Stability and Bifurcation of a Class of Discrete-Time Cohen-Grossberg Neural Networks with Delays

A class of discrete-time Cohen-Grossberg neural networkswith delays is investigated in this paper. By analyzing the corresponding characteristic equations, the asymptotical stability of the null solution and the existence of Neimark-Sacker bifurcations are discussed. By applying the normal form theory and the center manifold theorem, the direction of the Neimark-Sacker bifurcation and the stability of bifurcating periodic solutions are obtained. Numerical simulations are given to illustrate the obtained results.


Introduction
Hopfield 1 proposed Hopfield neural networks in 1982, and in 1983, Cohen and Grossberg 2 proposed Cohen-Grossberg neural networks, which include Hopfield neural networks.These network models have been successfully applied to signal processing, pattern recognition, optimization, and associative memories.The analysis of the dynamical behaviors is a necessary step for practical design of neural networks because its applications heavily depend on the dynamical behaviors, and considerable work has been done to develop the dynamics such as stability and periodicity 3-14 .In order to obtain a deep and clear understanding of the dynamics of complicated neural networks with time delays, researchers have focused on the studying of simple systems.This is indeed very useful since the complexity found may be carried over to large networks.
In applications of continuous-time neural networks with or without delays to some practical problems, such as computer simulation, experimental, or computational purposes, it is usual to formulate a discrete-time system which is a discrete version of the continuoustime system, while their discrete-time counterparts have only been in the spotlight since 2000.
Bifurcation analysis for some models has been undertaken 15-22 ; the stability of the equilibrium and Neimark-Sacker bifurcation for a Cohen-Grossberg system without delays are discussed in 15 .To the best of the author's knowledge, no similar results have been obtained for discrete-time Cohen-Grossberg system with delays.
The objective of this paper is to study the following two-neuron discrete-time Cohen-Grossberg neural networks with discrete delays: The rest of this paper is organized as follows: the asymptotical stability and bifurcation are analyzed for the system in Section 2. Based on the normal form method and the center manifold theorem, the formula for determining the direction of Neimark-Sacker bifurcation, stability of bifuricating periodic solutions of the model are derived in Section 3.An example is given in Section 4 to demonstrate the main results.Conclusions are finally drawn in Section 5.

Stability Analysis and Existence of Bifurcations
Throughout this paper, we assume that x 2 n ; we transform system 1.1 into the following system: where k k 1 k 2 .Furthermore, system 2.1 can be transformed into the following system of k 2 difference equations without delays: n , j 1, 2, . . ., k.

2.2
In the following discussion, for convenience, we denote The Jacobian matrix of system 2.2 at the fixed point 0, . . ., 0 is defined as follows: The associated characteristic equation of system 2.2 is Suppose that λ e iθ is a root of the characteristic equation and substitute e iθ into 2.5 ; then we have cos θ − α 1 i sin θ cos θ − α 2 i sin θ cos kθ i sin kθ b.

2.6
When θ ∈ 0, π , separating the real and imaginary parts of 2.6 , we have where in which cot −1 denotes the inverse of the cotangent function restricted to the interval 0, π .Since for θ ∈ 0, π and we know that h θ : 0, π → 0, k 2 π is an increasing bijective function.
We know from the second equation in 2.7 that h θ jπ.Denote θ j h −1 jπ , j 1, 2, . . ., k 1 ; then we have from the first equation in 2.7 that Obviously, if θ is a root of 2.6 , −θ is also the root of 2.6 .So, we only need to consider the characteristic roots e iθ of 2.5 in 0, π .Furthermore, from the value b j given above, we have On the other hand, we have from 2.5 that where

