Abstract

We provide geometric algorithms for checking the stability of matrix difference equations 𝑥𝑛=𝐴𝑥𝑛𝑚+𝐵𝑥𝑛𝑘 with two delays 𝑚,𝑘 such that the matrix 𝐴𝐵𝐵𝐴 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 equation𝑥𝑛=𝑎𝑥𝑛𝑚+𝑏𝑥𝑛𝑘(1.1) with real coefficients 𝑎,𝑏 is basically solved [14].

The stability of matrix (1.1) with special 2×2 matrices 𝑎,𝑏 and 𝑚=1 was studied in [5, 6]. The case 𝑚=1, 𝑎=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 𝑎=𝛼𝐼, 𝛼, 0𝛼1. The representation of the solutions of (1.1) with commuting matrices 𝑎,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 𝑎,𝑏 has not been studied yet.

In this paper we provide geometric algorithms for checking the stability of (1.1) with two delays 𝑚,𝑘+, 𝑘>𝑚1, for two cases: (1) 𝑎,𝑏 are complex numbers, (2) 𝑎,𝑏 are simultaneously triangularizable matrices. The results of this paper are based on the 𝐷-decomposition method (parameter plane method) [10, 11].

Matrices 𝑎,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 𝑚=1 is studied along with other values 𝑚+. The case 𝑚=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 𝑎 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 𝑘,𝑚,𝑎,𝜌

Consider the scalar variant of (1.1). The characteristic polynomial for (1.1) is𝑓(𝜆)=𝜆𝑘𝑎𝜆𝑘𝑚𝑏.(2.1) If 𝑘=𝑑𝑘1,𝑚=𝑑𝑚1, 𝑑>1, then the trajectory of (1.1) splits into 𝑑 independent trajectories, and degree of polynomial (2.1) gets smaller after the substitution 𝜆𝑑=𝜇: 𝑓1(𝜆)=𝜇𝑘1𝑎𝜇𝑘1𝑚1𝑏.(2.2) Therefore, often (but not always) we will assume that the delays 𝑚,𝑘 are relatively prime.

Definition 2.1. 𝐷-decomposition curve for given 𝑘,𝑚+, 𝑎, 𝜌+ is a curve on the complex plane of the variable 𝑏 defined by the equation 𝑏(𝜔)=𝜌𝑘exp(𝑖𝑘𝜔)|𝑎|𝜌𝑘𝑚exp(𝑖(𝑘𝑚)𝜔),𝜔.(2.3)

Parameter 𝜔 moves along the interval of length 2𝜋, 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 𝑎. Starting from Section 5 we will get rid of this restriction. Obviously, if we assume in (2.1) that 𝑏=𝑏(𝜔) and 𝑎, 𝑎0, then (2.1) will have a root 𝜆=𝜌exp(𝑖𝜔). Hodograph (2.3) splits the complex plane into the connected components. This decomposition is called the 𝐷-decomposition [10]. If we put 𝑎, 𝑎0, 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 𝑎, 𝑎0, 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 𝑘,𝑚 are coprime, then hodograph (2.3) is invariant under the rotation by 2𝜋/𝑚.

Proof. For coprime 𝑘,𝑚 there exist 𝑠,𝑡+ such that 𝑘𝑠𝑚𝑡=1.(2.4) From (2.3), (2.4) it follows that exp𝑖2𝜋𝑚𝑏(𝜔)=𝑏𝜔+2𝜋𝑠𝑚.(2.5) Lemma 2.2 is proved.

From now and further we will assume that 𝜋<𝑎𝑟𝑔𝑧𝜋 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(𝜔).

Lemma 2.3. Let 𝑘,𝑚 be coprime, 𝑘>𝑚>1, |𝑎|<𝜌𝑚. Let 𝜔1[0;𝜋/𝑚] be the least positive root of the equation arg𝑏(𝜔)=𝜋/𝑚, and let 0<𝜔2𝜔1. Then, for any 𝜔, from 𝜔arg𝑏(𝜔)=arg𝑏2,(2.6) it follows that |𝑏(𝜔)||𝑏(𝜔2)|.

