# The Stability Cone for a Difference Matrix Equation  with Two Delays

Authors: S. A. Ivanov; M. M. Kipnis; V. V. Malygina
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

We provide geometric algorithms for checking the stability of matrix difference equationsxn=Axn−m+Bxn−k with two delays
m,k such that the matrix
AB−BA is nilpotent. We give examples of how our results can be applied to the study of the stability of neural networks.


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## Body

## 1. Introduction

The problem of the stability of the equation(1.1)xn=axn-m+bxn-k
with real coefficients a,b is basically solved [1–4].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∈ℝ, 0⩽α⩽1. 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∈ℤ+, k>m⩾1, 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].Matricesa,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∈ℤ+. 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 Section2, 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) is(2.1)f(λ)=λk-aλk-m-b.
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: (2.2)f1(λ)=μk1-aμk1-m1-b.
Therefore, often (but not always) we will assume that the delays m,k are relatively prime.Definition 2.1.
D-decomposition curve for givenk,m∈ℤ+,a∈ℂ,ρ∈ℝ+is a curve on the complex plane of the variablebdefined by the equation(2.3)b(ω)=ρkexp⁡(ikω)-|a|ρk-mexp⁡(i(k-m)ω),ω∈R.Parameterω 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 ofa. Starting from Section 5 we will get rid of this restriction. Obviously, if we assume in (2.1) that b=b(ω) and a∈ℝ, a≥0, 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∈ℝ, a≥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 a∈ℝ, a⩾0, 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).
Ifk,m are coprime, then hodograph (2.3) is invariant under the rotation by  2π/m.Proof.
For coprimek,m there exist s,t∈ℤ+ such that
(2.4)ks-mt=1.
From (2.3), (2.4) it follows that
(2.5)exp⁡(i2πm)b(ω)=b(ω+2πsm).
Lemma 2.2 is proved.From now and further we will assume that-π<argz⩽π 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 hodographb(ω).Lemma 2.3.
Letk,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<ω2⩽ω1. Then, for any  ω∈ℝ, from
(2.6)argb(ω)=argb(ω2),
it follows that |b(ω)|⩾|b(ω2)|.Proof.
The function|b(ω)| is 2π/m-periodic, increasing in [(2j-2)π/m,(2j-1)π/m] for any j∈ℤ and decreasing in [(2j-1)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∈ℝ such that
(2.7)ω=2πsm+δ,|δ|<ω2.
From (2.3), (2.7) it follows that
(2.8)Argb(ω)=(k-m)ω+Arg(ρmexp⁡(imω)-|a|)=2πksm+Arg(ρmexp⁡(imδ)-|a|)+(k-m)δ=2πksm+Argb(δ).
Since k,m are coprime and s<m, let us find natural numbers j,q such that ks=mj+q, m>q⩾1. Then (2.8) implies
(2.9)Argb(ω)=2πqm+Argb(δ),
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)=iρk-m(kρm-|a|(k-m)) takes place. If(3.1)|a|<ρmkk-m,
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∈ℤ,0⩽j<m (see Lemma 2.2). We also call them the basic ovals. Here is a formal definition.Definition 3.1.
Letk,mbe coprime,k>m⩾1,j∈ℤ,0⩽j<m, and let (2.4), (3.1) hold. The basic ovalLjfor (1.1) is a closed curve given by (2.3), where the variableωruns from(-ω1+2πjs/m)to(ω1+2πjs/m), whereω1is the least positive root of the equation(3.2)argb(ω)=π.From Lemma2.2 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,…,(m-1).Considering Definition3.1, we get the following. For existence of the basic oval it is necessary that |a|<ρmk/(k-m). 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/(k-m)) 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 ovalLj decreases as |a| increases from 0, and it shrinks to the point b=-exp(2πj/m)ρk/(k-1) as |a| reaches ρmk/(k-m) (Figure 1 for m>1 and Figure 2 for m=1).D-decomposition curve and three basic ovals for m=3,k=5: (a) |a|<ρm, (b) |a|=ρm, (c) ρm<|a|<ρmk/(k-m), and (d) |a|=ρmk/(k-m).
