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

Abstract

A class of discrete-time Cohen-Grossberg neural networks with 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.

1. 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 continuous-time 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: 𝑥₁(𝑛+1)=𝑥₁(𝑛)−𝑎₁𝑥₁(𝑛)𝑏₁𝑥₁(𝑛)−𝑑₁𝑓₂𝑥₂𝑛−𝑘₂,𝑥₂(𝑛+1)=𝑥₂(𝑛)−𝑎₂𝑥₂(𝑛)𝑏₂𝑥₂(𝑛)−𝑑₂𝑓₁𝑥₁𝑛−𝑘₁, (1.1) 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.

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.

2. Stability Analysis and Existence of Bifurcations

Throughout this paper, we assume that (H₁)𝑏₁(0)=𝑏₂(0)=0, 𝑓₁(0)=𝑓₂(0)=0;(H₂)0<𝑎_𝑖(0)𝑏^′_𝑖(0)<1, 𝑖=1,2.

Let 𝑢₁(𝑛)=𝑥₁(𝑛−𝑘₁), 𝑢₂(𝑛)=𝑥₂(𝑛); we transform system (1.1) into the following system: 𝑢₁(𝑛+1)=𝑢₁(𝑛)−𝑎₁𝑢₁(𝑛)𝑏₁𝑢₁(𝑛)−𝑑₁𝑓₂𝑢₂(),𝑢𝑛−𝑘₂(𝑛+1)=𝑢₂(𝑛)−𝑎₂𝑢₂(𝑛)𝑏₂𝑢₂(𝑛)−𝑑₂𝑓₁𝑢₁(𝑛), (2.1) where 𝑘=𝑘₁+𝑘₂.

Furthermore, system (2.1) can be transformed into the following system of 𝑘+2 difference equations without delays: 𝑢₁⁽⁰⁾(𝑛+1)=𝑢₁⁽⁰⁾(𝑛)−𝑎₁𝑢₁⁽⁰⁾(𝑛)𝑏₁𝑢₁⁽⁰⁾(𝑛)−𝑑₁𝑓₂𝑢₂^(𝑘)(𝑛),𝑢₂⁽⁰⁾(𝑛+1)=𝑢₂⁽⁰⁾(𝑛)−𝑎₂𝑢₂⁽⁰⁾(𝑛)𝑏₂𝑢₂⁽⁰⁾(𝑛)−𝑑₂𝑓₁𝑢₁⁽⁰⁾(𝑛),𝑢₂^(𝑗)(𝑛+1)=𝑢₂^(𝑗−1)(𝑛),𝑗=1,2,…,𝑘. (2.2) In the following discussion, for convenience, we denote 𝑏=𝑎₁(0)𝑎₂(0)𝑑₁𝑑₂𝑓^′₁(0)𝑓^′₂(0),𝛼_𝑖=1−𝑎_𝑖(0)𝑏^′_𝑖(0),𝑎_𝑖=𝑎_𝑖(0),𝑎^′_𝑖=𝑎^′_𝑖(0),𝑎_𝑖^′′=𝑎_𝑖^′′(0),𝑏_𝑖=𝑏_𝑖(0),𝑏^′_𝑖=𝑏^′_𝑖(0),𝑏_𝑖^′′=𝑏_𝑖^′′(0),𝑏_𝑖^′′′=𝑏_𝑖^′′′(0),𝑓_𝑖=𝑓_𝑖(0),𝑓^′_𝑖=𝑓^′_𝑖(0),𝑓_𝑖^′′=𝑓_𝑖^′′(0),𝑓_𝑖^′′′=𝑓_𝑖^′′′(0) (2.3) for 𝑖=1,2.