Proof. The function |𝑏(𝜔)| is 2𝜋/𝑚-periodic, increasing in [(2𝑗2)𝜋/𝑚,(2𝑗1)𝜋/𝑚] for any 𝑗 and decreasing in [(2𝑗1)𝜋/𝑚,2𝑗𝜋/𝑚]. In addition, |𝑏(𝜔+𝜋/𝑚)|=|𝑏(𝜔+𝜋/𝑚)| for any 𝜔. Let us assume, in order to get a contradiction, that (2.6) and |𝑏(𝜔)|<|𝑏(𝜔2)| are true for some 𝜔>0. Then there exists a positive integer 𝑠<𝑚 and 𝛿 such that 𝜔=2𝜋𝑠𝑚||𝛿||+𝛿,<𝜔2.(2.7) From (2.3), (2.7) it follows that Arg𝑏(𝜔)=(𝑘𝑚)𝜔+Arg(𝜌𝑚=exp(𝑖𝑚𝜔)|𝑎|)2𝜋𝑘𝑠𝑚+Arg(𝜌𝑚exp(𝑖𝑚𝛿)|𝑎|)+(𝑘𝑚)𝛿=2𝜋𝑘𝑠𝑚+Arg𝑏(𝛿).(2.8) Since 𝑘,𝑚 are coprime and 𝑠<𝑚, let us find natural numbers 𝑗,𝑞 such that 𝑘𝑠=𝑚𝑗+𝑞, 𝑚>𝑞1. Then (2.8) implies Arg𝑏(𝜔)=2𝜋𝑞𝑚+Arg𝑏(𝛿),(2.9) which contradicts (2.6) and the inequality |arg𝑏(𝛿)|<𝜋/𝑚 following from (2.7). Lemma 2.3 is proved.

3. Basic Ovals

For hodograph (2.3) the equality 𝑏𝜔(0)=𝑖𝜌𝑘𝑚(𝑘𝜌𝑚|𝑎|(𝑘𝑚)) takes place. If𝜌|𝑎|<𝑚𝑘,𝑘𝑚(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 [𝜔1,𝜔1] under the map 𝑏(𝜔) defined by (2.3). Here 𝜔1(0,𝜋/𝑘] is the least positive root of the equation arg𝑏(𝜔)=𝜋. 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 2𝜋𝑗/𝑚, 𝑗,0𝑗<𝑚 (see Lemma 2.2). We also call them the basic ovals. Here is a formal definition.

Definition 3.1. Let 𝑘,𝑚 be coprime, 𝑘>𝑚1, 𝑗, 0𝑗<𝑚, 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 𝜔 runs from (𝜔1+2𝜋𝑗𝑠/𝑚) to (𝜔1+2𝜋𝑗𝑠/𝑚), where 𝜔1 is the least positive root of the equation arg𝑏(𝜔)=𝜋.(3.2)

From Lemma 2.2 and formula (2.5) it follows that all 𝑚 basic ovals can be obtained from 𝐿0 by rotation by the angles 2𝜋𝑗/𝑚, 𝑗=0,1,,(𝑚1).

Considering Definition 3.1, we get the following. For existence of the basic oval it is necessary that |𝑎|<𝜌𝑚𝑘/(𝑘𝑚). If 𝑚>1 and |𝑎|>𝜌𝑚, then the complex number 0 is outside any oval, and the intersection of all ovals is empty. If 𝑚=1, then for fixed 𝑘, 𝜌, |𝑎|[0,𝜌𝑚𝑘/(𝑘𝑚)) the basic oval 𝐿0 is unique. That is why the results related to the stability of (1.1) are different for 𝑚=1 and 𝑚>1.

The basic oval 𝐿𝑗 decreases as |𝑎| increases from 0, and it shrinks to the point 𝑏=exp(2𝜋𝑗/𝑚)𝜌𝑘/(𝑘1) as |𝑎| reaches 𝜌𝑚𝑘/(𝑘𝑚) (Figure 1 for 𝑚>1 and Figure 2 for 𝑚=1).

Lemma 3.2. Let 𝑘,𝑚 be coprime, 𝑘>𝑚1, 𝑗, 0𝑗<𝑚, 𝑎, and 0𝑎<𝜌𝑚𝑘/(𝑘𝑚). If the complex number 𝑏 lies outside the basic oval 𝐿𝑗, then characteristic polynomial (2.1) has a root 𝜆 such that |𝜆|>𝜌.

