The Stability Cone for a Difference Matrix Equation with Two Delays

We provide geometric algorithms for checking the stability of matrix difference equations 𝑥𝑛=𝐴𝑥𝑛−𝑚


Introduction
The problem of the stability of the equation 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 Ddecomposition 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 abovementioned 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 Section 2, we introduce the curve of Ddecomposition 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.

D-Decomposition Curve for Given k, m, a, ρ
Consider the scalar variant of 1.1 .The characteristic polynomial for 1.1 is If k dk 1 , m dm 1 , d > 1, then the trajectory of 1.1 splits into d independent trajectories, and degree of polynomial 2.1 gets smaller after the substitution λ d μ: Therefore, often but not always we will assume that the delays m, k are relatively prime.

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 ∈ R, a ≥ 0, then 2.1 will have a root λ ρ exp 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 b 1 , b 2 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. From now and further we will assume that −π < argz π for any complex z, while Arg z will be assumed as multivalued function, and the equality Arg z v will mean that one of the values of Arg z equals to v.
The following lemma asserts that some part of the complex plane is free from points of the hodograph b ω .

2.8
Since k, m are coprime and s < m, let us find natural numbers j, q such that ks mj q, m > q 1.Then

Basic Ovals
For hodograph 2.3 the equality b ω 0 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 −ω 1 , ω 1 under the map b ω defined by 2.3 .Here ω 1 ∈ 0, π/k is the least positive root of the equation arg b 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.Let k, m be coprime, k > m 1, j ∈ Z, 0 j < m, and let 2.4 , 3.1 hold.The basic oval L j for 1.1 is a closed curve given by 2.3 , where the variable ω runs from −ω 1 2πjs/m to ω 1 2πjs/m , where ω 1 is the least positive root of the equation arg b ω π.

3.2
From Lemma 2.2 and formula 2.5 it follows that all m basic ovals can be obtained from L 0 by rotation by the angles 2πj/m, j 0, 1, . . ., m − 1 .
Considering Definition 3.1, we get the following.For existence of the basic oval it is necessary that |a| < ρ m k/ k − m .If m > 1 and |a| > ρ m , 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, ρ m k/ k − m the basic oval L 0 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 L j decreases as |a| increases from 0, and it shrinks to the point b − exp 2πj/m ρ k / k − 1 as |a| reaches ρ m k/ k − m Figure 1 for m > 1 and Figure 2 for m 1 .Lemma 3.2.Let k, m be coprime, k > m 1, j ∈ Z, 0 j < m, a ∈ R, and 0 a < ρ m k/ k − m .If the complex number b lies outside the basic oval L j , 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 L j .Having changed ρ to R > ρ in Definition 3.1 let us consider the system of ovals L j R .If R → ∞, then the ovals L j R include a circle of an arbitrarily large radius.Therefore, there exists R 0 such that the point b is inside the oval L j R 0 .The ovals L j and L j R 0 are homotopic, therefore, there exists R 1 ∈ ρ, R 0 such that b lies on the curve L j R 1 , which means the existence of a root λ of characteristic polynomial 2.1 such that |λ| R 1 > ρ.Lemma 3.2 is proved.Definition 4.1.Equation 1.1 is said to be ρ-stable if for any of its solutions x n the sequence |x n |/ρ n is bounded, and asymptotically ρ-stable if for any of its solutions x n one has

Localization of Roots of Characteristic Polynomial
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 ρstable.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 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 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.
Taking into account the inequality a < R m k/ k − m , let us consider m basic ovals L j R , j 0, 1, . . ., m − 1 having ρ replaced by R in Definition 4.1.Since R m < a, the system of ovals L j R has no intersections.Hence, for any complex number b there exists j ∈ Z, 0 j < m such that b lies outside the oval The equation f 2 λ 0 transforms into

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:

Stability Cones for Matrix Equation 1.1 with Simultaneously Triangularizable Matrices
Let us consider a matrix equation 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.with complex entries a jj , b jj , 1 j l.Let us construct the points M j u 1j , u 2j , u 3j 1 j l in R 3 in the following way:

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 M j 1 j l lie inside the ρstability cone for given k, m.All the points M j with u 3j 0, u 2 1j u 2 2j < ρ k 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.Then 6.1 is ρ-asymptotically stable if and only if all the points M j 1 j l lie inside the ρstability cone for the given k, m, ρ.
If some point M j lies outside the ρ-stability cone, then 6.1 is ρ-unstable.
Proof.Let us make the change y n Sx n .Then 6.1 transforms to the following one: The characteristic polynomial for 6.6 has the form It coincides with the characteristic polynomial of diagonal system 6.3 .Therefore, from statement 3 of Theorem 5.1 for m > 1 and from statement 2 of Theorem 5.2 for m 1 we obtain asymptotic ρ-stability if all the points M j 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 M j lies outside the cone.Theorem 6.3 is proved.

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 y j n 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 T , and let us linearize system 7.1 in new variables about zero.We get 6.1 with the circulant 17 matrices Here α df y * /dy, β dg y * /dy, γ dh y * /dy.Let us introduce a matrix P a lines permutation operator : Then B γP , A αI βP l−1 , therefore, diagonalization P generates simultaneous diagonalization of A, B. The eigenvalues of P are 1, ε, ε 2 , . . ., ε l−1 , where ε exp i2π/l .Therefore, Granting 7.4 , by Theorem 6.3, we can build points M j u 1j , u 2j , u 3j in R 3 for system 6.1 , 7.2 :

7.5
We get the following consequence of Theorem 6.3.
Corollary 7.1.If for every j 1 j l the point M j u 1j , u 2j , u 3j 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 M j 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 M j u 1j , u 2j , u 3j defined by 7.5 lie on the closed curve

7.6
If l → ∞, then the points M j 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 ρstable 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 l 0 such that system 6.1 , 7.2 is ρ-unstable for any l > l 0 .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 ρ 0 α such that system 6.1 , 7.2 is asymptotically ρ-stable if ρ < ρ 0 α and ρ-unstable if ρ > ρ 0 α .Corollaries 7.1 and 7.2 allow us to find ρ 0 α by means of construction of different ρ-stability cones.Table 1 shows how ρ 0 depends on α.The value L α ln ρ 0 α is a Lyapunov exponent 18 for system 6.1 , 7.   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 M 1 , M 2 , M 3 for l 3. The upper part of the stability cone corresponding to u 3 > 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 M 1 u 1 , u 2 , u 3 defined by 7.5 lies on the cone surface.If γ > 0.4, then the point M 1 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.

