Neimark-Sacker Bifurcation Analysis for a Discrete-Time System of Two Neurons

and Applied Analysis 3 theory for discrete-time system developed by Kuznetsov 25 . In Section 4, numerical simulations aimed at justifying the theoretical analysis will be reported. 2. Stability and Existence of Neimark-Sacker Bifurcation In this section, we discuss the global and local stability of the equilibrium 0, 0 of system 1.1 . In order to prove our results, we need the following hypothesis: H1 fi : R → R is globally Lipschitz with Lipschitz constant Li > 0, i 1, 2 , that is, ∣ fi u − fi v ∣ ∣ ≤ Li|u − v| for u, v ∈ R. 2.1 Theorem 2.1. Let Δ |a11|L1 |a22|L2 2 4|a12||a21|L1L2. Suppose that hypothesis (H1) and the inequality ∣ ∣ ∣2β |a11|L1 |a22|L2 ± √ Δ ∣ ∣ ∣ < 2 2.2 are satisfied, then x1 n , x2 n → 0, 0 as n → ∞. Proof. It follows from system 1.1 that (|x1 n 1 | |x2 n 1 | ) ≤ ( β |a11|L1 |a12|L1 |a21|L2 β |a22|L2 )(|x1 n | |x2 n | ) . 2.3


Introduction
Since one of the models with electric circuit implementation was proposed by Hopfield 1 , the dynamical behaviors including stability, instability, periodic oscillatory, bifurcation, and chaos of the continuous-time neural networks have received increasing interest due to their promising potential applications in many fields, such as signal processing, pattern recognition, optimization, and associative memories see 2-5 . For computer simulation, experimental or computational purposes, it is common to discrete the continuous-time neural networks. Certainly, the discrete-time analog inherits the dynamical characteristics of the continuous-time neural networks under mild or no restriction on the discretional step size and also remains functionally similar to the continuoustime system and any physical or biological reality that the continuous-time system has. We refer to 6, 7 for related discussions on the importance and the need for discrete-time analog to reflect the dynamics of their continuous-time counterparts. Recently, Zhao et al. 8 discussed the stability and Hopf bifurcation on discrete-time Hopfield neural networks 2 Abstract and Applied Analysis with delay. Yu and Cao 9 studied the stability and Hopf bifurcation on a four-neuron BAM neural network with time delays. Xiao and Cao 10 considered the stability and pitchfork bifurcation, flip bifurcation, and Neimark-Sacker bifurcation. Yuan et al. 11 investigated the stability and Neimark-Sacker bifurcation of a discrete-time neural network. Yuan et al. 12 made a discussion on the stability and Neimark-Sacker bifurcation on a discrete-time neural network. For more knowledge about neural networks, one can see 13-18 . It will be pointed that two neurons have the same transfer function f in 11 and two neurons have different transfer functions f in 12 i.e., the transfer function of the first neuron is f 1 and the transfer function of the second neuron is f 2 . In this paper, we assume that there are same transfer function f 1 in the first equation and there are same transfer function f 2 in the second equation, then we obtain the following discrete-time neural network model with self-connection in the form: where x i i 1, 2 denotes the activity of the ith neuron, β ∈ 0, 1 is internal delay of neurons, the constants a ij i, j 1, 2 denote the connection weights, f i : R → R is a continuous transfer function, and f i 0 0 i 1, 2 . The discrete-time system 1.1 can be regarded as a discrete analogy of the differential systemẋ 1.2 or the system with a piecewise constant arguments: where μ > 0 and · denotes the greatest integer function. For the method of discrete analogy, we refer to 19-21 . The motivation of this research is system 1.1 which includes the discrete version of system 1.2 and 1.3 . On the other hand, the wide application of differential equations with piecewise constant argument in certain biomedical models 22 and much progress have been made in the study of system with the piecewise arguments since the pioneering work of Cooke and Wiener 23 and Shah and Wiener 24 .
In this paper, we investigate the nonlinear dynamical behavior of a discrete-time system of two neurons, namely, 1.1 , and prove that Neimark-Sacker bifurcation will occur in the discrete-time system. Using techniques developed by Kuznetsov to discrete-time systems 25 , we obtain the stability of the bifurcating periodic solution and the direction of Neimark-Sacker bifurcation.
The organization of this paper is as follows. In Section 2, we will discuss the stability of the trivial solutions and the existence of Neimark-Sacker bifurcation. In Section 3, a formula for determining the direction of Neimark-Sacker bifurcation and the stability of bifurcating periodic solution will be given by using the normal form method and the center manifold theory for discrete-time system developed by Kuznetsov 25 . In Section 4, numerical simulations aimed at justifying the theoretical analysis will be reported.