Proof. Let us fix 𝑘,𝑚,𝑎,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 𝑅, 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 𝜆 of characteristic polynomial (2.1) such that |𝜆|=𝑅1>𝜌. Lemma 3.2 is proved.

4. Localization of Roots of Characteristic Polynomial (2.1) for Real Nonnegative 𝑎 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 (𝑥𝑛) the sequence (|𝑥𝑛|/𝜌𝑛) is bounded, and asymptotically 𝜌-stable if for any of its solutions (𝑥𝑛) one has lim𝑛|𝑥𝑛|/𝜌𝑛=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 𝑘,𝑚 in (1.1) has no influence on 𝜌-stability. It is not the case if 𝜌1. It is easy to see that the equation𝑥𝑛=𝑎𝑥𝑛𝑚𝑑+𝑏𝑥𝑛𝑘𝑑(4.1) is asymptotically 𝜌-stable if and only if (1.1) is asymptotically 𝜌𝑑-stable. It implies the following important observation: proportional increase of both delays 𝑘,𝑚 with the conservation of the coefficients 𝑎,𝑏 in (1.1) preserves asymptotic 𝜌-stability if 𝜌>1 and may not preserve it if 𝜌<1.

Definition 4.2. Let 𝑘,𝑚 be coprime, 𝑘>𝑚1, 𝑎, and 0|𝑎|<𝜌𝑚𝑘/(𝑘𝑚). The stability domain 𝐷(𝑘,𝑚,𝑎,𝜌) is defined to be a set of all complex numbers 𝑏 such that for any 𝑗(0𝑗<𝑚) the number 𝑏 lies inside the basic oval 𝐿𝑗.
Under the same conditions if 𝑘,𝑚 are not coprime and 𝑑=𝑔𝑐𝑑(𝑘,𝑚), let us put 𝐷(𝑘,𝑚,𝑎,𝜌)=𝐷(𝑘/𝑑,𝑚/𝑑,𝑎,𝜌𝑑).

Theorems 4.35.2 will justify the name “stability domain” for 𝐷(𝑘,𝑚,𝑎,𝜌) a little later. Evidently, for coprime 𝑘,𝑚 such that 𝑘>𝑚>1 the domain 𝐷(𝑘,𝑚,𝑎,𝜌) has the following properties. If 0|𝑎|<𝜌𝑚, then 𝐷(𝑘,𝑚,𝑎,𝜌) is the connected domain on the complex plane, containing 0, whose boundary is the 𝐷-decomposition curve (2.3). If |𝑎|=𝜌𝑚, then the domain 𝐷(𝑘,𝑚,𝑎,𝜌) degenerates into the point 𝑏=0. If 𝜌𝑚<|𝑎|<𝜌𝑚𝑘/(𝑘𝑚), then the domain 𝐷(𝑘,𝑚,𝑎,𝜌) is empty. If |𝑎|𝜌𝑚𝑘/(𝑘𝑚), then the domain 𝐷(𝑘,𝑚,𝑎,𝜌) is not defined in view of the fact that the basic ovals (Definition 3.1) are not defined.

If 𝑚=1, then the domain 𝐷(𝑘,1,𝑎,𝜌) is a set of points lying inside the oval 𝐿0. If 0|𝑎|<𝜌, then 𝐷(𝑘,1,𝑎,𝜌) includes 0. If 𝜌|𝑎|<𝜌𝑘/(𝑘1), then 𝐷(𝑘,1,𝑎,𝜌) is nonempty and does not contain 0. If |𝑎|=𝜌𝑘/(𝑘1), then 𝐷(𝑘,1,𝑎,𝜌) degenerates into the point 𝑏=𝜌𝑘/(𝑘1). Finally, if |𝑎|>𝜌𝑘/(𝑘1), then the domain 𝐷(𝑘,1,𝑎,𝜌) is not defined.

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

Theorem 4.3. Let 𝑘,𝑚 be coprime, 𝑘>𝑚>1, 𝑎+, 𝜌>0. (1)If 𝑎>𝜌𝑚, then for any 𝑏 (1.1) is 𝜌-unstable.(2)If 𝑎=𝜌𝑚, then for any complex 𝑏0 (1.1) is 𝜌-unstable; for 𝑏=0 it is 𝜌-stable (nonasymptotically).(3)If 0𝑎<𝜌𝑚, then (1.1) is asymptotically 𝜌-stable if and only if the complex number 𝑏 lies inside the stability domain 𝐷(𝑘,𝑚,a,𝜌). (4)If 0𝑎<𝜌𝑚, then (1.1) is 𝜌-stable if and only if the complex number 𝑏 lies either inside or on the boundary of 𝐷(𝑘,𝑚,𝑎,𝜌).