The Jacobian matrix of system (2.2) at the fixed point (0,…,0) is defined as follows: 𝛼𝐴=¹00⋯0𝑎₁𝑑₁𝑓^′₂𝑎₂𝑑₂𝑓^′₁𝛼₂0⋯00010⋯00001⋯00⋮⋮⋮⋯⋮⋮000⋯10. (2.4) The associated characteristic equation of system (2.2) is 𝜆−𝛼₁𝜆−𝛼₂𝜆^𝑘=𝑏. (2.5) Suppose that 𝜆=𝑒^𝑖𝜃 is a root of the characteristic equation and substitute 𝑒^𝑖𝜃 into (2.5); then we have cos𝜃−𝛼₁+𝑖sin𝜃cos𝜃−𝛼₂+𝑖sin𝜃(cos𝑘𝜃+𝑖sin𝑘𝜃)=𝑏. (2.6)

When 𝜃∈(0,𝜋), separating the real and imaginary parts of (2.6), we have 𝑏=1+𝛼²₁−2𝛼₁cos𝜃1+𝛼²₂−2𝛼₂(𝜃),(𝜃)cos𝜃cosℎsinℎ=0, (2.7) where ℎ(𝜃)=cot⁻¹cos𝜃−𝛼₁sin𝜃+cot⁻¹cos𝜃−𝛼₂sin𝜃+𝑘𝜃 (2.8) in which cot⁻¹ denotes the inverse of the cotangent function restricted to the interval (0,𝜋). Since ℎ^′(𝜃)=11+cos𝜃−𝛼₁/sin𝜃²×1−𝛼₁cos𝜃sin²𝜃+11+cos𝜃−𝛼₂/sin𝜃²×1−𝛼₂cos𝜃sin²𝜃+𝑘>0 (2.9) for 𝜃∈(0,𝜋) and lim_{𝜃→𝜋−}ℎ(𝜃)=(𝑘+2)𝜋,lim_𝜃→0+ℎ(𝜃)=0, (2.10) 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 𝑏=𝑏_𝑗=(−1)^𝑗1+𝛼²₁−2𝛼₁cos𝜃_𝑗1+𝛼²₂−2𝛼₂cos𝜃_𝑗,𝑗=1,2,…,𝑘+1. (2.11)

When 𝜃=0, 𝑏=𝑏₀=(1−𝛼₁)(1−𝛼₂)>0.

When 𝜃=𝜋,𝑏=𝑏_𝑘+2=(−1)^𝑘+2(1+𝛼₁)(1+𝛼₂).

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 ⋯<𝑏₅<𝑏₃<𝑏₁<0<𝑏₀<𝑏₂<𝑏₄⋯. (2.12)

On the other hand, we have from (2.5) that 𝑑|𝜆|²=𝑑𝑏𝑑𝜆𝜆𝑑𝑑𝑏=𝜆𝜆+𝑑𝑏𝜆𝑑𝜆=((𝜆𝑑𝑏2Re𝜆𝑃))|𝑃(𝜆)|², (2.13) where 𝑃(𝜆)=𝜆^𝑘−1[𝜆(𝜆−𝛼₁)+𝜆(𝜆−𝛼₂)+𝑘(𝜆−𝛼₁)(𝜆−𝛼₂)].

It follows that 𝑑|𝜆|²𝑑𝑏_{𝑏=𝑏𝑗}=𝑏_𝑗1𝑃𝑒^{𝑖𝜃𝑗2}𝑘+1−𝛼₁cos𝜃_𝑗1+𝛼²₁−2𝛼₁cos𝜃_𝑗+1−𝛼₂cos𝜃_𝑗1+𝛼²₂−2𝛼₂cos𝜃_𝑗. (2.14) Hence we have 𝑑|𝜆|sign²𝑑𝑏_{𝑏=𝑏𝑗}=sign𝑏_𝑗=(−1)^𝑗. (2.15)

Since the roots of the characteristic equations (2.5) are 𝛼₁, 𝛼₂, 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 (2.2) is asymptotically stable if and only if 𝑏∈(𝑏₁,𝑏₀).

At the critical points 𝑏=𝑏₀ and 𝑏=𝑏₁, bifurcation phenomena occur.