2.14
Hence we have sign Since the roots of the characteristic equations 2.5 are α 1 , α 2 , and 0 when b 0, they are obviously inside the unit circle, so the null solution of system 2.2 is asymptotically stable.As the parameter b varies, the number of roots of the characteristic equation out of the circle can change if a root appears on or crosses the unit circle.According to 2.12 and 2.15 , we obtain that the null solution of system 2.
, the characteristic equation of system 2.2 has a simple root λ 1 on the unit circle, and all the other roots are inside the unit circle.Therefore, a Fold bifurcation occurs at the origin in system 2.2 20, 21 .
When b b 1 , the characteristic equation of system 2.2 has a simple pair of roots e ±iθ 1 on the unit circle, and all the other roots are inside the unit circle.Since θ 1 h −1 π , we have that θ 1 satisfies the following equation: It follows from 2.16 that So, θ 1 satisfies 2.16 if and only if it satisfies 2.18 .Let 19 y 1 and y 2 are easy to be proved to be convex function; furthermore, y 1 0 y 2 0 0, y 1 0 > y 2 0 , and y 1 π/ k 2 0 < y 2 π/ k 2 ; then curves y 1 and y 2 have a unique intersection in positive quadrant for θ in 0, π/ k 2 , that is, 2.18 has a unique solution θ θ

2.20
As θ 1 ∈ 0, π/ k 2 , so e ±iθ 1 / 1, e ±iθ 1 is not a root of order 1, 2, 3, or 4 of the unity.We also know from 2.15 that d|λ|/db | b b 1 / 0. Hence, a Neimark-Sacker bifurcation occurs at the origin in system 2.2 at b b 1 23 .From the discussions above, we have the following results.  1 α 1 1 α 2 , all roots of the characteristic equation are outside the unit circle, and system 1.1 may be chaostic as shown in Figures 4 and 5 .Furthermore, under some conditions which have more restrictions than H 2 and b ∈ −b 1 , b 0 , the null solution of system 1.1 is globally asymptotically stable.

Theorem 2.3. Under assumption (H 1 ), the null solution of system (1) is globally asymptotically stable if the following conditions hold.
H 3 There exist constants a i and a i such that 0 < a i ≤ a i • ≤ a i for i 1, 2.
H 4 There exist constants b i and is an M-matrix, and there exists a vector p p i 1×2 > 0 such that pA > 0 24 , that is, Hence, we can choose a constant λ > 1 such that

2.25
Define a Lypunov function V t by

2.26
Then we obtain from 2.24 and 2.25 that where m 0 min{1/a 1 , 1/a 2 } and M 0 is a positive constant.Thus It is easy to know that if condition H 4 holds, then H 2 and |b| ≤ b 0 hold, consequently, the null solution of system 1.1 is asymptotically stable.Then the null solution of system 1.1 is globally asymptotically stable.
Remark 2.4.The conditions under which the null solution is globally asymptotically stable in Theorem 2.3 are less restrictive than ones in Theorem 2.3 in 14 for discrete-time Cohen-Grossberg system while they are more restrictive than ones for the corresponding continuous Cohen-Grossberg system because |d 1 ||d 2 |L 1 L 2 ≤ b 1 b 2 implies the globally asymptotical stability of zero solution for the corresponding continuous system according to Corollary 2 in 8 .

Direction and Stability of Neimark-Sacker Bifurcation for the Model
In Section 2, we obtain some conditions under which system 1.1 undergoes Neimark-Sacker bifurcation at the origin.In this section, we investigate the properties of the Neimark-Sacker bifurcation by applying the normal form theory and the center manifold theorem for discrete time system developed by Kuznetsov 23 , namely, to determine the direction of Neimark-Sacker bifurcation and the stability of periodic solutions bifurcating from the origin of system 1.1 .We still discuss system 2.2 .From the discussions in Section 2, we know that if b b 1 , system 2.2 undergoes a Neimark-Sacker bifurcation at the origin and the associated characteristic equation of system 2.2 has a pair simple imaginary roots e ±iθ 1 .
Denote λ 1 e iθ 1 .Let q ∈ C k 2 be an eigenvector of A corresponding to eigenvalue λ 1 .Then Aq λq.Again, let p ∈ C k 2 be an eigenvector of A T corresponding to its eigenvalue λ 1 .
By direct calculation, we obtain where p and q satisfy p, q 1. System 2.2 can be written as

3.5
When b b 1 , the restriction of system 2.2 to its two-dimensional center manifold at the critical parameter value can be transformed into the normal form written in complex coordinates 23, 25 : where p and q are defined by 3.1 and B B 1 , B 2 , 0, . . ., 0 T , C C 1 , C 2 , 0, . . ., 0 T with B 1 , B 2 , C 1 , and C 2 defined by 3.5 .Theorem 3.1 see 25 .The direction and stability of the Neimark-Sacker bifurcation are determined by the sign of Re d .If Re d < 0, then the bifurcation is supercritical, that is, the closed invariant curve bifurcating from the origin is asymptotically stable.If Re d > 0, then the bifurcation is subcritical, that is, the closed invariant curve bifurcating from the origin is unstable.
, and f 1 f 2 , system 1.1 reduces to the following discrete time Hopfield neural networks:
Proof.Since f 0 0, it is easy to know from 3.5 that B 1 0, B 2 0, and 3.9 Substituting 3.9 into d in 3.6 , we can obtain
Similar results for continuous system corresponding to 3.8 are obtained in 16 .