Proof. (1) Let 𝑎>𝜌𝑚. Let us find 𝑅 such that 𝜌𝑚<𝑅𝑚𝑅<𝑎<𝑚𝑘.𝑘𝑚(4.2) Taking into account the inequality 𝑎<𝑅𝑚𝑘/(𝑘𝑚), let us consider 𝑚 basic ovals 𝐿𝑗(𝑅), 𝑗=0,1,,𝑚1 having 𝜌 replaced by 𝑅 in Definition 4.1. Since 𝑅𝑚<𝑎, the system of ovals 𝐿𝑗(𝑅) has no intersections. Hence, for any complex number 𝑏 there exists 𝑗,0𝑗<𝑚 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 𝑎=𝜌𝑚. If 𝑏=0, then statement 2 of Theorem 4.3 is obvious. Let 𝑏0. If Re𝑏0, then 𝑏 lies outside the oval 𝐿0. If Re𝑏0 and 𝑚 is even, then 𝑏 lies outside the oval 𝐿𝑚/2. If Re𝑏0 and 𝑚 is odd, then 𝑏 lies either outside the oval 𝐿(𝑚1)/2 or outside the oval 𝐿(𝑚+1)/2. In any case (1.1) is 𝜌-unstable by Lemma 3.2. Statement 2 is proved.
(3) Let 0𝑎<𝜌𝑚. Let the number 𝑏 be inside the domain 𝐷(𝑘,𝑚,𝑎,𝜌). Then for any 𝑗(0𝑗<𝑚) 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(𝑘,𝑚,𝑎,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(𝑘,𝑚,𝑎,𝜌), then the conclusion of statement 4 of Theorem 4.3 is a straightforward consequence of Lemma 3.2. If 𝑏 lies inside 𝐷(𝑘,𝑚,𝑎,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(𝑘,𝑚,𝑎,𝜌). Then for any root 𝜆 of polynomial (2.1) either |𝜆|<𝜌 or |𝜆|=𝜌. In the latter case in view of the inequality 0𝑎<𝜌𝑚<𝜌𝑚𝑘/𝑘𝑚 we have 𝑑𝑓𝑑𝜆=𝜆𝑘𝑚1(𝑘𝜆𝑚𝑎(𝑘𝑚))0,(4.3) hence the root 𝜆 such that |𝜆|=𝜌 is simple. Theorem 4.3 is proved.

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

Theorem 4.4. Let 𝑘>𝑚=1, 𝑎+, 𝜌>0. (1)If 𝑎𝜌𝑘/(𝑘1), then for all complex numbers 𝑏 (1.1) is 𝜌-unstable.(2)If 0𝑎<𝜌𝑘/(𝑘1), then (1.1) is asymptotically 𝜌-stable if and only if the complex number 𝑏 lies inside the domain 𝐷(𝑘,1,𝑎,𝜌).(3)If 0𝑎<𝜌𝑘/(𝑘1), then (1.1) is 𝜌-stable if and only if the complex number 𝑏 lies inside 𝐷(𝑘,1,𝑎,𝜌).