When 𝑏=𝑏₀=(1−𝛼₁)(1−𝛼₂), 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=𝑏₁, the characteristic equation of system (2.2) has a simple pair of roots 𝑒^±𝑖𝜃1 on the unit circle, and all the other roots are inside the unit circle. Since 𝜃₁=ℎ⁻¹(𝜋), we have that 𝜃₁ satisfies the following equation: 𝜋=cot⁻¹cos𝜃−𝛼₁sin𝜃+cot⁻¹cos𝜃−𝛼₂sin𝜃+𝑘𝜃. (2.16) It follows from (2.16) that −cot𝑘𝜃=cotcot⁻¹cos𝜃−𝛼₁sin𝜃+cot⁻¹cos𝜃−𝛼₂,sin𝜃 (2.17) which leads to sin(𝑘+2)𝜃−𝛼₁+𝛼₂sin(𝑘+1)𝜃+𝛼₁𝛼₂sin𝑘𝜃=0. (2.18) So, 𝜃₁ satisfies (2.16) if and only if it satisfies (2.18).

Let 𝑦₁=sin(𝑘+2)𝜃,𝑦₂=𝛼₁+𝛼₂sin(𝑘+1)𝜃−𝛼₁𝛼₂sin𝑘𝜃, (2.19)𝑦₁ and 𝑦₂ are easy to be proved to be convex function; furthermore, 𝑦₁(0)=𝑦₂(0)=0, 𝑦^′₁(0⁺)>𝑦^′₂(0⁺), and 𝑦₁(𝜋/(𝑘+2))=0<𝑦₂(𝜋/(𝑘+2)); then curves 𝑦₁ and 𝑦₂ have a unique intersection in positive quadrant for 𝜃 in (0,𝜋/(𝑘+2)), that is, (2.18) has a unique solution 𝜃=𝜃₁ in (0,𝜋/(𝑘+2)). Then 𝑏₁=−1+𝛼²₁−2𝛼₁cos𝜃₁1+𝛼²₂−2𝛼₂cos𝜃₁. (2.20) As 𝜃₁∈(0,𝜋/(𝑘+2)), so 𝑒^±𝑖𝜃1≠1, 𝑒^±𝑖𝜃1 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=𝑏1≠0. Hence, a Neimark-Sacker bifurcation occurs at the origin in system (2.2) at 𝑏=𝑏₁ [23].

From the discussions above, we have the following results.

Theorem 2.1. Under assumptions (H₁)-(H₂), we have that (1)if 𝑏∈(𝑏₁,𝑏₀), the null solution of system (1.1) is asymptotically stable,(2)if 𝑏=𝑏₀, a Fold bifurcation occurs at the origin in system (1.1),(3)if 𝑏=𝑏₁, 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=𝑏₁,where 𝑏₀=1−𝛼₁1−𝛼₂,𝑏₁=−1+𝛼²₁−2𝛼₁cos𝜃₁1+𝛼²₂−2𝛼₂cos𝜃₁ (2.21) in which 𝜃₁ is the unique solution in (0,𝜋/(𝑘+2)) of the equation sin(𝑘+2)𝜃−(𝛼₁+𝛼₂)sin(𝑘+1)𝜃+𝛼₁𝛼₂sin𝑘𝜃=0.

Remark 2.2. Similar to the case when 𝑏=𝑏₁, 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+𝛼₁)(1+𝛼₂), 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|=(1+𝛼₁)(1+𝛼₂), 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 (H₂) and 𝑏∈(−𝑏₁,𝑏₀), the null solution of system (1.1) is globally asymptotically stable.

