2. The Roots of the Polynomial P

doi:10.1155/2009/938492

Research Article

Stability Results for a Class of Difference Systems with Delay

Eva Kaslik

Department of Mathematics and Computer Science, West University of Timisoara, Bd. C. Coposu 4, 300223 Timisoara, Romania

Correspondence should be addressed to Eva Kaslik,[email protected] Received 2 November 2009; Accepted 13 December 2009

Recommended by Mariella Cecchi

Considering the linear delay difference system xn1 axn Bxn−k, wherea∈0,1,Bis a p×preal matrix, andkis a positive integer, the stability domain of the null solution is completely characterized in terms of the eigenvalues of the matrixB. It is also shown that the stability domain becomes smaller as the delay increases. These results may be successfully applied in the stability analysis of a large class of nonlinear difference systems, including discrete-time Hopfield neural networks.

Copyrightq2009 Eva Kaslik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper, we will characterize the stability region of the null solution for the following class of linear delay difference systems:

xn1 axn Bxn−k ∀n≥k, 1.1

wherea∈0,1,Bis ap×preal matrix, andkis a positive integer.

Similar linear difference systems have been recently investigated by Levitskaya1 focusing on the special casea1and by Kipnis and Komissarova2 studying the special casea −1. Two-dimensional systems of this form have been thoroughly investigated by Matsunaga, in the casea 13,4and in the general casea ∈R5. The common starting point of all these results is the well-known papers of Kuruklis 6 and Papanicolaou 7, which focus on the scalar difference equation

xn1−axn bxn−k 0, 1.2

wherea, b ∈Randk∈N^∗. A more recent discussion of the stability properties of this scalar difference equation, based on relatively simple arguments, is presented in 8. As for the nondelayed casek0, we refer to the recent paper in9and the references therein.

System 1.1 can be regarded as the linearization at the origin of the following nonlinear delay difference system:

xn1 Fxn,xn−k ∀n≥k, 1.3

where the functionF :Ω×Ω → Ω with 0∈Ω⊂R^psatisfiesF0,0 0,Dx1F0,0 a·Id where Id denotes thep-dimensional identity matrix, andD_x2F0,0 B.

In particular, discrete-time delayed Hopfield-type neural networks described by

xn1 axn Tgxn−k ∀n≥k 1.4

belong to the class of nonlinear difference systems1.3. In this context,a∈0,1is the self- regulating parameter of the neurons,T ∈ R^p×p is the interconnection matrix,g_i : R → R, i ∈ {1,2, . . . , p} are the neuron input-output activation functions satisfyinggi0 0, and g : R^p → R^pis defined bygx g1x1, g2x2, . . . , gpxp^T. In this framework, stability and bifurcation results have been obtained in10for the two-dimensional case, in11for the case of a single-directional ring of four neurons, and in12for a bidirectional ring ofp neurons. Moreover, coexistence of chaos and periodic orbits for a network of this type with two identical neurons and no self-connections has been observed in13.

The stability of the null solution of system1.1will be investigated by analyzing the distribution of the roots of the corresponding characteristic equation with respect to the unit circle. Based on2, Corollary 2.2,1.1is asymptotically stable if and only if all roots of the equation

det

B−z^kz−aI

0 1.5

lie inside the unit disk. This means thatzis a root of the characteristic equation of1.1if and only ifz^kz−ais an eigenvalue of the matrixB.

In the followings, we will denote byλ_i,i∈ {1,2, . . . , p}the eigenvalues ofB. Based on the previous remark, we obtain that the characteristic equation of1.1is

p i1

z^kz−a−λ_i

0. 1.6

The null solution of1.1is asymptotically stable if and only if all the roots of the characteristic equation 1.6 are inside the unit circle. Therefore, in order to characterize the asymptotic stability of the null solution of1.1, we first need to analyze the distribution of the roots of the polynomialP_λz z^kz−a−λ, whereλis a complex parameter, with respect to the unit circle. This requires a generalization of the results first obtained by Kuruklis6, for the case whenλis a real parameter.