Proof. (1) Let 𝑎>𝜌𝑘/(𝑘𝑚), and let 𝑏 be a given complex number. Let us find 𝑅>𝜌 such that 𝜌𝑘/(𝑘𝑚)<𝑎<𝑅𝑘/(𝑘𝑚) 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 𝜆 of polynomial (2.1) such that |𝜆|>𝑅>𝜌, so 𝜌-instability of (1.1) is proved. Let 𝑎=𝜌𝑘/(𝑘1). Then the previous arguments also prove 𝜌-instability provided that 𝑏𝜌𝑘/(𝑘1). However, if 𝑏=𝜌𝑘/(𝑘1), then under the assumption that 𝑎=𝜌𝑘/(𝑘1) the number 𝜆=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 0𝑎<𝜌𝑘/(𝑘1). Since 𝐷(𝑘,1,𝑎,𝜌) is the domain of inner points of the oval 𝐿0, it is connected. The function |𝑏(𝜔)| (see (2.3)) increases as 𝜔 moves either from 0 to 𝜋 or from 0 to (𝜋). 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 0𝑎<𝜌. Then the point 𝑏=0 lies inside 𝐿0. If 𝑏=0, then polynomial (2.1) has the (𝑘1)-multiple root 𝜆=0 and the simple root 𝜆=𝑎. This gives the asymptotic 𝜌-stability, in view of 𝑎<𝜌.
CASE 2. Let 𝜌𝑎<𝜌𝑘/(𝑘1). Let us consider the point 𝑏=𝜌𝑘𝑎𝜌𝑘1 at the boundary of 𝐿0 and consider characteristic polynomial (2.1) with given 𝑏: 𝑓2(𝜆)=𝜆𝑘𝑎𝜆𝑘1𝜌𝑘+𝑎𝜌𝑘1.(4.4) The equation 𝑓2(𝜆)=0 transforms into 𝜆𝜌𝜆1𝜌𝑘1𝑎𝜌1𝑘2𝑗=0𝜆𝜌𝑘𝑗2=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 0𝑎𝜌<𝜌/(𝑘1) [14].
Let us return to (2.1), and let us figure out in what direction the root 𝜆=𝜌 moves as the coefficient 𝑏 moves from the point 𝑏=𝜌𝑘𝑎𝜌𝑘1 toward the interior of 𝐿0 so that d𝑏, d𝑏<0. From (2.1) it follows that for 𝜆=𝜌 we have d𝜆=𝜌d𝑏𝑘+2,𝑘𝜌𝑎(𝑘1)(4.6) and in view of 𝑎<𝜌𝑘/(𝑘1) we get d𝜆/d𝑏>0. Therefore, d𝑏<0 implies 𝜆<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 𝑎,𝑏

Let us change the variables in (1.1) so that it has no influence on (asymptotic) 𝜌-stability:𝑥𝑛=𝑦𝑛𝑖𝑛exp𝑚.arg𝑎(5.1) Equation (1.1) changes to𝑦𝑛=𝛼𝑦𝑛𝑚+𝛽𝑦𝑛𝑘,(5.2) where𝑘𝛼=|𝑎|,𝛽=𝑏exp𝑖𝑚.arg𝑎(5.3) The characteristic polynomial for (5.2) has the form𝜓(𝜇)=𝜇𝑘𝛼𝜇𝑘𝑚𝛽.(5.4) It is related to (2.1) by the change 𝜇=𝜆exp(𝑖(1/𝑚)arg𝑎), 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 𝑦𝑛𝑚, 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 𝑎, 𝑏.

Theorem 5.1. Let 𝑘,𝑚 be coprime, 𝑘>𝑚>1, 𝑎, 𝜌>0.
(1)If |𝑎|>𝜌𝑚, then for any complex 𝑏 (1.1) is 𝜌-unstable.(2)If |𝑎|=𝜌𝑚, then for any 𝑏0 (1.1) is 𝜌-unstable; for 𝑏=0 it is 𝜌-stable (nonasymptotically).(3)If |𝑎|<𝜌𝑚, then (1.1) is asymptotically 𝜌-stable if and only if the complex number 𝛽=𝑏exp(𝑖(𝑘/𝑚)arg𝑎) lies inside the domain 𝐷(𝑘,𝑚,𝑎,𝜌).(4)If |𝑎|<𝜌𝑚, then (1.1) is 𝜌-stable if and only if the complex number 𝛽=𝑏exp(𝑖(𝑘/𝑚)arg𝑎) lies either inside 𝐷(𝑘,𝑚,𝑎,𝜌) or on its boundary.

Theorem 5.2. Let 𝑘>𝑚=1, 𝑎, 𝜌>0.
(1)If |𝑎|𝜌𝑘/(𝑘1), then for any complex 𝑏 (1.1) is 𝜌-unstable.(2)If |𝑎|<𝜌𝑘/(𝑘1), then (1.1) is asymptotically 𝜌-stable if and only if the complex number 𝛽=𝑏exp(𝑖𝑘arg𝑎) lies inside the domain 𝐷(𝑘,1,𝑎,𝜌).(3)If |𝑎|<𝜌𝑘/(𝑘1), then (1.1) is 𝜌-stable if and only if the complex number 𝛽=𝑏exp(𝑖𝑘arg𝑎) lies either inside 𝐷(𝑘,1,a,𝜌) or on its boundary.