Theorem 2.3. Under assumption (H₁), the null solution of system (1) is globally asymptotically stable if the following conditions hold. (H₃) There exist constants 𝑎_𝑖 and 𝑎_𝑖 such that 0<𝑎_𝑖≤𝑎_𝑖(⋅)≤𝑎_𝑖 for 𝑖=1,2.(H₄) There exist constants 𝑏_𝑖 and 𝐿_𝑖 such that 𝑏^′_𝑖(⋅)≥𝑏_𝑖>0 and |𝑓^′_𝑖(⋅)|≤𝐿_𝑖 for 𝑖=1,2.(H₅)0<𝑎_𝑖𝑏_𝑖<1 for i=1,2, and 𝑎₁𝑎₂|𝑑₁||𝑑₂|𝐿₁𝐿₂≤𝑎₁𝑎₂𝑏₁𝑏₂.

Proof. Since 𝑎₁𝑎₂𝑑₁𝑑₂𝐿₁𝐿₂≤𝑎₁𝑎₂𝑏₁𝑏₂, the following matrix 𝑎𝐴=¹𝑏₁−𝑎₁𝑑₁𝐿₂−𝑎₂𝑑₂𝐿₁𝑎₂𝑏₂, (2.22) is an 𝑀-matrix, and there exists a vector 𝑝=(𝑝_𝑖)_1×2>0 such that 𝑝𝐴>0 ([24]), that is, 𝑝₁𝑎₁𝑏₁−𝑝₂𝑎₂𝑑₂𝐿₁>0,𝑝₂𝑎₂𝑏₂−𝑝₁𝑎₁𝑑₁𝐿₂>0. (2.23) Hence, we can choose a constant 𝜆>1 such that 𝑝₁1−𝜆+𝑎₁𝑏₁𝜆−𝑝₂𝑎₂𝑑₂𝐿₁𝜆>0,𝑝₂1−𝜆+𝑎₂𝑏₂𝜆−𝑝₁𝑎₁𝑑₁𝐿₂𝜆^𝑘+1>0. (2.24) Let 𝑈₁(𝑛)=𝜆^𝑛|𝑢₁(𝑛)|, 𝑈₂(𝑛)=𝜆^𝑛|𝑢₂(𝑛)|; then we have from (1.1) that 𝑈₁(𝑛)=𝜆^𝑛+1𝑢₁(𝑛+1)≤𝜆1−𝑎₁𝑏₁𝑈₁(𝑛)+𝑎₁𝑑₁𝐿₂𝜆^𝑘+1𝑈₂(,𝑈𝑛−𝑘)₂(𝑛)=𝜆^𝑛+1𝑢₂(𝑛+1)≤𝜆1−𝑎₂𝑏₂𝑈₂(𝑛)+𝑎₂𝑑₂𝐿₁𝜆𝑈₁(𝑛). (2.25) Define a Lypunov function 𝑉(𝑡) by 𝑉(𝑡)=²_𝑖=1𝑝_𝑖𝑈_𝑖(𝑛)+𝑝₁𝑎₁𝑑₁𝐿₂𝜆^{𝑘+1𝑛−1}_{𝑙=𝑛−𝑘}𝑈₂(𝑙). (2.26) Then we obtain from (2.24) and (2.25) that Δ𝑉(𝑛)=𝑉(𝑛+1)−𝑉(𝑛)≤−𝑝₁1−𝜆+𝑎₁𝑏₁𝜆−𝑝₂𝑎₂𝑑₂𝐿₁𝜆𝑈₁(𝑛)−𝑝₂1−𝜆+𝑎₂𝑏₂𝜆−𝑝₁𝑎₁𝑑₁𝐿₂𝜆^𝑘+1𝑈₂(𝑛)≤0, (2.27) which implies 𝑉(𝑡)≤𝑉(0).
Note that 𝑉(𝑡)≥𝜆^𝑛𝑚₀𝑢₁(𝑡)+𝑢₂(𝑡,𝑉(0)=_2𝑖=1)𝑝_𝑖𝑈_𝑖(0)+𝑝₁𝑎₁𝑑₁𝐿₂𝜆^𝑘+1−1_𝑙=−𝑘𝑈₂(𝑙)=^def𝑀₀, (2.28) where 𝑚₀=min{1/𝑎₁,1/𝑎₂} and 𝑀₀ is a positive constant.
Thus 𝑢₁(𝑡)+𝑢₂(𝑡)≤𝑀₀/𝑚₀1𝜆^𝑛, (2.29) where 𝜆>1. Therefore, lim_𝑛→+∞𝑢_𝑖(𝑡)→0, 𝑖=1,2, that is, the null solution of system (1.1) is globally attractive. Note that 𝑏₁≤−𝑏₀, which implies that 𝑏∈(−𝑏₁,𝑏₀) if |𝑏|≤𝑏₀. It is easy to know that if condition (H₄) holds, then (H₂) and |𝑏|≤𝑏₀ 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 |𝑑₁||𝑑₂|𝐿₁𝐿₂≤𝑏₁𝑏₂ implies the globally asymptotical stability of zero solution for the corresponding continuous system according to Corollary 2 in [8].