Stability and Existence of Neimark-Sacker Bifurcation
In this section, we discuss the global and local stability of the equilibrium 0, 0 of system 1.1 . In order to prove our results, we need the following hypothesis: Clearly, the eigenvalues of M are given by which implies that |λ 1,2 | < 1. Thus the eigenvalues of M are inside the unit circle and x 1 n , x 2 n → 0, 0 as n → ∞.
Next, we will analyze the local stability of the equilibrium 0, 0 . For most of models in the literature, including the ones 20, 26, 27 , the transfer function f is f u tanh cu . However, we only make the following assumption on functions f i : H2 f i ∈ C 1 R and f i 0 0 i 1, 2 .

Abstract and Applied Analysis
For the sake of simplicity and the need of discussion, we define the following parameters:

2.7
Proof. Under H2 , using Taylor expansion, we can expand the right-hand side of system 1, 1 into first-, second-, third-, and other higher-order terms about the equilibrium 0, 0 , and we have The associated characteristic equation of its linearized system is In order to make the equilibrium 0, 0 be locally asymptotically stable, it is necessary and sufficient that all the roots of 2.10 are inside the unit circle. Hence, we will discuss the following two cases.
Abstract and Applied Analysis 5 Case 1 P 1 − P 2 2 ≥ D . In this case, the roots of 2.10 are given by Obviously, we obtained that the modulus of eigenvalues λ 1,2 are less than 1 if and only if P 1 , P 2 , D ∈ X 1 X 2 , where

2.12
Thus, we obtain that the eigenvalues λ 1,2 are inside the unit circle when P 1 , P 2 , D ∈ X 1 X 2 is satisfied.
In this case, the characteristic equation of 2.10 has a pair of conjugate complex roots: It is easy to verify that |λ 1,2 | < 1 if and only if P 1 , P 2 , D ∈ X 3 . Combining case 1 with case 2 yields that the the eigenvalues λ 1,2 are inside the unit circle for P 1 , P 2 , D ∈ X 0 X 1 X 2 X 3 and the zero solution of 1.1 is asymptotically stable.
In what follows, we will choose D as the bifurcation parameter to study the Neimark-Sacker bifurcation at 0, 0 . For P 1 − P 2 2 < D, we denote Then the eigenvalues of 2.10 are conjugate complex λ D and λ D . The modulus of eigenvalue When the parameter D passes through such critical value of D * 4 1 − P 1 P 2 , a Neimark-Sacker bifurcation may be expected. Obviously, we have |λ D | < 1 for P 1 − P 2 2 < D < D * .

2.16
Since the modulus of eigenvalue |λ D * | 1, we know that D * is a critical value which destroies the stability of 0, 0 . The following lemma is helpful to study bifurcation of 0, 0 .

6
Abstract and Applied Analysis Lemma 2.3. If (H2) and 0 < P 1 P 2 < 2 are satisfied, then where λ D and D D * are given by 2.14 and 2.15 , respectively.
By Lemma 2.2 in 28 , we obtain the following results.
ii If D > D * , then the equilibrium 0, 0 is unstable.
iii The Neimark-Sacker bifurcation occurs at D D * . That is, system 1.1 has a unique close invariant curve bifurcating from the equilibrium 0, 0 .
Proof. Obviously, we have |λ| < 1 for P 1 − P 2 2 < D < D * and |λ| > 1 for D > D * , which means i and ii are true. The conclusions in Lemma 2.3 indicate the transversality condition for the Neimark-Sacker bifurcation is satisfied, so the Neimark-Sacker bifurcation occurs at D D * . Conclusion iii follows.