Example 5.3. Let 𝑚=6, 𝑎=0.8+0.9𝑖, 𝜌=1.15 in (1.1). Let 6𝑘12. For every given value 𝑘 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 |𝑎|=1.204, arg𝑎0.844. If 𝑘=6, then to find 𝜌-stability domain one does not need to use Theorems 4.35.2. The domain is a circle given in Figure 4(a). Since 6,7 are coprime, then for 𝑘=7, by Theorem 5.1, (1.1) 𝜌-stable if and only if 𝑏exp(𝑖(7/6)arg𝑎)𝐷(7,6,𝑎,𝜌). The corresponding “curved hexagon” is shown in Figure 4(a). Similarly for 𝑘=11 the condition 𝑏exp(𝑖(11/6)arg𝑎)𝐷(11,6,𝑎,𝜌) is necessary and sufficient for asymptotic stability of (1.1). The corresponding “curved hexagon” is shown in Figure 4(b). For 𝑘=8 the stability criterion is the condition 𝑏exp(𝑖(4/3)arg𝑎)𝐷(4,3,𝑎,𝜌2). The corresponding “curved triangle” is shown in Figure 4(a). Similarly the “digon” exp(𝑖(3/2)arg𝑎)𝐷(3,2,𝑎,𝜌3) for 𝑘=9 is shown in Figure 4(a), and the “curved triangle” exp(𝑖(5/3)arg𝑎)𝐷(5,3,𝑎,𝜌2) for 𝑘=10 is shown in Figure 4(b). For 𝑘=12, according to Theorem 5.2, the stability criterion for (1.1) is 𝑏exp(𝑖2arg𝑎)𝐷(2,1,𝑎,𝜌6) (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 equation𝑥𝑛=𝐴𝑥𝑛𝑚+𝐵𝑥𝑛𝑘,(6.1)𝑥+𝑙; 𝐴,𝐵𝑙×𝑙. The characteristic equation for (6.1) is𝜓(𝜆)=det𝐼𝜆𝑘𝐴𝜆𝑘𝑚.𝐵(6.2)

Definition 6.1. Matrix equation (6.1) is called 𝜌-stable if for every solution (𝑥𝑛) the sequence (|𝑥𝑛|/𝜌𝑛) is bounded. Equation (6.1) is called asymptotically 𝜌-stable if lim𝑛|𝑥𝑛|/𝜌𝑛=0 holds for every solution (𝑥𝑛).

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,𝐵. It is known [12] that if the matrix 𝐴𝐵𝐵𝐴 is nilpotent, then 𝐴,𝐵 can be simultaneously triangularized.

Definition 6.2. If 𝑘>𝑚>1, then the 𝜌-stability cone for given 𝑘,𝑚,𝜌 is a set of points 𝑀=(𝑢1,𝑢2,𝑢3)3 such that 0𝑢31 and the intersection of the set with any plane 𝑢3=𝑎(0𝑎1) is the stability domain 𝐷(𝑘,𝑚,𝑎,𝜌). If 𝑘>𝑚=1, then the 𝜌-stability cone for given 𝑘,𝜌 is a set of points 𝑀=(𝑢1,𝑢2,𝑢3)3 such that 0𝑢3𝑘/(𝑘1) and the intersection of the set with the plane 𝑢3=𝑎(0𝑎𝑘/(𝑘1)) is the domain 𝐷(𝑘,1,𝑎,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 𝑘=5,𝑚=1 at different heights 𝑢3=𝑎, and the ones in Figure 3(b) as sections of the stability cone for 𝑘=5,𝑚=2, and so on.

The stability cones for 𝑚>1 are the intersections of 𝑚 conical surfaces formed by the basic ovals as the parameter 𝑎 changes from 0 to 𝑘/(𝑘𝑚) (Figure 5).

