Neimark-Sacker Bifurcation Analysis for a Discrete-Time System of Two Neurons
A class of discrete-time system modelling a network with two neurons is considered. First, we investigate the global stability of the given system. Next, we study the local stability by techniques developed by Kuznetsov to discrete-time systems. It is found that Neimark-Sacker bifurcation (or Hopf bifurcation for map) will occur when the bifurcation parameter exceeds a critical value. A formula determining the direction and stability of Neimark-Sacker bifurcation by applying normal form theory and center manifold theorem is given. Finally, some numerical simulations for justifying the theoretical results are also provided.
Since one of the models with electric circuit implementation was proposed by Hopfield , 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 continuous-time 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.  discussed the stability and Hopf bifurcation on discrete-time Hopfield neural networks with delay. Yu and Cao  studied the stability and Hopf bifurcation on a four-neuron BAM neural network with time delays. Xiao and Cao  considered the stability and pitchfork bifurcation, flip bifurcation, and Neimark-Sacker bifurcation. Yuan et al.  investigated the stability and Neimark-Sacker bifurcation of a discrete-time neural network. Yuan et al.  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 in  and two neurons have different transfer functions in  (i.e., the transfer function of the first neuron is and the transfer function of the second neuron is ). In this paper, we assume that there are same transfer function in the first equation and there are same transfer function in the second equation, then we obtain the following discrete-time neural network model with self-connection in the form: where denotes the activity of the th neuron, is internal delay of neurons, the constants denote the connection weights, is a continuous transfer function, and .
The discrete-time system (1.1) can be regarded as a discrete analogy of the differential system or the system with a piecewise constant arguments: where 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  and much progress have been made in the study of system with the piecewise arguments since the pioneering work of Cooke and Wiener  and Shah and Wiener .
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 , 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 . 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 of system (1.1). In order to prove our results, we need the following hypothesis:
(H1) is globally Lipschitz with Lipschitz constant , that is,
Theorem 2.1. Let . Suppose that hypothesis (H1) and the inequality are satisfied, then as .
Proof. It follows from system (1.1) that
Clearly, the eigenvalues of are given by
which implies that . Thus the eigenvalues of are inside the unit circle and as .
Next, we will analyze the local stability of the equilibrium . For most of models in the literature, including the ones [20, 26, 27], the transfer function is . However, we only make the following assumption on functions :
(H2) and .
For the sake of simplicity and the need of discussion, we define the following parameters:
Theorem 2.2. The zero solution of (1.1) is asymptotically stable if (H2) is satisfied and , where
Proof. Under (H2), using Taylor expansion, we can expand the right-hand side of system into first-, second-, third-, and other higher-order terms about the equilibrium , and we have
where , and
The associated characteristic equation of its linearized system is
In order to make the equilibrium 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.
Case 1 (). In this case, the roots of (2.10) are given by
Obviously, we obtained that the modulus of eigenvalues are less than 1 if and only if , where Thus, we obtain that the eigenvalues are inside the unit circle when is satisfied.
Case 2 ().In this case, the characteristic equation of (2.10) has a pair of conjugate complex roots: It is easy to verify that if and only if .
Combining case 1 with case 2 yields that the the eigenvalues are inside the unit circle for and the zero solution of (1.1) is asymptotically stable.
In what follows, we will choose as the bifurcation parameter to study the Neimark-Sacker bifurcation at . For , we denote Then the eigenvalues of (2.10) are conjugate complex and . The modulus of eigenvalue is . Clearly, if and only if When the parameter passes through such critical value of , a Neimark-Sacker bifurcation may be expected. Obviously, we have Since the modulus of eigenvalue , we know that is a critical value which destroies the stability of . The following lemma is helpful to study bifurcation of .
Proof. Under the assumption , we have which implies (i) holds. On the other hand, for some if and only if the argument . Since , it follows that . Hence the condition (ii) of Lemma 2.3 is also satisfied. The proof is complete.
By Lemma 2.2 in , we obtain the following results.
Theorem 2.4. Suppose that (H2) and are satisfied, then one has the following.(i)If , then the equilibrium is asymptotically stable.(ii)If , then the equilibrium is unstable.(iii)The Neimark-Sacker bifurcation occurs at . That is, system (1.1) has a unique close invariant curve bifurcating from the equilibrium .
Proof. Obviously, we have for and for , 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 . Conclusion (iii) follows.
3. Direction and Stability of Neimark-Sacker Bifurcation
In the above section, we have shown that Neimark-Sacker bifurcation occurs at some value for system (1.1) under condition (H2) and . In this section, by employing the normal form method and the center manifold theory for discrete-time system developed by Kuznetsov , we will study the direction and stability of Neimark-Sacker bifurcation. In what follows, we make the following further assumption:
Now system (1.1) can be rewritten as where and Denote Suppose that is an eigenvector of corresponding to eigenvalue given by (2.14) and is an an eigenvector of corresponding to eigenvalue . Then By direct calculation, we obtain that where For the eigenvector , to normalize , let We have , where means the standard scalar product in . Any vector can be represented for near as For some complex , obviously, Thus, system (3.1) can be transformed for near into the following form: where that can be written as is a smooth function with and We know that in (3.1) can be expanded as It follows that By (3.11)–(3.16) and the following formulas: we obtain where Noting that , we can compute the coefficient which determines the direction of the appearance of the invariant curve in system (1.1) exhibiting the Neimark-Sacker bifurcation: We calculate every term, respectively,(i) where (ii) where (iii) Thus
Theorem 3.1. Suppose that condition (H3) holds and , then the direction of the Neimark-Sacker bifurcation and stability of bifurcating periodic solution can be determined by the sign of . In fact, if , then the Neimark-Sacker bifurcation is supercritical (subcritical) and the bifurcating periodic solution is asymptotically stable (unstable), where is given by (2.15).
Remark 3.2. This method is introduced by Kuznetsov in .
4. Numerical Examples
In this section, we give numerical simulations to support our theoretical analysis. Let , and in system (1.1); namely, system (1.1) has the following form: By the simple calculation, we obtain Choose so that . 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 , then . By Theorem 2.4, we know that a Neimark-Sacker bifurcation occurs when . By a series of complicated computation, we obtain . 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.
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.
This work is supported by National Natural Science Foundation of China (no. 10961008), Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J2100), Governor Foundation of Guizhou Province (2012), and Doctoral Foundation of Guizhou University of Finance and Economics (2010).
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