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

Authors: Qiming Liu; Rui Xu; Zhiping Wang
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## 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.


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## Body

## 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:(1.1)x1(n+1)=x1(n)-a1(x1(n))[b1(x1(n))-d1f2(x2(n-k2))],x2(n+1)=x2(n)-a2(x2(n))[b2(x2(n))-d2f1(x1(n-k1))],
where xi(n) denote the state variable of the ith neuron; ai(·) represent amplification functions which are positive for R; fi(·) denote the signal functions of the ith neuron; bi(·) are appropriately behaved functions; di(·) are connection weights of the neural networks; discrete delays ki 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 Section2. 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<ai(0)bi′(0)<1, i=1,2.Letu1(n)=x1(n-k1), u2(n)=x2(n); we transform system (1.1) into the following system: (2.1)u1(n+1)=u1(n)-a1(u1(n))[b1(u1(n))-d1f2(u2(n-k))],u2(n+1)=u2(n)-a2(u2(n))[b2(u2(n))-d2f1(u1(n))],
where k=k1+k2.Furthermore, system (2.1) can be transformed into the following system of k+2 difference equations without delays: (2.2)u1(0)(n+1)=u1(0)(n)-a1(u1(0)(n))[b1(u1(0)(n))-d1f2(u2(k)(n))],u2(0)(n+1)=u2(0)(n)-a2(u2(0)(n))[b2(u2(0)(n))-d2f1(u1(0)(n))],u2(j)(n+1)=u2(j-1)(n),j=1,2,…,k.
In the following discussion, for convenience, we denote (2.3)b=a1(0)a2(0)d1d2f1′(0)f2′(0),αi=1-ai(0)bi′(0),ai=ai(0),ai′=ai′(0),ai′′=ai′′(0),bi=bi(0),bi′=bi′(0),bi′′=bi′′(0),bi′′′=bi′′′(0),fi=fi(0),fi′=fi′(0),fi′′=fi′′(0),fi′′′=fi′′′(0)
for i=1,2.The Jacobian matrix of system (2.2) at the fixed point (0,…,0) is defined as follows: (2.4)Â=(α100⋯0a1d1f2′a2d2f1′α20⋯00010⋯00001⋯00⋮⋮⋮⋯⋮⋮000⋯10).
The associated characteristic equation of system (2.2) is (2.5)(λ-α1)(λ-α2)λk=b.
Suppose that λ=eiθ is a root of the characteristic equation and substitute eiθ into (2.5); then we have (2.6)(cos⁡θ-α1+isinθ)(cos⁡θ-α2+isinθ)(cos⁡kθ+isinkθ)=b.Whenθ∈(0,π), separating the real and imaginary parts of (2.6), we have (2.7)b=(1+α12-2α1cos⁡θ)(1+α22-2α2cos⁡θ)cos⁡h(θ),sinh(θ)=0,
where (2.8)h(θ)=cot-1(cos⁡θ-α1sinθ)+cot-1(cos⁡θ-α2sinθ)+kθ
in which cot-1 denotes the inverse of the cotangent function restricted to the interval (0,π). Since (2.9)h′(θ)=11+((cos⁡θ-α1)/sinθ)2×1-α1cos⁡θsin2θ+11+((cos⁡θ-α2)/sinθ)2×1-α2cos⁡θsin2θ+k>0
for θ∈(0,π) and (2.10)lim⁡θ→π-h(θ)=(k+2)π,lim⁡θ→0+h(θ)=0,
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 θj=h-1(jπ),j=1,2,…,(k+1); then we have from the first equation in (2.7) that (2.11)b=bj=(-1)j(1+α12-2α1cos⁡θj)(1+α22-2α2cos⁡θj),j=1,2,…,k+1.Whenθ=0, b=b0=(1-α1)(1-α2)>0.Whenθ=π,b=bk+2=(-1)k+2(1+α1)(1+α2).Obviously, ifθ is a root of (2.6), -θ is also the root of (2.6). So, we only need to consider the characteristic roots eiθ of (2.5) in [0,π]. Furthermore, from the value bj given above, we have (2.12)⋯<b5<b3<b1<0<b0<b2<b4⋯.On the other hand, we have from (2.5) that (2.13)d|λ|2db=dλλ¯db=λdλ¯db+λ¯dλdb=2Re(λP(λ))|P(λ)|2,
where P(λ)=λk-1[λ(λ-α1)+λ(λ-α2)+k(λ-α1)(λ-α2)].It follows that(2.14)d|λ|2db|b=bj=bj1|P(eiθj)|2(k+1-α1cos⁡θj1+α12-2α1cos⁡θj+1-α2cos⁡θj1+α22-2α2cos⁡θj).
Hence we have (2.15)sign⁡{d|λ|2db|b=bj}=sign⁡{bj}=(-1)j.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 pointsb=b0 and b=b1, bifurcation phenomena occur.Whenb=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]).Whenb=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: (2.16)π=cot-1(cos⁡θ-α1sinθ)+cot-1(cos⁡θ-α2sinθ)+kθ.