Let us consider the simple case of a diagonal system𝑦𝑛𝑎diag11,,𝑎𝑙𝑙𝑦𝑛𝑚𝑏diag11,,𝑏𝑙𝑙𝑦𝑛𝑘=0(6.3) with complex entries 𝑎𝑗𝑗,𝑏𝑗𝑗, 1𝑗𝑙. Let us construct the points 𝑀𝑗=(𝑢1𝑗,𝑢2𝑗,𝑢3𝑗)   (1𝑗𝑙) in 3 in the following way:𝑢1𝑗𝑏=Re𝑗𝑗𝑘exp𝑖𝑚arg𝑎𝑗𝑗,𝑢2𝑗𝑏=Im𝑗𝑗𝑘exp𝑖𝑚arg𝑎𝑗𝑗,𝑢3𝑗=||𝑎𝑗𝑗||.(6.4)

It follows from the definition of the 𝜌-stability cone and from Theorems 5.15.2 that (6.3) is asymptotically 𝜌-stable if and only if all the points 𝑀𝑗(1𝑗𝑙) lie inside the 𝜌-stability cone for given 𝑘,𝑚. All the points 𝑀𝑗 with 𝑢3𝑗=0, 𝑢21𝑗+𝑢22𝑗<𝜌𝑘 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 𝑘>𝑚1, let the numbers 𝑘,𝑚 be coprime, and 𝜌>0. Let 𝐴,𝐵,𝑆𝑙×𝑙, and 𝑆1𝐴𝑆=𝐴𝑇, and 𝑆1𝐵𝑆=𝐵𝑇, where 𝐴𝑇 and 𝐵𝑇 are lower triangle matrices with elements 𝑎𝑗𝑠,𝑏𝑗𝑠(1𝑗,𝑠𝑙). Let one construct the points 𝑀𝑗=(𝑢1𝑗,𝑢2𝑗,𝑢3𝑗), (1𝑗𝑙) by the formulas (cf. (6.4)) 𝑢1𝑗=||𝑏𝑗𝑗||cosarg𝑏𝑗𝑗𝑘𝑚arg𝑎𝑗𝑗,𝑢2𝑗=||𝑏𝑗𝑗||sinarg𝑏𝑗𝑗𝑘𝑚arg𝑎𝑗𝑗,𝑢3𝑗=||𝑎𝑗𝑗||.(6.5) Then (6.1) is 𝜌-asymptotically stable if and only if all the points 𝑀𝑗(1𝑗𝑙) lie inside the 𝜌-stability cone for the given 𝑘,𝑚,𝜌.
If some point 𝑀𝑗 lies outside the 𝜌-stability cone, then (6.1) is 𝜌-unstable.

Proof. Let us make the change 𝑦𝑛=𝑆𝑥𝑛. Then (6.1) transforms to the following one: 𝑦𝑛=𝐴𝑇𝑦𝑛𝑚+𝐵𝑇𝑦𝑛𝑘.(6.6) The characteristic polynomial for (6.6) has the form 𝜓(𝜆)=𝑙𝑗=1𝜆𝑘𝑎𝑗𝑗𝜆𝑘𝑚𝑏𝑗𝑗.(6.7) It coincides with the characteristic polynomial of diagonal system (6.3). Therefore, from statement 3 of Theorem 5.1 (for 𝑚>1) and from statement 2 of Theorem 5.2 (for 𝑚=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 𝑚>1) and statement 3 of Theorem 5.2 (for 𝑚=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 𝑘-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)𝑛𝑚𝑦+𝑔(𝑙)𝑛𝑚𝑦+(2)𝑛𝑘,𝑦𝑛(2)𝑦=𝑓(2)𝑛𝑚𝑦+𝑔(1)𝑛𝑚𝑦+(3)𝑛𝑘,𝑦𝑛(𝑙)𝑦=𝑓(𝑙)𝑛𝑚𝑦+𝑔(𝑙1)𝑛𝑚𝑦+(1)𝑛𝑘,(7.1) where 𝑓,𝑔, are sufficiently smooth real-valued functions of a real variable. Let us assume that there is a real number 𝑦 such that the stationary sequences 𝑦1𝑛=𝑦,,𝑦𝑙𝑛=𝑦 form a solution of (7.1). Let us introduce the variables 𝑥𝑛(𝑗)=𝑦𝑛(𝑗)𝑦 and the vector 𝑥𝑛=(𝑥𝑛(1),,𝑥𝑛(𝑙))𝑇, and let us linearize system (7.1) in new variables about zero. We get (6.1) with the circulant [17] matrices𝐴=𝛼00𝛽𝛽𝛼000𝛽𝛼0000𝛼,𝐵=0𝛾0000𝛾00000𝛾000.(7.2) Here 𝛼=𝑑𝑓(𝑦)/𝑑𝑦, 𝛽=𝑑𝑔(𝑦)/𝑑𝑦, 𝛾=𝑑(𝑦)/𝑑𝑦. Let us introduce a matrix 𝑃 (a lines permutation operator):𝑃=0100001000001000.(7.3) Then 𝐵=𝛾𝑃, 𝐴=𝛼𝐼+𝛽𝑃𝑙1, therefore, diagonalization 𝑃 generates simultaneous diagonalization of 𝐴,𝐵. The eigenvalues of 𝑃 are 1, 𝜀, 𝜀2,,𝜀𝑙1, where 𝜀=exp(𝑖2𝜋/𝑙). Therefore,𝐴𝑇=𝛼𝐼+𝛽diag1,𝜀𝑙1,𝜀𝑙2,,𝜀,𝐵𝑇=𝛾diag1,𝜀,𝜀2,,𝜀𝑙1.(7.4) Granting (7.4), by Theorem 6.3, we can build points 𝑀𝑗=(𝑢1𝑗,𝑢2𝑗,𝑢3𝑗) in 3 for system (6.1), (7.2):𝑢1𝑗=𝛾cos2𝜋(𝑗1)𝑙𝑘𝑚arg𝛼+𝛽exp2𝜋𝑖(𝑗1)𝑙,𝑢2j=𝛾sin2𝜋(𝑗1)𝑙𝑘𝑚arg𝛼+𝛽exp2𝜋𝑖(𝑗1)𝑙,𝑢3𝑗=||||𝛼+𝛽exp2𝜋𝑖(𝑗1)𝑙||||.(7.5) We get the following consequence of Theorem 6.3.