2. The Roots of the Polynomial P

z z

z − a − λ, λ ∈ C

In the followings, we will consider the following two important functions:

c_a,kθ cosk1θ−acoskθ,

sa,kθ sink1θ−asinkθ, 2.1

and the curveΓin the complex plane given by the following parametric equation:

Γ:uθ ca,kθ isa,kθ, θ∈−π, π. 2.2 Lemma 2.1. Letk∈Nanda∈0,1. The functions_a,khas exactlyk2 roots in the interval0, π, more precisely, as follows:

iθ_a,k⁰ 0 is a root,

iiifk ≥1, then there is one rootθ^j_a,k in every interval2j−1π/2k1, jπ/k1⊂ j−1π/k, jπ/k,j ∈ {1,2, . . . , k},

iiiθ_a,k^k1πis a root.

Moreover,−1^jsa,kθ>0 for anyθ∈θ_a,k^j , θ^j1_a,k,j ∈ {0,1, . . . , k}.

Proof. Obviously, 0 andπare solutions of the equations_a,kθ 0 for anyk∈N.

Consideringk ≥ 1, on the intervalj−1π/k, jπ/k,j ∈ {1,2, . . . , k}, the equation sa,kθ 0 becomes

sink1θ

sinkθ a. 2.3

The functionh:j−1π/k, jπ/k → Rdefined byhθ sink1θ/sinkθis differentiable and

hθ k1cosk1θsinkθ−kcoskθsink1θ sin²kθ

k1sin2k1θ−sinθ−ksin2k1θsinθ 2sin²kθ

sin2k1θ−2k1sinθ

2sin²kθ .

2.4

Therefore, the sign ofhθdepends on the sign ofgθ sin2k1θ−2k1sinθ. We have gθ 2k1cos2k1θ−cosθ −22k1sink1θsinkθ. 2.5

One can easily verify that the only root ofgin the intervalj−1π/k, jπ/kisθjπ/k 1. Moreover,gθ<0 for anyθ < θandgθ>0 for anyθ > θ. Therefore, the functiong

is decreasing on the intervalj−1π/k, θand increasing onθ, jπ/k. Asgj−1π/k

−2ksinj−1π/k ≤ 0 andgjπ/k −2ksinjπ/k ≤ 0, it results thatgθ < 0 for any θ∈j−1π/k, jπ/k.

Hence, the functionhis strictly decreasing on the intervalj−1π/k, jπ/k. Hence, the equations_a,kθ 0 has a single rootθ^j_a,kin the intervalj−1π/k, jπ/k. Moreover, as h2j−1π/2k1 1,hjπ/k1 0 anda∈0,1, we obtain that this single rootθ_a,k^j belongs to the interval2j−1π/2k1, jπ/k1.

Moreover, forj∈ {0,1, . . . , k}we have

sa,k

jπ k1

−asin jkπ

k1

−asin

jπ− jπ k1

a−1^jsin jπ

k1

. 2.6

Sincesa,khas constant sign on the intervalθ^j_a,k, θ^j1_a,kandθ^j_a,k < jπ/k1< θ^j1_a,k, it follows that, for anyθ∈θ^j_a,k, θ_a,k^j1, we have

signsa,kθ sign s_a,k jπ

k1

sign a−1^jsin jπ

k1

−1^j. 2.7

Lemma 2.2. Letk∈Nanda∈0,1. The functionca,khas exactlyk2 roots in the interval0, π, namely

iϕ⁰_a,k0 is a root;

iiif k ≥ 1, then there is one root ϕ^j_a,k in every interval θ_a,k^j , jπ/k 1 ⊂ j − 1π/k, jπ/k,j ∈ {1,2, . . . , k};

iiiϕ^k1_a,k πis a root.

Moreover, the function−1^jca,kθis strictly decreasing on the intervalϕ^j_a,k, ϕ^j1_a,k,j∈ {0,1, ..., k}, and

ca,k

θ^j_a,k

−1^j

1a²−2acosθ_a,k^j ∀j∈ {0,1, . . . , k1}. 2.8

Proof. The first part of the proof is similar to the proof ofLemma 2.1. Sincec_a,k θ −k 1sink1θaksinkθ, the equationc_a,kθ 0 becomes

hθ ak

k1, 2.9

wherehis the function defined in the proof ofLemma 2.1. We have shown thathis strictly decreasing on the interval j−1π/k, jπ/k; hence, the equation c_a,kθ 0 has a single root on the intervalj −1π/k, jπ/k. Moreover, as hθ_a,k^j a,hjπ/k 1 0 and ak/k1∈0, a, we obtain that this single rootϕ^j_a,kbelongs to the intervalθ_a,k^j , jπ/k1.