(a)(b)(c)(d)D-decomposition curve and the basic oval L0 for m=1,k=5: (a) |a|<ρ, (b) |a|=ρ, (c) ρ<|a|<ρk/(k-1), and (d) |a|=ρk/(k-1).
(a)(b)(c)(d)Lemma 3.2.
Letk,m be coprime,  k>m⩾1,  j∈ℤ,  0⩽j<m,  a∈ℝ, and  0⩽a<ρmk/(k-m). 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 fixk,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 beρ-stable if for any of its solutions(xn)the sequence(|xn|/ρn)is bounded, and asymptoticallyρ-stable if for any of its solutions(xn)one haslimn→∞|xn|/ρn=0.Ifρ=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 ρ≠1. It is easy to see that the equation(4.1)xn=axn-md+bxn-kd
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.
Letk,mbe coprime,k>m⩾1,a∈ℂ, and0⩽|a|<ρmk/(k-m). The stability domainD(k,m,a,ρ)is defined to be a set of all complex numbersbsuch that for anyj(0⩽j<m)the numberblies inside the basic ovalLj.
Under the same conditions ifk,mare not coprime andd=gcd(k,m), let us putD(k,m,a,ρ)=D(k/d,m/d,a,ρd).Theorems4.3–5.2 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/(k-m), then the domain D(k,m,a,ρ) is empty. If |a|⩾ρmk/(k-m), 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.Ifm=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/(k-1), then D(k,1,a,ρ) is nonempty and does not contain 0. If |a|=ρk/(k-1), then D(k,1,a,ρ) degenerates into the point b=-ρk/(k-1). Finally, if |a|>ρk/(k-1), 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.
Letk,m be coprime,  k>m>1,  a∈ℝ+,  ρ>0. (1)
Ifa>ρm, then for any b (1.1) is ρ-unstable.(2)
Ifa=ρm, then for any complex b≠0 (1.1) is ρ-unstable; for b=0 it is ρ-stable (nonasymptotically).(3)
If0⩽a<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)
If0⩽a<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) Leta>ρm. Let us find R∈ℝ such that
(4.2)ρm<Rm<a<Rmkk-m.
Taking into account the inequality a<Rmk/(k-m), let us consider m basic ovals Lj(R), j=0,1,…,m-1 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∈ℤ,0⩽j<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) Leta=ρm. 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 m is even, then b lies outside the oval Lm/2. If Reb⩽0 and m is odd, then b lies either outside the oval L(m-1)/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) Let0⩽a<ρm. Let the number b be inside the domain D(k,m,a,ρ). Then for any j(0⩽j<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.
Ifb 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) Ifb 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 0⩽a<ρm<ρmk/k-m we have
(4.3)dfdλ=λk-m-1(kλm-a(k-m))≠0,
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.
Letk>m=1,  a∈ℝ+,  ρ>0.  (1)
Ifa⩾ρk/(k-1), then for all complex numbers b (1.1) is ρ-unstable.(2)
If0⩽a<ρk/(k-1), 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)
If0⩽a<ρk/(k-1), then (1.1) is ρ-stable if and only if the complex number b lies inside D(k,1,a,ρ).Proof.
(1) Leta>ρk/(k-m), and let b be a given complex number. Let us find R>ρ such that ρk/(k-m)<a<Rk/(k-m) 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/(k-1). Then the previous arguments also prove ρ-instability provided that b≠-ρk/(k-1). However, if b=-ρk/(k-1), then under the assumption that a=ρk/(k-1) 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) Let0⩽a<ρk/(k-1). 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. Let0⩽a<ρ. Then the point b=0 lies inside L0. If b=0, then polynomial (2.1) has the (k-1)-multiple root λ=0 and the simple root λ=a. This gives the asymptotic ρ-stability, in view of a<ρ.