3. 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=𝑏₁, 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^±𝑖𝜃1.

Denote 𝜆₁=𝑒^𝑖𝜃1. Let 𝑞∈𝐶^𝑘+2 be an eigenvector of 𝐴 corresponding to eigenvalue 𝜆₁. Then 𝐴𝑞=𝜆𝑞. Again, let 𝑝∈𝐶^𝑘+2 be an eigenvector of 𝐴^𝑇 corresponding to its eigenvalue 𝜆₁.

By direct calculation, we obtain 𝑎𝑞=¹𝑑₁𝑓^′₂𝜆₁−𝛼₁,𝜆^𝑘₁,𝜆₁^𝑘−1,…,𝜆₁,1^𝑇,𝑝=𝐸₁𝜆₁𝑎₁𝑑₁𝑓^′₂,1𝜆₁^𝑘−1(𝜆₁−𝛼₂),1𝜆₁^𝑘−1,1𝜆₁^𝑘−21,…,𝜆₁,1^𝑇, (3.1) in which 𝐸₁=1𝑘+2+𝛼₁/𝜆₁−𝛼₁+𝛼₂/𝜆₁−𝛼₂, (3.2) where 𝑝 and 𝑞 satisfy ⟨𝑝,𝑞⟩=1.

System (2.2) can be written as 𝑢(𝑛+1)=𝐴𝑢(𝑛)+𝐹(𝑢(𝑛)), (3.3) where 𝑢(𝑛)=(𝑢₁⁽⁰⁾(𝑛), 𝑢₂⁽⁰⁾(𝑛),…,𝑢₂^(𝑘)(𝑛))^𝑇, 𝐹(𝑢(𝑛))=(𝐹₁(𝑢(𝑛)), 𝐹₂(𝑢(𝑛)),0,…,0)^𝑇 in which 𝐹_𝑖(𝑖=1,2) can be expanded in the form 𝐹₁(𝜉)=−𝑎₁𝑏₁^′′2+𝑎^′₁𝑏^′₁𝜉²₁+𝑎₁𝑑₁𝑓₂^′′2𝜉²_𝑘+2+𝑎^′₁𝑑₁𝑓^′₂𝜉₁𝜉_𝑘+2−𝑎₁𝑏₁^′′′6+𝑎^′₁𝑏₁^′′2+𝑎₁^′′2𝑏^′₁𝜉³₁+𝑎₁𝑑₁𝑓₂^′′′6𝜉³_𝑘+2+𝑎^′₁𝑑₁𝑓₂^′′2𝜉₁𝜉²_𝑘+2+𝑎₁^′′2𝑑₁𝑓^′₂𝜉²₁𝜉_𝑘+2𝐹+h.o.t.,₂(𝜉)=−𝑎₂𝑏₂^′′2+𝑎^′₂𝑏^′₂𝜉²₂+𝑎₂𝑑₂𝑓₁^′′2𝜉²₁+𝑎^′₂𝑑₂𝑓^′₁𝜉₁𝜉₂−𝑎₂𝑏₂^′′′6+𝑎^′₂𝑏₂^′′2+𝑎₂^′′2𝑏^′₂𝜉³₂+𝑎₂𝑑₂𝑓₁^′′′6𝜉³₁+𝑎^′₂𝑑₂𝑓₁^′′2𝜉₂𝜉²₁+𝑎₂^′′2𝑑₂𝑓^′₁𝜉₁𝜉²₂+h.o.t.. (3.4) and 𝐴 is defined by (2.4).