Moreover, forj∈ {0,1, . . . , k}we have

c_a,k jπ

k1

aksin jkπ

k1

aksin

jπ− jπ k1

ak−1^j1sin jπ

k1

. 2.10

Sincec_a,k has constant sign on the intervalϕ^j_a,k, ϕ^j1_a,kandϕ^j_a,k < jπ/k1< ϕ^j1_a,k, it follows that, for anyθ∈ϕ^j_a,k, ϕ^j1_a,k, we have

sign c_a,kθ

sign c_a,k jπ

k1

sign ak−1^j1sin jπ

k1

−1^j1, 2.11

and hence, the function−1^jca,kθis strictly decreasing on the intervalϕ^j_a,k, ϕ^j1_a,k.

Fromc_a,kθ²s_a,kθ² 1a²−2acosθ, we easily obtain thatc_a,kθ_a,k^{j 2} 1a²− 2acosθ_a,k^j . We also observe thatc_a,kθ²s_a,kθ²is strictly increasing on0, π, and hence,

c_a,kθc_a,kθ s_a,kθs_a,kθ>0 ∀θ∈0, π. 2.12

It follows that c_a,kθ_a,k^j c_a,k θ^j_a,k > 0. Since θ_a,k^j ∈ ϕ^j−1_a,k, ϕ^j_a,k for any j ∈ {1,2, . . . , k 1} and −1^j−1c_a,kθ is strictly decreasing on the interval ϕ^j−1_a,k, ϕ^j_a,k, we obtain that

−1^j−1c_a,kθ_a,k^j < 0, and hence, −1^jca,kθ^j_a,k > 0. Taking into account that ca,kθ_a,k^{j 2} 1a²−2acosθ_a,k^j , it follows thatca,kθ^j_a,k −1^j

1a²−2acosθ^j_a,k. Remark 2.3. Properties of the curveΓdefined by2.2:

awe can easily see that

|uθ|²ca,kθ²sa,kθ²1a²−2acosθ 2.13

and hence,|uθ|is strictly decreasing on the interval−π,0and increasing on the interval0, π. The curve piecesΓ|_−π,0andΓ|_0,πare, therefore, simple curves;

bmoreover

d dθ

tan

arguθ

d dθ

s_a,kθ c_a,kθ

k

1a²−2acosθ

1−acosθ

c_a,kθ² >0. 2.14

Therefore, asθincreases from−πtoπ, the corresponding pointuθfrom the curve Γmoves anticlockwise around the origin;

cthe curve piecesΓ|_−π,0andΓ|_0,πare symmetrical with respect to the real axis, that is,uθ∈Γ|_0,πif and only ifuθ u−θ∈Γ|_−π,0;

dthe curveΓintersects the real axis at the points

u θ^j_a,k

c_a,k θ^j_a,k

−1^j

1a²−2acosθ_a,k^j , j∈ {0,1, . . . , k1}, 2.15

which are at the same time the intersection points of the curve piecesΓ|_−π,0and Γ|_0,π.

In what follows, we will consider the following curve pieces:

Γ^j_a,k Γ|_−θ^j

a,k,−θ^j−1_a,k∪θ_a,k^j−1,θ^j_a,k, j ∈ {1,2, . . . , k1}. 2.16 Based on the previous remarks, one can easily see that these are closed curves.

For everyj ∈ {1,2, . . . , k1}, letΔ^j_a,k denote the domainopen and connected set, containing the originof the complex plane inclosed by the curveΓ^j_a,k.

Remark 2.4. For the curvesΓ^j_a,kand the inclosed domainsΔ^j_a,k, the following properties hold:

a Γ^j_a,k∩Γ^j1_a,k {ca,kθ_a,k^j }for anyj∈ {1,2, . . . , k},

b∂Δ^j_a,k Γ^j_a,kfor anyj ∈ {1,2, . . . , k1}here,∂Sdenotes the boundary of the setS;

c Δ¹_a,k⊂Δ²_a,k ⊂ · · · ⊂Δ^k1_a,k.

Using all these preliminary notations and results, the following proposition is obtained.

