## Stability Analysis Including Monostability and Multistability in Dynamical System and Applications

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# Periodic Oscillation Analysis for a Coupled FHN Network Model with Delays

**Academic Editor:**Chunhua Feng

#### Abstract

The existence of periodic oscillation for a coupled FHN neural system with delays is investigated. Some criteria to determine the oscillations are given. Simple and practical criteria for selecting the parameters in this network are provided. Some examples are also presented to illustrate the result.

#### 1. Introduction

Recently, several researchers have studied the dynamics of coupled FHN neural systems [1ā5]. Wang et al. have investigated the following model [6]:

The effects of time delay on bifurcation and synchronization in the two synaptically coupled FHN neurons have been investigated. The authors found that time delay can control the occurrence of bifurcation in the coupled FHN neural model and synchronization is sometimes related to bifurcation transition. Fan and Hong introduced second time delay in model (1) as follows [7]:

Letbe a parameter. The authors have shown that there is a critical value of the parameter; the steady state of model (2) is stable when the parameter is less than the critical value and unstable when the parameter is greater than the critical value. Thus, the zero equilibrium loses its stability when the parameter passes through the critical value, and a Hopf bifurcation occurs and oscillations induced by the Hopf bifurcation appeared. Zhen and Xu generated models (1) and (2) to a three coupled FHN neurons network with time delay as follows [8]: where, represent the synaptic strength of self-connection and neighborhood interaction, respectively, and is a sufficiently smooth sigmoid amplification function such as and . The method of Lyapunov functional is used to obtain the synchronization conditions of the neural system. Noting that, for each neuron of model (3), the synaptic strength of self-connection and neighborhoodinteraction are the same under the same restrictive condition, the dynamics of (3) are completely characterized by the following system: whereis a completely synchronous solution of system (4). The Bautin bifurcation of synchronous solution for this neural system (4) in which,are regarded as the bifurcating parameters is investigated. However, generally speaking, the synaptic strength of self-connection, neighborhoodinteraction for each neuron, and the time delays are different. Therefore, in this paper, we will discuss the following model: where () are constants. () represent the time delays in signal transmission. System (5) can be rewritten as follows:

It is known that if all solutions of system (6) are bounded and there exists a unique unstable equilibrium point of system (6), then this particular instability will force system (6) to generate a limit cycle, namely, a periodic oscillation [9]. We will provide some restrictive conditions which are easy to check to ensure the existence of periodic oscillation. It was pointed out that bifurcating method to determine the periodic solution of system (6) is very difficult.

In the following, we first assume that () are continuous bounded monotone increasing functions, satisfying

For example, activation functions , , and satisfy condition (7). From assumption (7), the linearization of system (6) about the zero point leads to the following:

The matrix form of system (8) is as follows: where, ,

#### 2. Preliminaries

Lemma 1. *Suppose that, , , (); then each solution of system (6) is bounded.*

*Proof. *Note that the activation functions are bounded continuous nonlinear functions. Therefore, there exist such that (). Since, (), this implies that there exist constants such that for any valueswe have (). From (6) we get

Noting that system (11) is the first-order linear system of equations with constant coefficients, the eigenvalues of system (11) are (). Since, , (), if or are complex numbers with if (). This implies that all solutions of system (11), as well as the system, (6) are bounded according to the theory of the first-order linear system of equations with constant coefficients.

According to [10], there is the same oscillatory behavior for systems (8) and (6). So, in order to investigate the periodic oscillatory behavior of system (6), we only need to deal with system (8).

Lemma 2. *Suppose that matrix () is a nonsingular matrix. Then, system (9) has a unique equilibrium point. *

*Proof. *An equilibrium pointis the solution of the following algebraic equation:

Assume that and are two equilibrium points of system (9); then we have

Since is a nonsingular matrix, implying that and system (9) has a unique equilibrium point. Obviously, this equilibrium point is exactly the zero point.