It follows that 𝐵₁𝑥,𝑦=^𝑛_𝑗,𝑘𝜕²𝐹₁(𝜉)𝜕𝜉_𝑗𝜕𝜉_𝑘𝜉=0𝑥_𝑗𝑦_𝑘=−𝑎₁𝑏₁^′′+2𝑎^′₁𝑏^′₁𝑥₁𝑦₁+𝑎₁𝑑₁𝑓₂^′′𝑥_𝑘+2𝑦_𝑘+2+𝑎^′₁𝑑₁𝑓^′₂𝑥₁𝑦_𝑘+2,𝐵₂𝑥,𝑦=^𝑛_𝑗,𝑘𝜕²𝐹₂(𝜉)𝜕𝜉_𝑗𝜕𝜉_𝑘𝜉=0𝑥_𝑗𝑦_𝑘=−𝑎₂𝑏₂^′′+2𝑎^′₂𝑏^′₂𝑥₂𝑦₂+𝑎₂𝑑₂𝑓₁^′′𝑥₁𝑦₁+𝑎₂^′′𝑑₂𝑓^′₁𝑥₁𝑦₂,𝐶₁𝑥,𝑦,𝑧=^𝑛_{𝑗,𝑘,𝑙}𝜕³𝐹₁(𝜉)𝜕𝜉_𝑗𝜕𝜉_𝑘𝜕𝜉_𝑙𝜉=0𝑥_𝑗𝑦_𝑘𝑧_𝑙=−𝑎₁𝑏₁^′′′+3𝑎^′₁𝑏₁^′′+3𝑎₁^′′𝑏^′₁𝑥₁𝑦₁𝑧₁+𝑎₁𝑑₁𝑓₂^′′′𝑥_𝑘+2𝑦_𝑘+2𝑧_𝑘+2+𝑎^′₁𝑑₁𝑓₂^′′𝑥₁𝑦_𝑘+2𝑧_𝑘+2+𝑎₁^′′𝑑₁𝑓^′₂𝑥₁𝑦₁𝑧_𝑘+2,𝐶₂𝑥,𝑦,𝑧=^𝑛_{𝑗,𝑘,𝑙}𝜕³𝐹₂(𝜉)𝜕𝜉_𝑗𝜕𝜉_𝑘𝜕𝜉_𝑙𝜉=0𝑥_𝑗𝑦_𝑘𝑧_𝑙=−𝑎₂𝑏₁^′′′+3𝑎^′₂𝑏₂^′′+3𝑎₂^′′𝑏^′₂𝑥₂𝑦₂𝑧₂+𝑎₂𝑑₂𝑓₁^′′′𝑥₁𝑦₁𝑧₁+𝑎^′₂𝑑₁𝑓₁^′′𝑥₂𝑦₁𝑧₁+𝑎₂^′′𝑑₁𝑓^′₁𝑥₁𝑦₂𝑧₂. (3.5) When 𝑏=𝑏₁, 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]: 𝑤⟶𝜆₁𝑤11+2𝑑|𝑤|²+𝑂|𝑤|⁴,𝑤∈𝐶 (3.6) in which 𝑑=𝜆₁𝑝,𝐶𝑞,𝑞,𝑞+2𝐵𝑞,1−𝐴⁻¹𝐵𝑞,𝑞+𝐵𝑞,𝜆²₁𝐼−𝐴⁻¹𝐵𝑞,𝑞, (3.7) where 𝑝 and 𝑞 are defined by (3.1) and 𝐵=(𝐵₁,𝐵₂,0,…,0)^𝑇, 𝐶=(𝐶₁,𝐶₂,0,…,0)^𝑇 with 𝐵₁, 𝐵₂, 𝐶₁, and 𝐶₂ 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.