Proposition 2.5. Considering the polynomialPλz z^kz−a−λ,λ∈C, the following hold.

aIfλ∈Δ¹_a,k, then all roots of the polynomialP_λzare inside the unit circle.

bIfλ∈Δ^j_a,k\Δ^j−1_a,k(withj∈ {2,3, . . . , k1}), then the polynomialP_λzhas exactlyk−j2 roots inside the unit circle andj−1 roots outside the unit circle.

cIfλ∈C\Δ^k1_a,k, then all roots of the polynomialP_λzare outside the unit circle.

dIfλ∈Γ^j_a,k\ {ca,kθ_a,k^j−1, ca,kθ^j_a,k}(withj ∈ {1,2, . . . , k1}), then the polynomialP_λz has exactly one simple root on the unit circle,k−j1 roots inside the unit circle andj−1 roots outside the unit circle.

eIfλca,kθ_a,k^j ,j ∈ {1,2, . . . , k}, then the polynomialPλzhas exactly two simple roots on the unit circle,k−jroots inside the unit circle, andj−1 roots outside the unit circle.

fIfλca,kθ_a,k⁰ 1−a, then the polynomialPλzhas the simple rootz1 on the unit circle andkroots inside the unit circle.

gIfλca,kθ_a,k^k1 −1^k11a, then the polynomialPλzhas the simple rootz−1 on the unit circle andkroots outside the unit circle.

Here,Sdenotes the closure of the setS.

Proof. The polynomialPλzhas a rootze^iθ,θ∈−π, πon the unit circle if and only if

λe^ikθ

e^iθ−a

c_a,kθ is_a,kθ, 2.17

that is, if and only ifλ∈Γ.

Furthermore, due to the properties of the curve Γ stated in Remark 2.3, we easily obtain that the polynomial P_λz has a unique root on the unit circle if and only if λ ∈ Γ\ {ca,kθ¹_a,k, ca,kθ²_a,k, . . . , ca,kθ_a,k^k }. We also obtain that ifλ ca,kθ^j_a,k,j ∈ {1,2, . . . , k}, then the polynomialP_λzhas exactly two roots on the unit circle, namely, z e^iθja,k and ze^−iθja,k.

Moreover, if the polynomialP_λzhas a root on the unit circle, then this root is simple.

Indeed, assuming that there existsθ∈−π, πsuch thatze^iθis a root ofP_λzandP_λz k1z^k−akz^k−10, we obtain thats_a,kθ c_a,kθ 0. But we can easily see that

s_a,kθ²c_a,kθ² k1²a²k²−2akk1cosθ≥k1²a²k²−2akk1 k1−ak²>0,

2.18

and hence, a contradiction is obtained.

To provea, we will use the argument principle for the investigation of the roots of the polynomialP_λz z^kz−a−λ. In other words, we will study the increase of the argument ofP_λzalong the unit circle. Consider the function

Gθ, λ Pλ

e^iθ

e^ikθ

e^iθ−a

−λuθ−λ, 2.19

whereuθ c_a,kθ is_a,kθ. We will estimate the increase of the argument ofGθ, λasθ increases from−πtoπ.

FromRemark 2.3, we know that|uθ|is strictly decreasing on the interval−π,0and increasing on the interval0, π, and asθ increases from−π toπ, the corresponding point uθmoves anticlockwise around the origin. Moreover,Remark 2.3provides that the locus of uθintersects the real axis 2k1times asθincreases on the interval−π, π. Hence, the increase of the argument ofuθasθincreases from−πtoπis 2k1πseeFigure 1.

The locus ofGθ, λis obtained by the translation of the locus ofuθby the vector

−Reλ,−Imλ. Ifλlies inside the domainΔ¹_a,k, then the increase of the argument ofGθ, λ is the same as the increase of the argument of uθ, that is, it is equal to 2k1π. The argument principle provides that all the roots of the polynomial Pλz are inside the unit circle, andais proved.

Letj ∈ {1,2, . . . , k1}. The next step of the proof is to show that when the complex parameterλ λ1iλ2 leaves the domainΔ^j_a,k by crossing its boundaryΓ^j_a,k at a valueλ c_a,kθisa,kθ, withθ∈−θ^j_a,k,−θ_a,k^j−1∪θ^j−1_a,k, θ^j_a,k, the rootzzλof the polynomialP_λz, which is equal toe^iθwhenλλ, crosses the unit circle.