CASE 2. Letρ⩽a<ρk/(k-1). Let us consider the point b=ρk-aρk-1 at the boundary of L0 and consider characteristic polynomial (2.1) with given b:
(4.4)f2(λ)=λk-aλk-1-ρk+aρk-1.
The equation f2(λ)=0 transforms into
(4.5)((λρ)-1)((λρ)k-1-((aρ)-1)∑j=0k-2(λρ)k-j-2)=0.
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 0⩽a-ρ<ρ/(k-1) [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=ρk-aρk-1 toward the interior of L0 so that db∈ℝ, db<0. From (2.1) it follows that for λ=ρ we have
(4.6)dλdb=ρ-k+2kρ-a(k-1),
and in view of a<ρk/(k-1) 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 Theorem4.4 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:(5.1)xn=ynexp⁡(inmarga).
Equation (1.1) changes to(5.2)yn=αyn-m+βyn-k,
where(5.3)α=|a|,β=bexp⁡(-ikmarga).
The characteristic polynomial for (5.2) has the form(5.4)ψ(μ)=μk-αμk-m-β.
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 yn-m, 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.
Letk,m be coprime,  k>m>1,  a∈ℂ,  ρ>0.
(1)
If|a|>ρm, then for any complex b (1.1) is ρ-unstable.(2)
If|a|=ρm, then for any b≠0 (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.
Letk>m=1,  a∈ℂ,  ρ>0.
(1)
If|a|⩾ρk/(k-1), then for any complex b (1.1) is ρ-unstable.(2)
If|a|<ρk/(k-1), 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/(k-1), 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.
Letm=6, a=0.8+0.9i, ρ=1.15 in (1.1). Let 6⩽k⩽12. 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, arga≃0.844. If k=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 k=7, by Theorem 5.1, (1.1) ρ-stable if and only if b∈exp(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 b∈exp(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 b∈exp(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 b∈exp(i·2arga)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 equation(6.1)xn=Axn-m+Bxn-k,x:ℤ+→ℂl; A,B∈ℂl×l. The characteristic equation for (6.1) is(6.2)ψ(λ)=det⁡(Iλk-Aλk-m-B).Definition 6.1.
Matrix equation (6.1) is calledρ-stable if for every solution(xn)the sequence(|xn|/ρn)is bounded. Equation (6.1) is called asymptoticallyρ-stable iflimn→∞|xn|/ρn=0holds 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 AB-BA is nilpotent, then A,B can be simultaneously triangularized.Definition 6.2.
Ifk>m>1, then theρ-stability cone for givenk,m,ρis a set of pointsM=(u1,u2,u3)∈ℝ3such that0⩽u3⩽1and the intersection of the set with any planeu3=a(0⩽a⩽1)is the stability domainD(k,m,a,ρ). Ifk>m=1, then theρ-stability cone for givenk,ρis a set of pointsM=(u1,u2,u3)∈ℝ3such that0⩽u3⩽k/(k-1)and the intersection of the set with the planeu3=a(0⩽a⩽k/(k-1))is the domainD(k,1,a,ρ).Let us define a stability cone as theρ-stability cone for ρ=1.Returning to Figure3, 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.Stability domainsD(k,m,a,ρ) for k=5, for different values of the coefficient a: (a) m=1, (b) m=2, (c) m=3, (d) m=4.
(a)(b)(c)(d)To Example5.3. Stability domains for m=6, a=0.8+0.9i, ρ=1.19; (a) k=6,7,8,9, (b) k=10,11,12.
(a)(b)The stability cones form>1 are the intersections of m conical surfaces formed by the basic ovals as the parameter a changes from 0 to k/(k-m) (Figure 5).Figure 5
The stability cone as an intersection of three surfaces formed by the basic ovals,m=3,k=4.Let us consider the simple case of a diagonal system(6.3)yn-diag⁡(a11,…,all)yn-m-diag⁡(b11,…,bll)yn-k=0
with complex entries ajj,bjj, 1⩽j⩽l. Let us construct the points Mj=(u1j,u2j,u3j)   (1⩽j⩽l) in ℝ3 in the following way:(6.4)u1j=Re(bjjexp⁡(-ikmargajj)),u2j=Im⁡(bjjexp⁡(-ikmargajj)),u3j=|ajj|.It follows from the definition of theρ-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 k,m. All the points Mj with u3j=0, u1j2+u2j2<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.