Corollary 7.1. If for every 𝑗(1𝑗𝑙) the point 𝑀𝑗=(𝑢1𝑗,𝑢2𝑗,𝑢3𝑗) defined by formulas (7.5) lies inside the 𝜌-stability cone for given 𝑘,𝑚,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 𝑘,𝑚,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=(𝑢1𝑗,𝑢2𝑗,𝑢3𝑗) defined by (7.5) lie on the closed curve𝑢1𝑘(𝑡)=𝛾cos𝑡𝑚,𝑢arg(𝛼+𝛽exp(𝑖𝑡))2𝑘(𝑡)=𝛾sin𝑡𝑚,𝑢arg(𝛼+𝛽exp(𝑖𝑡))3||||(𝑡)=𝛼+𝛽exp(𝑖𝑡),0𝑡2𝜋.(7.6)

If 𝑙, 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,𝐵. If any point of the curve (7.6) lies inside the 𝜌-stability cone for given 𝑘,𝑚,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 𝑘,𝑚,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 𝑘=4, 𝑚=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 𝑘,𝑚,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 𝑙2. 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).

Example 7.4. Let us consider the neuron chain shown in Figure 6. Let us fix the parameters: 𝑘=4, 𝑚=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=𝑟, =0.1, 𝑟=0,1,,5, and the points 𝑀1, 𝑀2, 𝑀3 for 𝑙=3. The upper part of the stability cone corresponding to 𝑢3>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 0𝛾<0.4. Therefore, if 0𝛾<0.4, then system (6.1), (7.2) is asymptotically stable for any 𝑙2. If 𝛾=0.4, then the point 𝑀1=(𝑢1,𝑢2,𝑢3) 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 0𝛾<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+𝐵<1 is sufficient for the asymptotic stability of matrix (6.1) [19], and it does not require simultaneous triangularization of the matrices 𝐴,𝐵. There are sufficient conditions for stability of nonautonomous scalar difference equations in [2022].

Stability cones for differential matrix equations ̇𝑥=𝐴𝑥+𝐵𝑥(𝑡𝜏) 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 ̇𝑥=𝑎𝑥(𝑡𝜏1)+𝑏𝑥(𝑡𝜏2) with delays 𝜏1,𝜏2 [25, 26] and scalar difference equations 𝑥𝑛=𝑥𝑛1+𝑎𝑥𝑛𝑚+𝑏𝑥𝑛𝑘 with delays 𝑘,𝑚 [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.