0.5 1 1.5 0.5

1 1.5

−0.5

−1

−1.5

−0.5

−1

−1.5 Im

Re

Figure 1: The curveΓdefined by2.2fork 3 anda 2/3. The blue part represents the curve piece Γ|_−π,0and the red part represents the curve pieceΓ|_0,π.

In the following computations, we rely on the fact that the roots of the polynomial P_λzwhich lie on the unit circle are simple.

Asz^k1−az^kλ1iλ2andz^k1−az^kλ1−iλ2, differentiating with respect toλ1and then with respect toλ2, we obtain

∂z

∂λ₁ 1

P_λz; ∂z

∂λ₁ 1

P_λz; ∂z

∂λ₂ i

P_λz; ∂z

∂λ₂ −i

P_λz. 2.20

Now we can evaluate that

∂|z|²

∂λ₁ z∂z

∂λ₁ z∂z

∂λ₁ z

P_λz z

P_λz 2 Re

zP_λz P_λz² ,

∂|z|²

∂λ₂ z∂z

∂λ₂ z∂z

∂λ₂ −iz P_λz iz

P_λz 2Im

zP_λz P_λz² .

2.21

Given the positive parametrization of the curveΓ, we obtain thatTθ c_a,k θ is_a,kθ is tangent toΓ^j_a,k at the pointλ ca,kθ isa,kθin the counterclockwise direction. Hence, Nθ s_a,kθ−ic_a,kθis an outward-pointing normal vector toΓ^j_a,kat the pointλca,kθ is_a,kθ.

We compute the directional derivative of|z|²atλ ca,kθ isa,kθin the direction ww₁iw₂|w|1as

∇w|z|²λ w1∂|z|²

∂λ₁ λ w2∂|z|²

∂λ₂ λ w1

2 Re e^iθP_λ

e^iθ P_λ

e^iθ2 w2

2Im e^iθP_λ

e^iθ P_λ

e^iθ2 2

w1s_a,kθ−w2c_a,kθ P_λ

e^iθ2

2 Rew·Nθ

P_λ e^iθ2

2|Nθ|cosw, Nθ

P_λ

e^iθ2 .

2.22

Whenλcrosses the curveΓ^j_a,kthrough the pointλ, from the inside of the domainΔ^j_a,kto the outside, in the directionw, we have cosw, Nθ>0, and hence,

∇w|z|²λ>0, 2.23

that is, |zλ| increases and the root zλ crosses the unit circle. This, together with the continuous dependence of the roots of the polynomialP_λzon the parameterλ, guarantees the validity of the statementsb–gand completes the proof.

3. Stability Results

3.1. Characterization of the Stability Domain

Based on the results presented in the previous section and the characteristic equation1.6, the following main result is obtained.

Proposition 3.1. The null solution of system 1.1 is asymptotically stable if and only if all eigenvalues of matrixBbelong to the domainΔ¹_a,k of the complex plane inclosed by the closed curve Γ¹_a,kgiven by the parametric equation

Γ¹_a,k : uθ c_a,kθ is_a,kθ, θ∈

−θ¹_a,k, θ_a,k¹

. 3.1

We know fromLemma 2.2that the functionca,k is strictly decreasing on the interval 0, θ¹_a,k⊂0, ϕ¹_a,k, and hence, invertible, allowing us to state the following remark.

Remark 3.2. The domainΔ¹_a,k can be expressed as

Δ¹_a,k

λ∈C : c_a,k θ¹_a,k

<Reλ<1−a ,|Imλ|< h_a,kReλ

, 3.2

whereh_a,k s_a,k ◦c_a,k⁻¹ andc_a,k⁻¹ denotes the inverse of the restriction ofc_a,k to the interval 0, θ¹_a,k.

3.2. Dependence of the Stability Domain on the Delay

Proposition 3.3. As the delaykincreases, the stability domain becomes smaller, that is, the domains Δ¹_a,ksatisfy

Δ¹_a,k1⊂Δ¹_a,k for anyk∈N. 3.3

IfD0,1−adenotes the open disk of the complex plane, centered at the origin, of radius 1−a, one has

∞ k0

Δ¹_a,k D0,1−a. 3.4

Proof. Letk∈N. Based onRemark 3.2, the proof of the fact thatΔ¹_a,k1 ⊂Δ¹_a,kwill consist of the following steps.