It follows from (2.16) that (2.17)-cot⁡kθ=cot(cot-1(cos⁡θ-α1sinθ)+cot-1(cos⁡θ-α2sinθ)),
which leads to (2.18)sin(k+2)θ-(α1+α2)sin(k+1)θ+α1α2sinkθ=0.
So, θ1 satisfies (2.16) if and only if it satisfies (2.18).Let(2.19)y1=sin(k+2)θ,y2=(α1+α2)sin(k+1)θ-α1α2sinkθ,y1 and y2 are easy to be proved to be convex function; furthermore, y1(0)=y2(0)=0, y1′(0+)>y2′(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 (2.20)b1=-(1+α12-2α1cos⁡θ1)(1+α22-2α2cos⁡θ1).
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)
ifb∈(b1,b0), the null solution of system  (1.1) is asymptotically stable,(2)
ifb=b0, a Fold bifurcation occurs at the origin in system  (1.1),(3)
ifb=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
(2.21)b0=(1-α1)(1-α2),b1=-(1+α12-2α1cos⁡θ1)(1+α22-2α2cos⁡θ1)
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 whenb=b1, if b=bj, j=2,3,…,k+1, a Neimark-Sacker bifurcation also occurs at the origin in system  (1.1) near b=bj except θj=2π/3 and π/2. If b=bk+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|>|bk+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 constantsa̲i and a¯i such that 0<a̲i≤ai(·)≤a¯i for i=1,2.(H4)
There exist constantsb̲i and Li such that bi′(·)≥b̲i>0 and |fi′(·)|≤Li for i=1,2.(H5)
0<a̲ib̲i<1 for i=1,2, and a¯1a¯2|d1||d2|L1L2≤a̲1a̲2b̲1b̲2.Proof.
Sincea¯1a¯2d1d2L1L2≤a̲1a̲2b̲1b̲2, the following matrix
(2.22)A=(a̲1b̲1-a¯1|d1|L2-a¯2|d2|L1a̲2b̲2),
is an M-matrix, and there exists a vector p=(pi)1×2>0 such that pA>0 ([24]), that is,
(2.23)p1a̲1b̲1-p2a¯2|d2|L1>0,p2a̲2b̲2-p1a¯1|d1|L2>0.
Hence, we can choose a constant λ>1 such that
(2.24)p1(1-λ+a̲1b̲1λ)-p2a¯2|d2|L1λ>0,p2(1-λ+a̲2b̲2λ)-p1a¯1|d1|L2λk+1>0.
Let U1(n)=λn|u1(n)|, U2(n)=λn|u2(n)|; then we have from (1.1) that
(2.25)U1(n)=λn+1|u1(n+1)|≤λ(1-a̲1b̲1)|U1(n)|+a¯1|d1|L2λk+1|U2(n-k)|,U2(n)=λn+1|u2(n+1)|≤λ(1-a̲2b̲2)|U2(n)|+a¯2|d2|L1λ|U1(n)|.
Define a Lypunov function V(t) by
(2.26)V(t)=∑i=12piUi(n)+p1a¯1|d1|L2λk+1∑l=n-kn-1|U2(l)|.
Then we obtain from (2.24) and (2.25) that
(2.27)ΔV(n)=V(n+1)-V(n)≤-[p1(1-λ+a̲1b̲1λ)-p2a¯2|d2|L1λ]U1(n)-[p2(1-λ+a̲2b̲2λ)-p1a¯1|d1|L2λk+1]U2(n)≤0,
which implies V(t)≤V(0).
Note that(2.28)V(t)≥λnm0(|u1(t)|+|u2(t)|),V(0)=∑i=12piUi(0)+p1a¯1|d1|L2λk+1∑l=-k-1|U2(l)|=defM0,
where m0=min{1/a¯1,1/a¯2} and M0 is a positive constant.
Thus(2.29)|u1(t)|+|u2(t)|≤M0/m0(1λ)n,
where λ>1. Therefore, limn→+∞ui(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 Theorem2.3 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≤b̲1b̲2 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 Section2, 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.Denoteλ1=eiθ1. Let q∈Ck+2 be an eigenvector of Â corresponding to eigenvalue λ1. Then Âq=λq. Again, let p∈Ck+2 be an eigenvector of ÂT corresponding to its eigenvalue λ¯1.By direct calculation, we obtain(3.1)q=(a1d1f2′λ1-α1,λ1k,λ1k-1,…,λ1,1)T,p=E1(λ¯1a1d1f2′,1λ¯1k-1(λ¯1-α2),1λ¯1k-1,1λ¯1k-2,…,1λ¯1,1)T,
in which (3.2)E1=1k+2+(α1/(λ1-α1))+(α2/(λ1-α2)),
where p and q satisfy 〈p,q〉=1.System (2.2) can be written as (3.3)u(n+1)=Âu(n)+F(u(n)),
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 Fi(i=1,2) can be expanded in the form (3.4)F1(ξ)=-(a1b1′′2+a1′b1′)ξ12+a1d1f2′′2ξk+22+a1′d1f2′ξ1ξk+2-(a1b1′′′6+a1′b1′′2+a1′′2b1′)ξ13+a1d1f2′′′6ξk+23+a1′d1f2′′2ξ1ξk+22+a1′′2d1f2′ξ12ξk+2+h.o.t.,F2(ξ)=-(a2b2′′2+a2′b2′)ξ22+a2d2f1′′2ξ12+a2′d2f1′ξ1ξ2-(a2b2′′′6+a2′b2′′2+a2′′2b2′)ξ23+a2d2f1′′′6ξ13+a2′d2f1′′2ξ2ξ12+a2′′2d2f1′ξ1ξ22+h.o.t..