If 𝑎₁(𝑥₁(𝑛))=1, 𝑎₂(𝑥₂(𝑛))=1, 𝑏₁(𝑥₁(𝑛))=𝑏̃₁𝑥₁(𝑛), 𝑏₂(𝑥₁(𝑛))=𝑏̃₂𝑥₁(𝑛), and 𝑓₁=𝑓₂, system (1.1) reduces to the following discrete time Hopfield neural networks: 𝑥₁(𝑛+1)=𝛼₁𝑥₁(𝑛)+𝑑₂𝑓𝑥₂𝑛−𝑘₂,𝑥₂(𝑛+1)=𝛼₂𝑥₂(𝑛)+𝑑₁𝑓𝑥₁𝑛−𝑘₁, (3.8) where 𝛼₁=1−𝑏̃₁ and 𝛼₂=1−𝑏̃₂.

Corollary 3.2. Suppose system (1.1) satisfies (H₁)–(H₃) and the following condition: (H₄)𝑓^′(0)≠0, 𝑓^′′(0)=0 and 𝑓^′′′(0)≠0,then system (3.8) undergoes a Neimark-Sacker bifurcation from the origin near 𝑏=𝑏₁. If 𝑓^′′′(0)/𝑓^′(0)<0(𝑓^′′′(0)/𝑓^′(0)>0), the Neimark-Sacker bifurcation is supercritical (subcritical) and the bifurcating periodic solutions are asymptotically stable (unstable).

Proof. Since 𝑓^′′(0)=0, it is easy to know from (3.5) that 𝐵₁=0,𝐵₂=0, and 𝐶₁𝑞,𝑞,𝑞=𝑎₁(0)𝑑₁𝑓^′′′(0),𝐶₂𝑞,𝑞,𝑞=𝑎₂(0)𝑑₂𝑓^′′′(0𝑎)¹(0)𝑑₁𝑓^′(0)𝜆₁−𝛼₁²𝑎₁(0)𝑑₁𝑓^′(0)𝜆₁−𝛼₁. (3.9) Substituting (3.9) into 𝑑 in (3.6), we can obtain Re𝑑=𝜆₁²𝑎1+¹(0)𝑑₁𝑓^′(0)𝜆₁−𝛼₁²Re𝐸₁𝑓^′′′(0)𝑓^′(0). (3.10) Note that signRe𝐸₁𝛼=signRe𝑘+2+¹𝜆₁−𝛼₁+𝛼₂𝜆₁−𝛼₂=sign𝑘+1−𝛼₁cos𝜃₁1+𝛼²₁−2𝛼₁cos𝜃₁+1−𝛼₂cos𝜃₁1+𝛼²₁−2𝛼₂cos𝜃₁=1>0. (3.11) Hence, sign{Re𝑑}=sign{𝑓^′′′(0)/𝑓^′(0)}. Thus the results hold from Theorem 3.1.

Similar results for continuous system corresponding to (3.8) are obtained in [16].

4. A Numerical Example

Consider the following discrete-time Cohen-Grossberg neural network with discrete delays: 𝑥₁(𝑛+1)=𝑥₁(𝑛)−0.22+cos𝑥₁(𝑛)0.6𝑥₁(𝑛)−𝑑₁tanh𝑥₂(),𝑥𝑛−3₂(𝑛+1)=𝑥₂(𝑛)−0.22+cos𝑥₂(𝑛)𝑥₂(𝑛)+sin𝑥₁(𝑛−2). (4.1) We can obtain that 𝜃₁=0.3467 and furthermore we obtain that 𝑏₁=−0.2896 in view of bisection method by using MATLAB. It is easy to know that 𝑏₀=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=𝑏₁, the Neimark-Sacker bifurcation occurs at the origin.