#### 3. Periodic Oscillation

Theorem 3. *Suppose that,, , (), and is a nonsingular matrix. Let,, , and denote the eigenvalues of matricesand, respectively. Assume that there is at least one , , and the following inequalities hold:
*

*Then, the trivial solution of system (8) is unstable, implying that there is a periodic oscillatory solution of system (6).*

*Proof. *From the assumptions, we know that system (8) has a unique equilibrium point and all solutions are bounded. We will prove that the unique equilibrium point is unstable. We first discuss the case that in system (8). The characteristic equation of system (8) is as follows:

Equation (15) is equal to

Therefore, we are led to an investigation of the nature of the roots for the following equations:

For some, consider equation

If is a solution of (18), then; from (18) we have

Using the formula , leads to the fact that

Equation (20) contradicts the first inequality of assumption (14). Then, we discuss the case thatin system (8). Similarly, if is a solution of the equation, we also have a contradiction with the second inequality of assumption (14). Since (), one can conclude that there exists a positive real part of the eigenvalue of system (8) for any () under the assumptions. This means that the trivial solution of system (8) is unstable, implying that there is a periodic oscillatory solution of system (6) based on Chafeeās criterion.

Theorem 4. *Suppose that,, , (), and is a nonsingular matrix. Let ( may equal zero) and ( may equal zero) () denote the eigenvalues of matricesand, respectively. If, for some, | as , then the trivial solution of system (8) is unstable, implying that system (6) has a periodic oscillatory solution.*

*Proof. *The assumptions guarantee that system (8) has a unique equilibrium point and all solutions are bounded. In this case, we first consider in system (8). Then, for some , let ; from (18) we have

We will show that and there is an eigenvalue which has positive real part of system (18). Let ; then is a continuous function of . If, then select suitable delaysuch that. Therefore,. Noting thatas, obviously, there exists a suitably large such that. By the continuity of, there exists a positive such that. If, since | (), then there exists a suitable delayand a positivesuch that and both hold. Then, and . Again, from the continuity of , there exists a positive such that. Thus, there is an eigenvalue of system (18) that has positive real part. Implying that the trivial solution of system (8) is unstable. Thus, the trivial solution of system (6) is also unstable. Based on the theory of delay differential equation, the oscillatory behavior of the solution will maintain as time delay increasing. Therefore, for any (), system (8), as well as system, (6) generates a periodic oscillatory solution. We select a suitable delay such that system (6) has a periodic oscillatory solution. This oscillation is said to be induced by time delay.

#### 4. Simulation Result

The parameter values are selected as, , ; , , ; , , ; , , ; , , ; , , , respectively. It is easy to check that the conditions of Lemmas 1 and 2 hold. The activation functions are selected as and , respectively.In this case,, and eigenvalues of matrices and are,, , ,, and , and,, , , and , , respectively. Since, there is a periodic oscillatory solution based on Theorem 4. Both in Figures 1 and 2, the time delays are selected as , , and . Then, we change delays as, , ; activation function is kept as ; periodic oscillatory solution also occurred (Figure 3). In Figure 4, the parameter values are selected as, , ; , , ; , , ; , , ; , , ; , , and , respectively. The activation function is . The eigenvalues of matricesandare , , , , , and , , , , , and , respectively. We see that periodic oscillatory solution appeared.

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**(b) Solid line: , dashed line: , dotted line:**

**(a) āSolid line: , dashed line: , dotted line:**

**(b) Solid line: , dashed line: , dotted line:**

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**(b) Solid line: , dashed line: , dotted line:**

#### 5. Conclusion

This paper discusses a three coupled FHN neurons model in which the synaptic strength of self-connection, neighborhoodinteraction for each neuron, and the time delays are different. Two theorems are provided to determine the periodic oscillatory behavior of the solutions based on Chafeeās criterion of limit cycle. Computer simulation suggested that those theorems only are sufficient conditions.

#### Acknowledgments

This research was supported by NNSF of China (11361010) and Scientific Research Foundation of the Education Department of Guangxi Province (201010LX463).

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#### Copyright

Copyright © 2013 Yuanhua Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.