and Â is defined by (2.4).It follows that(3.5)B1(x,y)=∑j,kn∂2F1(ξ)∂ξj∂ξk|ξ=0xjyk=-(a1b1′′+2a1′b1′)x1y1+a1d1f2′′xk+2yk+2+a1′d1f2′x1yk+2,B2(x,y)=∑j,kn∂2F2(ξ)∂ξj∂ξk|ξ=0xjyk=-(a2b2′′+2a2′b2′)x2y2+a2d2f1′′x1y1+a2′′d2f1′x1y2,C1(x,y,z)=∑j,k,ln∂3F1(ξ)∂ξj∂ξk∂ξl|ξ=0xjykzl=-(a1b1′′′+3a1′b1′′+3a1′′b1′)x1y1z1+a1d1f2′′′xk+2yk+2zk+2+a1′d1f2′′x1yk+2zk+2+a1′′d1f2′x1y1zk+2,C2(x,y,z)=∑j,k,ln∂3F2(ξ)∂ξj∂ξk∂ξl|ξ=0xjykzl=-(a2b1′′′+3a2′b2′′+3a2′′b2′)x2y2z2+a2d2f1′′′x1y1z1+a2′d1f1′′x2y1z1+a2′′d1f1′x1y2z2.
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]: (3.6)w⟶λ1w(1+12d|w|2)+O(|w|4),w∈C
in which (3.7)d=λ¯1〈p,Ĉ(q,q,q¯)+2B̂(q,(1-Â)-1B̂(q,q¯))+B̂(q¯,(λ12I-Â)-1B̂(q,q¯))〉,
where p and q are defined by (3.1) and B̂=(B1,B2,0,…,0)T, Ĉ=(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 ofRe(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.Ifa1(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: (3.8)x1(n+1)=α1x1(n)+d2f(x2(n-k2)),x2(n+1)=α2x2(n)+d1f(x1(n-k1)),
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.
Sincef′′(0)=0, it is easy to know from (3.5) that B1=0,B2=0, and
(3.9)C1(q,q,q¯)=a1(0)d1f′′′(0),C2(q,q,q¯)=a2(0)d2f′′′(0)(a1(0)d1f′(0)λ1-α1)2a1(0)d1f′(0)λ¯1-α1.
Substituting (3.9) into d in (3.6), we can obtain
(3.10)Red=|λ1|2[1+(a1(0)d1f′(0)|λ1-α1|)2]Re{E¯1f′′′(0)f′(0)}.
Note that
(3.11)sign⁡{ReE¯1}=sign⁡{Re[k+2+α1λ1-α1+α2λ1-α2]}=sign⁡{k+1-α1cos⁡θ11+α12-2α1cos⁡θ1+1-α2cos⁡θ11+α12-2α2cos⁡θ1}=1>0.
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:(4.1)x1(n+1)=x1(n)-0.2(2+cos⁡(x1(n)))[0.6x1(n)-d1tanh(x2(n-3))],x2(n+1)=x2(n)-0.2(2+cos⁡(x2(n)))[x2(n)+sin(x1(n-2))].
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 Theorem2.1, 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.Sincef2′′′(0)=-2,f1′′′(0)=-1, f2′(0)=1, f2′(0)=1, f′′(0)=0, and a1′(0)=a2′(0)=0, B1=B2=0, sign {ReE1}=1>0, we have (4.2)sign⁡{Red}=sign⁡{Re[f2′′′(0)f2′(0)+f1′′′(0)f1′(0)(a1(0)d1f′(0)|λ1-α1|)2]}=-1<0.
So the Neimark-Sacker bifurcation is supercritical and the bifurcating periodic solution is stable.Case 1.
Letd1=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) 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.
(a)(b)Case 2.
Letd1=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) 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)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)(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)(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.
(a)(b)Case 3.
Letd1=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 usingb=a1(0)a2(0)d1d2f1′(0)f2′(0) as a bifurcation parameter, we show that this system undergoes Neimark-Sacker bifurcations at the critical parameter (5.1)b=b1=-(1+α12-2α1cos⁡θ1)(1+α22-2α2cos⁡θ1),
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)b1′(0) and α2=1-a2(0)b2′(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.



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*Source: 403873-2011-01-18.xml*