Since 𝑓₂^′′′(0)=−2,𝑓₁^′′′(0)=−1, 𝑓^′₂(0)=1, 𝑓^′₂(0)=1, 𝑓^′′(0)=0, and 𝑎^′₁(0)=𝑎^′₂(0)=0, 𝐵₁=𝐵₂=0, sign {Re𝐸₁}=1>0, we have {}𝑓signRe𝑑=signRe^2′′′(0)𝑓^′₂(0)+𝑓₁^′′′(0)𝑓^′₁(0)𝑎₁(0)𝑑₁𝑓^′(0)𝜆₁−𝛼₁²=−1<0. (4.2) So the Neimark-Sacker bifurcation is supercritical and the bifurcating periodic solution is stable.

Case 1. Let 𝑑₁=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 (0.02,0.02) and (0.6,0.6), respectively.

(a)

(b)

Figure 1 

(a) Phase plot in space (𝑥1,𝑥2) for system (4.1) with initial condition (0.02,0.02) and 𝑑1=0.806. (b) Phase plot in space (𝑥1,𝑥2) for system (4.1) with initial condition (0.6,0.6) and 𝑑1=0.806.

Case 2. Let 𝑑₁=0.806 and 𝑑₁=0.85, respectively; then 𝑏=−0.2902 and 𝑏=−0.306, respectively, and 𝑏<𝑏₁=−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 (0.02,0.02) and (0.6,0.6), respectively.

(a)

(b)

Figure 2 

(a) Phase plot in space (𝑥1,𝑥2) for system (4.1) with initial condition (0.02,0.02) and 𝑑1=0.85. (b) Phase plot in space (𝑥1,𝑥2) for system (4.1) with initial condition (0.6,0.6) and 𝑑1=0.85.

(a)

(b)

Figure 3 

Bifurcation diagram in (𝑑1,𝑥1)-plane for system (4.1) with step size of 0.01 for 𝑑1 and initial condition (0.001,0.001), in which there are 100 points for each 𝑑1.

(a)

(b)

Figure 4 

(a) Phase plot in space (𝑥1,𝑥2) for system (4.1) with initial condition (0.6,0.6) and 𝑑1=0.79. (b) Phase plot in space (𝑥1,𝑥2) for system (4.1) with initial condition (0.02,0.02) and 𝑑1=0.79.

(a)

(b)

Figure 5 

(a) Phase plot in space (𝑥1,𝑥2) for system (4.1) with initial condition (0.02,0.02) and 𝑑1=3.9. (b) Phase plot in space (𝑥1,𝑥2) for system (4.1) with initial condition (0.02,0.02) and 𝑑1=−3.4.

Case 3. Let 𝑑₁=3.9 and 𝑑₁=−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).

5. Conclusions

A class of Cohen-Grossberg neural networks with discrete delays is analyzed in this paper. By using 𝑏=𝑎₁(0)𝑎₂(0)𝑑₁𝑑₂𝑓^′₁(0)𝑓^′₂(0) as a bifurcation parameter, we show that this system undergoes Neimark-Sacker bifurcations at the critical parameter 𝑏=𝑏₁=−1+𝛼²₁−2𝛼₁cos𝜃₁1+𝛼²₂−2𝛼₂cos𝜃₁, (5.1) where 𝜃₁ is the unique solution in (0,𝜋/(𝑘+2)) of the equation sin(𝑘+2)𝜃−(𝛼₁+𝛼₂)sin(𝑘+1)𝜃+𝛼₁𝛼₂sin𝑘𝜃=0 in which 𝛼₁=1−𝑎₁(0)𝑏^′₁(0) and 𝛼₂=1−𝑎₂(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.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant no. 11071254.