Step 1. We will prove thatca,kθ¹_a,k < c_a,k1θ¹_a,k1. Sinceca,kθ¹_a,k −

1a²−2acosθ¹_a,k, this reduces to show thatθ¹_a,k1< θ¹_a,k.

Indeed, assuming the contrary that is, θ¹_a,k < θ_a,k1¹ and taking into account that s_a,k1θ > 0 for anyθ ∈ 0, θ¹_a,k1, it follows thats_a,k1θ¹_a,k > 0. On the other hand, we can easily see that

s_a,k1θ sink2θ−asink1θs_a,kθcosθc_a,kθsinθ 3.5

and hence,

s_a,k1 θ¹_a,k

c_a,k θ_a,k¹

sinθ¹_a,k<0. 3.6

So the assumption is contradicted. Therefore, it follows thatca,kθ¹_a,k< c_a,k1θ_a,k1¹ .

Step 2. We will prove thath_a,k1t< ha,ktfor anyt∈ca,k1θ_a,k1¹ ,1−a. Since

c_a,kθ²s_a,kθ²1a²−2acosθ, 3.7

makingθc⁻¹_a,kt, it follows that

ha,kt²1a²−t²−2acosc⁻¹_a,kt. 3.8

Therefore, in order to prove thath_a,k1t < ha,kt, it is sufficient to prove that c⁻¹_a,k1t <

c⁻¹_a,ktfor anyt∈c_a,k1θ_a,k1¹ ,1−a. Sincec_a,kis decreasing on0, θ¹_a,k, this is equivalent to show thatc_a,k1θ< ca,kθfor anyθ∈0, θ_a,k1¹ . Indeed, for anyθ∈0, θ_a,k1¹ we have

c_a,k1θ cosk2θ−acosk1θca,kθcosθ−sa,kθsinθ < ca,kθ. 3.9

Finally, we will prove that_∞

k0Δ¹_a,kD0,1−a.

We remark that D0,1−a ⊂ Δ¹_a,k, for any k ∈ N. Indeed, it is easy to see that if λ ∈ ∂Δ¹_a,k Γ¹_a,k, there existsθ ∈ −θ¹_a,k, θ_a,k¹ such that λ ca,kθ isa,kθ, and hence,

|λ|√

1a²−2acosθ≥1−a. Thereforeλ /∈D0,1−aand it follows thatD0,1−a⊂Δ¹_a,k. We obtain thatD0,1−a⊂_∞

k0Δ¹_a,k. On the other hand, ifλ∈_∞

k0Δ¹_a,k, then it follows fromRemark 3.2that

c_a,k θ¹_a,k

<Reλ<1−a, |Imλ|< h_a,kReλ ∀k∈N. 3.10

From the first inequality, since c⁻¹_a,k is decreasing see Lemma 2.2, it follows that 0 <

c⁻¹_a,kReλ< θ_a,k¹ . From the second inequality, we obtain

|λ|²Reλ²Imλ²< Reλ²h_a,kReλ²1a²−2acos

c⁻¹_a,kReλ

<1a²−2acosθ¹_a,k,

3.11

and hence

|λ|<

1a²−2acosθ¹_a,k ∀k∈N. 3.12

FromLemma 2.1we know that 0 < θ¹_a,k < π/k1, and hence, lim_{k→ ∞}θ¹_a,k 0. Passing to the limit whenk → ∞in the previous inequality, we obtain that|λ|< 1−a, and therefore, λ∈D0,1−a. It results that_∞

k0Δ¹_a,k⊂D0,1−aand the proof is complete.

The results presented inProposition 3.3are exemplified inFigure 2.

0.2 0.2

0.4

−0.2

−0.4

−0.6

−0.8

−1

−0.2

−0.4 Im

Re

Figure 2: The stability domains fora2/3 andk∈ {1,2, . . . ,8}. We haveΔ¹_a,8⊂Δ¹_a,7⊂ · · · ⊂Δ¹_a,2⊂Δ¹_a,1.