A Numerical Example
Consider the following discrete-time Cohen-Grossberg neural network with discrete delays:

4.1
We can obtain that θ 1 0.3467 and furthermore we obtain that b 1 −0.2896 in view of bisection method by using MATLAB.It is easy to know that b 0 0.256.
According to Theorem 2.1, the null solution of system 4.1 is asymptotically stable when b ∈ −0.2896, 0.256 , and when b b 1 , the Neimark-Sacker bifurcation occurs at the origin. Since So the Neimark-Sacker bifurcation is supercritical and the bifurcating periodic solution is stable.
Case 1.Let d 1 0.79 and b −0.2844 ∈ −0.2896, 0.256 ; then the null solution of system 4.1 is asymptotically stable.Figure 1 shows the dynamic behavior of system 4.1 with initial conditions 0.02,0.02and 0.6,0.6 , respectively.Case 2. Let d 1 0.806 and d 1 0.85, respectively; then b −0.2902 and b −0.306, respectively, and b < b 1 −0.2896, thus Neimark-Sacker bifurcation occurs at the origin and the bifurcating periodic solutions are stable.Figures 2 and 3 show the dynamic behaviors of system 4.1 with initial conditions 0.02,0.02and 0.6,0.6 , respectively.Case 3. Let d 1 3.9 and d 1 −3.4,respectively; then system 4.1 may be chaostic.Figures 4  and 5 show the dynamic behaviors of system 4.1 with initial condition 0.02,0.02 .where θ 1 is the unique solution in 0, π/ k 2 of the equation sin k 2 θ− α 1 α 2 sin k 1 θ α 1 α 2 sin kθ 0 in which α 1 1 − a 1 0 b 1 0 and α 2 1 − a 2 0 b 2 0 .The direction of Neimark-Sacker bifurcation and the stability of the bifurcating periodic solutions are investigated by applying the normal form theory and the center manifold theorem for discrete time system.In addition, system may be chaostic when |b| is large enough.The presented numerical simulations demonstrated the theoretical results.

Conclusions
2 is asymptotically stable if and only if b ∈ b 1 , b 0 .At the critical points b b 0 and b b 1 , bifurcation phenomena occur.When b b 0

Theorem 2 . 1 .
Under assumptions (H 1 )-(H 2 ), we have that 1 if b ∈ b 1 , b 0 , the null solution of system 1.1 is asymptotically stable, 2 if b b 0 , a Fold bifurcation occurs at the origin in system 1.1 , 3 if b b 1 , a Neimark-Sacker bifurcation occurs at the origin in system 1.1 , that is, a unique closed invariant curve bifurcates from the origin near b b 1 , where

Remark 2 . 2 .
Similar to the case when b b 1 , if b b j , j 2, 3, . . ., k 1, a Neimark-Sacker bifurcation also occurs at the origin in system 1.1 near b b j except θ j 2π/3 and π/2.If b b k 2 −1 k 1 α 1 1 α 2 , the characteristic equation has a simple root λ −1 on the unit circle, and a flip bifurcation occurs in system 1.1 .If |b| > |b k 2 |

Figure 1 :
Figure 1: a Phase plot in space x 1 , x 2 for system 4.1 with initial condition 0.02,0.02and d 1 0.806.b Phase plot in space x 1 , x 2 for system 4.1 with initial condition 0.6,0.6 and d 1 0.806.
1 where x i n denote the state variable of the ith neuron; a i • represent amplification functions which are positive for R; f i • denote the signal functions of the ith neuron; b i • are appropriately behaved functions; d i • are connection weights of the neural networks; discrete delays k i correspond to the finite speed of the axonal signal transmission, i 1, 2.