Letk>m⩾1, let the numbers  k,m be coprime, and  ρ>0. Let  A,B,S∈ℝl×l, and  S-1AS=AT, and  S-1BS=BT, where  AT and  BT are lower triangle matrices with elements  ajs,bjs(1⩽j,s⩽l). Let one construct the points  Mj=(u1j,u2j,u3j),  (1⩽j⩽l) by the formulas (cf. (6.4))
(6.5)u1j=|bjj|cos⁡(argbjj-kmargajj),u2j=|bjj|sin(argbjj-kmargajj),u3j=|ajj|.
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 k,m,ρ.
If some pointMj lies outside the ρ-stability cone, then (6.1) is ρ-unstable.Proof.
Let us make the changeyn=Sxn. Then (6.1) transforms to the following one:
(6.6)yn=ATyn-m+BTyn-k.
The characteristic polynomial for (6.6) has the form
(6.7)ψ(λ)=∏j=1l(λk-ajjλk-m-bjj).
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.Figure 6
A neural net.Letyn(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 equations(7.1)yn(1)=f(yn-m(1))+g(yn-m(l))+h(yn-k(2)),yn(2)=f(yn-m(2))+g(yn-m(1))+h(yn-k(3)),⋯yn(l)=f(yn-m(l))+g(yn-m(l-1))+h(yn-k(1)),
where f,g,h 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 yn1=y*,…,ynl=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] matrices(7.2)A=(α00⋯ββα0⋯00βα⋯0⋯⋯⋯⋯⋯000⋯α),B=(0γ0⋯000γ⋯0000⋯0⋯⋯⋯⋯⋯γ00⋯0).
Here α=df(y*)/dy, β=dg(y*)/dy, γ=dh(y*)/dy. Let us introduce a matrix P (a lines permutation operator):(7.3)P=(010⋯0001⋯0000⋯0⋯⋯⋯⋯⋯100⋯0).
Then B=γP, A=αI+βPl-1, therefore, diagonalization P generates simultaneous diagonalization of A,B. The eigenvalues of P are 1, ɛ, ɛ2,…,ɛl-1, where ɛ=exp(i2π/l). Therefore,(7.4)AT=αI+βdiag⁡(1,ɛl-1,ɛl-2,…,ɛ),BT=γdiag⁡(1,ɛ,ɛ2,…,ɛl-1).
Granting (7.4), by Theorem 6.3, we can build points Mj=(u1j,u2j,u3j) in ℝ3 for system (6.1), (7.2):(7.5)u1j=γcos⁡(2π(j-1)l-(km)arg(α+βexp⁡(-2πi(j-1)l))),u2j=γsin(2π(j-1)l-(km)arg(α+βexp⁡(-2πi(j-1)l))),u3j=|α+βexp⁡(-2πi(j-1)l)|.
We get the following consequence of Theorem 6.3.Corollary 7.1.
If for everyj(1⩽j⩽l) 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 pointsMj=(u1j,u2j,u3j) defined by (7.5) lie on the closed curve(7.6)u1(t)=γcos⁡(t-(km)arg(α+βexp⁡(-it))),u2(t)=γsin(t-(km)arg(α+βexp⁡(-it))),u3(t)=|α+βexp⁡(-it)|,0⩽t⩽2π.Ifl→∞, 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 Figure6. 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 Corollaries7.1 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 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.To Example7.3. (a) The curves (7.6) for α=0.1+rh,h=0.2,r=0,1,…,5. (b) The stability cone covers the curves (7.6) for α<0.5. If α=0.5, then the point M1 is on the surface of the cone. If α>0.5, then all the curves (7.6) are outside the cone either entirely or partially.
(a)(b)Now let us consider Example7.3 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).Table 1
ρ0 versus α in Example 7.3.
α0.10.30.50.70.91.11.31.5ρ0(α)0.8700.9381.0001.0561.1081.1561.2011.243Example 7.4.
Let us consider the neuron chain shown in Figure6. 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=rh, h=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 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=(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 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.To Example7.4. The stability cone and the curves (7.6) for γ=rh,h=0.1,r=0,1,…,5 in two projections.
(a)(b)


## 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 [20–22].Stability cones for differential matrix equationsẋ=Ax+Bx(t-τ) 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 equationsẋ=ax(t-τ1)+bx(t-τ2) with delays τ1,τ2 [25, 26] and scalar difference equations xn=xn-1+axn-m+bxn-k 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.



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