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 equations ẋ 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 equations ẋ ax t − τ 1 bx t − τ 2 with delays τ 1 , τ 2 25, 26 and scalar difference equations x n x n−1 ax n−m bx n−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.

Definition 2 . 1 .
D-decomposition curve for given k, m ∈ Z , a ∈ C, ρ ∈ R is a curve on the complex plane of the variable b defined by the equation b ω ρ k exp ikω − |a|ρ k−m exp i k − m ω , ω ∈ R.
then statement 2 of Theorem 4.3 is obvious.Let b / 0. If Re b 0, then b lies outside the oval L 0 .If Re b 0 and m is even, then b lies outside the oval L m/2 .If Re b 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 Let 0 a < ρ m .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 L j .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 L j 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 0 a < ρ m < ρ m k/k−m we have df dλ λ k−m−1 kλ m − a k − m / 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 ∈ R , ρ > 0.1 If a ρk/ k − 1 , then for all complex numbers b 1.1 is ρ-unstable.2If 0 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 .3If 0 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 Let a > 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 L 0 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 / −ρ k / k−1 .However, if b −ρ k / 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 Let 0 a < ρk/ k − 1 .Since D k, 1, a, ρ is the domain of inner points of the oval L 0 , 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 L 0 .To complete the proof of asymptotical ρ-stability of 1.1 at any point of L 0 it is sufficient to prove that there exists at least one point b 0 inside the oval L 0 such that the equation is asymptotically ρ-stable for b b 0 .CASE 1.Let 0 a < ρ.Then the point b 0 lies inside L 0 .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. Let ρ a < ρk/ k−1 .Let us consider the point b ρ k −aρ k−1 at the boundary of L 0 and consider characteristic polynomial 2.1 with given b:

4 . 5 Figure 3 :
Figure 3: Stability domains D 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.

x n y n exp i n m arg a . 5 . 1 Equation 1 . 1 Theorem 5 . 1 .
changes to y n αy n−m βy n−k , αμ k−m − β. 5.4 It is related to 2.1 by the change μ λ exp −i 1/m arg a , 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 y n−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.Let k, m be coprime, k > m > 1, a ∈ C, ρ > 0. 1 If |a| > ρ m , then for any complex b 1.

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 arg a 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 arg a 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 arg a D 4, 3, a, ρ 2 .The corresponding "curved triangle" is shown in Figure 4 a .Similarly the "digon" exp i 3/2 arg a D 3, 2, a, ρ 3 for k 9 is shown in Figure 4 a , and the "curved triangle" exp i 5/3 arg a D 5, 3, a, ρ 2 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 • 2 arg a D 2, 1, a, ρ 6 Figure 4 b .The corresponding "stability oval" is shown in Figure 4 a .

Definition 6 . 2 .
If k > m > 1, then the ρ-stability cone for given k, m, ρ is a set of points M u 1 , u 2 , u 3 ∈ R 3 such that 0 u 3 1 and the intersection of the set with any plane u 3 a 0 a 1 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 u 1 , u 2 , u 3 ∈ R 3 such that 0 u 3 k/ k − 1 and the intersection of the set with the plane u 3 a 0 a k/ k − 1 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 u 3 a, and the ones in Figure 3 b as sections of the stability cone for k 5, m 2, and so on.

Figure 5 :
Figure 5: The stability cone as an intersection of three surfaces formed by the basic ovals, m 3, k 4.

Theorem 6 . 3 .
Let k > m 1, let the numbers k, m be coprime, and ρ > 0. Let A, B, S ∈ R l×l , and S −1 AS A T , and S −1 BS B T , where A T and B T are lower triangle matrices with elements a j s , b j s 1 j, s l .Let one construct the points M j u 1j , u 2j , u 3j , 1 j l by the formulas (cf.6.4 ) u 1j b jj cos arg b jj − k m arg a jj , u 2j b jj sin arg b jj − k m arg a jj , u 3j a jj .
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 y 1 n y * , . . ., y l n y * form a solution of 7.1 .Let us introduce the variables x j n y j n − y * and the vector x n x 1 n , . . ., x l n

2 . 7 . 4 .
Example 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
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 0 a < ρ m , 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 > ρ m .Let us find R ∈ R such that 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, arg a 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 1 is ρ-unstable. 2 If |a| ρ m , then for any b / 0 1.1 is ρ-unstable; for b 0 it is ρ-stable (nonasymptotically). 3 If |a| < ρ m , then 1.1 is asymptotically ρ-stable if and only if the complex number β b exp −i k/m arg a lies inside the domain D k, m, a, ρ .4 If |a| < ρ m , then 1.1 is ρ-stable if and only if the complex number β b exp −i k/m arg a lies either inside D k, m, a, ρ or on its boundary.Theorem 5.2.Let k > 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 b exp −ik arg a 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 b exp −ik arg a lies either inside D k, 1, a, ρ or on its boundary.