝛼=0.1,𝛼=0.3 are hidden inside the cone. The point 𝑀1 corresponding to the values 𝛼=0.5 is on the surface of the cone, while all other points of the curve (7.6) for 𝛼=0.5 lie inside the cone. Therefore, according to Corollaries 7.1 and 7.2, if 𝛼<0.5, then system (6.1), (7.2) is asymptotically stable for any 𝑙⩾2. In our interpretation it means that the neuron configuration in Figure 6 is stable for any number of neurons. If 𝛼>0.5, 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.

Now let us consider Example 7.3 without the condition 𝜌=1. Under assumptions of Example 7.3, for any value 𝛼 there exists a number 𝜌0(𝛼) such that system (6.1), (7.2) is asymptotically 𝜌-stable if 𝜌<𝜌0(𝛼) and 𝜌-unstable if 𝜌>𝜌0(𝛼). Corollaries 7.1 and 7.2 allow us to find 𝜌0(𝛼) by means of construction of different 𝜌-stability cones. Table 1 shows how 𝜌0 depends on 𝛼. The value 𝐿(𝛼)=ln𝜌0(𝛼) is a Lyapunov exponent [18] for system (6.1), (7.2).


𝛼 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
𝜌 0 ( 𝛼 ) 0.870 0.938 1.000 1.056 1.108 1.156 1.201 1.243

Example 7.4. Let us consider the neuron chain shown in Figure 6. Let us fix the parameters: π‘˜=4, π‘š=3, 𝜌=1, 𝛽=0.1, 𝛼=0.5. In this example we demonstrate how the change of the parameter 𝛾 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 𝛾=π‘Ÿβ„Ž, β„Ž=0.1, π‘Ÿ=0,1,…,5, and the points 𝑀1, 𝑀2, 𝑀3 for 𝑙=3. The upper part of the stability cone corresponding to 𝑒3>0.7 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 0⩽𝛾<0.4. Therefore, if 0⩽𝛾<0.4, then system (6.1), (7.2) is asymptotically stable for any 𝑙⩾2. If 𝛾=0.4, then the point 𝑀1=(𝑒1,𝑒2,𝑒3) defined by (7.5) lies on the cone surface. If 𝛾>0.4, then the point 𝑀1 lies outside the cone, and this shows the instability of system (6.1), (7.2). In our interpretation, for 0⩽𝛾<0.4, the neuron chain is stable no matter how many neurons are in the chain, and for 𝛾>0.4 it is unstable even if it consists only of two neurons.

8. Conclusion

The condition ‖𝐴‖+‖𝐡‖<1 is sufficient for the asymptotic stability of matrix (6.1) [19], and it does not require simultaneous triangularization of the matrices 𝐴,𝐡. There are sufficient conditions for stability of nonautonomous scalar difference equations in [20–22].

Stability cones for differential matrix equations Μ‡π‘₯=𝐴π‘₯+𝐡π‘₯(π‘‘βˆ’πœ) with one delay 𝜏 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 Μ‡π‘₯=π‘Žπ‘₯(π‘‘βˆ’πœ1)+𝑏π‘₯(π‘‘βˆ’πœ2) with delays 𝜏1,𝜏2 [25, 26] and scalar difference equations π‘₯𝑛=π‘₯π‘›βˆ’1+π‘Žπ‘₯π‘›βˆ’π‘š+𝑏π‘₯π‘›βˆ’π‘˜ with delays π‘˜,π‘š [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.

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