3.3. Some Particular Cases

Corollary 3.4. If all the eigenvalues of the matrix B are real, then the null solution of system 1.1 is asymptotically stable if and only if all eigenvalues of matrix B belong to the interval

−

1a²−2acosθ¹_a,k,1−a.

For example, the previous corollary covers the case whenB is a symmetric matrix.

In particular, for the 1-dimensional case p 1 we obtain the result of Kuruklis6and Papanicolaou7, whena ∈ 0,1. For the 2-dimensional case p 2, if the matrixBhas two real eigenvalues, then we obtain the result of Matsunaga5, whena ∈ 0,1. On the other hand, if the matrixB∈R^2×2has two complex eigenvalues, then we obtain the following simple formulation.

Corollary 3.5. In the case of a 2-dimensional system of the form1.1, where the matrixBhas two complex conjugated eigenvaluesλ_1,2 β₁±iβ₂, the null solution is asymptotically stable if and only if

−

1a²−2acosθ¹_a,k< β₁ <1−a, β₂< h_a,k β₁

, 3.13

whereha,ksa,k◦c⁻¹_a,kandc_a,k⁻¹ denotes the inverse of the restriction ofca,kto the interval0, θ_a,k¹ .

4. Conclusions and Future Directions

In this paper, we have characterized the stability domain of the null solution of the linear delay difference system 1.1, in terms of the eigenvalues of the matrix B. We have also studied the dependence of the stability domain on the delay, showing that the stability domain becomes smaller as the delay increases. These results have potential applications

in the stability analysis of many nonlinear discrete-time dynamical systems arising from practical problems, such as discrete-time Hopfield neural networks. Investigating the bifurcations occurring in such nonlinear dynamical systems at the boundary of the stability domain may constitute a direction for future research.

References

1 I. S. Levitskaya, “A note on the stability oval forx_n1xnax_n−k,” Journal of Difference Equations and Applications, vol. 11, no. 8, pp. 701–705, 2005.

2 M. Kipnis and D. Komissarova, “Stability of a delay difference system,” Advances in Difference Equations, vol. 2006, Article ID 31409, 9 pages, 2006.

3 H. Matsunaga, “A note on asymptotic stability of delay difference systems,” Journal of Inequalities and Applications, vol. 2005, no. 2, pp. 119–125, 2005.

4 H. Matsunaga, “Exact stability criteria for delay differential and difference equations,” Applied Mathematics Letters, vol. 20, no. 2, pp. 183–188, 2007.

5 H. Matsunaga, “Stability regions for a class of delay difference systems,” in Differences and Differential Equations, vol. 42 of Fields Institute Communications, pp. 273–283, American Mathematical Society, Providence, Ri, USA, 2004.

6 S. A. Kuruklis, “The asymptotic stability ofx_n1−axnbx_n−k 0,” Journal of Mathematical Analysis and Applications, vol. 188, no. 3, pp. 719–731, 1994.

7 V. G. Papanicolaou, “On the asymptotic stability of a class of linear difference equations,” Mathematics Magazine, vol. 69, no. 1, pp. 34–43, 1996.

8 S. S. Cheng and Y.-Z. Lin, “Exact stability regions for linear difference equations with three parameters,” Applied Mathematics E-Notes, vol. 6, pp. 49–57, 2006.

9 B. Zheng and L. Wang, “Spectral radius and infinity norm of matrices,” Journal of Mathematical Analysis and Applications, vol. 346, no. 1, pp. 243–250, 2008.

10 E. Kaslik and St. Balint, “Bifurcation analysis for a two-dimensional delayed discrete-time Hopfield neural network,” Chaos, Solitons & Fractals, vol. 34, no. 4, pp. 1245–1253, 2007.

11 S. Guo, X. Tang, and L. Huang, “Bifurcation analysis in a discrete-time single-directional network with delays,” Neurocomputing, vol. 71, no. 7–9, pp. 1422–1435, 2008.

12 E. Kaslik, “Dynamics of a discrete-time bidirectional ring of neurons with delay,” in Proceedings of the International Joint Conference on Neural Networks, pp. 1539–1546, IEEE Computer Society Press, Atlanta, Ga, USA, June 2009.

13 Y. Huang and X. Zou, “Co-existence of chaos and stable periodic orbits in a simple discrete neural network,” Journal of Nonlinear Science, vol. 15, no. 5, pp. 291–303, 2005.