Direction and Stability of Neimark-Sacker Bifurcation
In the above section, we have shown that Neimark-Sacker bifurcation occurs at some value D D * for system 1.1 under condition H2 and P 1 − P 2 2 < D, 0 < P 1 P 2 < 2. In this section, by employing the normal form method and the center manifold theory for discrete-time system developed by Kuznetsov 25 , we will study the direction and stability of Neimark-Sacker bifurcation. In what follows, we make the following further assumption: H3 f ∈ C 3 R . Now system 1.1 can be rewritten as Abstract and Applied Analysis Suppose that q ∈ C 2 is an eigenvector of A D corresponding to eigenvalue λ D given by 2.14 and p ∈ C 2 is an an eigenvector of A T D corresponding to eigenvalue λ D . Then

A D q D λ D q D , A T D q D λ D q D .
3.4 By direct calculation, we obtain that For the eigenvector q, to normalize p, let

3.7
We have q, p 1, where · means the standard scalar product in C 2 : q, p p 1 q 1 p 2 q 2 . Any vector x ∈ R 2 can be represented for D near D * as x zq D zq D . 3.8 For some complex z, obviously, Abstract and Applied Analysis Thus, system 3.1 can be transformed for D near D * into the following form: where λ D that can be written as λ D 1 ϕ D e iθ D ϕ D is a smooth function with ϕ D * 0 and We know that F i i 1, 2 in 3.1 can be expanded as

3.12
It follows that x j x k a 11 f 1 0 x 1 y 1 a 12 f 1 0 x 2 y 2 , 3.13 x j x k a 21 f 2 0 x 1 y a 22 f 2 0 x 2 y 2 , 3.14 3.15

3.16
By 3.11 -3.16 and the following formulas: g 20 D * p, B q, q , g 11 D * p, B q, q , g 02 D * p, B q, q , g 21 D * p, B q, q, q ,

3.17
Abstract and Applied Analysis 9 we obtain

3.19
Noting that e −iθ D * λ D * , we can compute the coefficient a D * which determines the direction of the appearance of the invariant curve in system 1.1 exhibiting the Neimark-Sacker bifurcation: 3.20 We calculate every term, respectively, i Re e −iθ D * g 21 2

3.25
Abstract and Applied Analysis Thus

Numerical Examples
In this section, we give numerical simulations to support our theoretical analysis. Let β 1/2, a 11 1, a 12 −1, a 22 −1, f 1 u sin u, and f 2 u arctan u/3 in system 1.1 ; namely, system 1.1 has the following form:  By the simple calculation, we obtain  Choose a 21 0.4 so that D 38/15 < D * 3, P 1 , P 2 , D ∈ X 0 . By Theorem 2.4, we know that the origin is asymptotically stable. The corresponding waveform and phase plots are shown in Figures 1, 2, 3, and 4. Choose a 21 4, then D 14/3 > D * 3. By Theorem 2.4, we know that a Neimark-Sacker bifurcation occurs when D D * 3. By a series of complicated computation, we obtain g 20 g 11 g 02 0, a D * ≈ −1.214 < 0. By Theorem 3.1, we know that the periodic solution is stable. The corresponding phase plot is shown in Figures 5, 6, 7, and 8.

Conclusions
The discrete-time delay system of neural networks provides some dynamical behaviors which enrich the theory of continuous system and have potential applications in neural networks. Although the system discussed in this paper is quite simple, it is potentially useful  applications as the complexity which has been carried over to the other models with delay. By choosing a proper bifurcation parameter, we have shown that a Neimark-Sacker bifurcation occurs when this parameter passes through a critical value. We have also determined the direction of the Neimark-Sacker bifurcation and the stability of periodic solutions by applying the normal form theory and the center manifold reduction. Our simulation results have verified and demonstrated the correctness of the theoretical results. Our work is a excellent complementary to the known results 11, 12 in the literatures.