STABILITY AND BIFURCATION IN A SIMPLIFIED FOUR-NEURON BAM NEURAL NETWORK WITH MULTIPLE DELAYS

We first study the distribution of the zeros of a fourth-degree exponential polynomial. Then we apply the obtained results to a simplified bidirectional associated memory (BAM) neural network with four neurons and multiple time delays. By taking the sum of the delays as the bifurcation parameter, it is shown that under certain assumptions the steady state is absolutely stable. Under another set of conditions, there are some critical values of the delay, when the delay crosses these critical values, the Hopf bifurcation occurs. Furthermore, some explicit formulae determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are obtained by applying the normal form theory and center manifold reduction. Numerical simulations supporting the theoretical analysis are also included.


Introduction
In recent years, the interest in investigating the dynamics of neural networks has been steadily increasing since Hopfield [11] constructed a simplified neural network model.Based on the Hopfield neural network, Marcus and Westervelt [14] argued that time delays always appear in the signal transmission and therefore they proposed a neural network model with delays.Hereafter, various dynamical models modeling delayed neural networks have been proposed and investigated widely to understand the dynamics behavior of the like-neurons (see [1,2,6,7]) and there has been an extensive literature on the local and global stability analysis of the delayed neural networks (see [6,15,19,23] and the references cited therein).It is well known that the study on dynamical systems not only involve a discussion of stability, but also involve many dynamical behaviors such as periodic phenomenon, bifurcation and chaos.In particular, the properties of periodic solutions are of great interest, which can arise through the Hopf bifurcation in delayed systems, see Hale [8], Liu and Yuan [13], Wei and Ruan [20] and Wei and Velarde [21].
As large-scale nonlinear dynamical systems, neural networks are complex while the dynamics of the delayed neural networks are even richer and more complicated.Neural networks with delays can exhibit very rich dynamics [22].Since an exhaustive analysis of the dynamics of large-scale dynamical systems is quite difficult, Babcock and Westervelt [1] suggested examining carefully the dynamical behaviors of the simple and lower-order networks.Consequently, in recent years, there has been an extensive literature about the studies of the dynamics of some special and simple lower-order neural networks (see [3-5, 7, 16, 17, 20, 22]).However, the studies of the dynamics regarding higher-order systems modelling the neural networks with more neurons and multiple delays are rare.To the best of our knowledge, there is a few literatures have investigated higher-order neural networks (see [18,21] for the case of three neurons and multiple delays and [4,12] for a ring of four neurons).In fact, in practical applications, the neural networks with more neurons and multiple delays are of great importance and frequency.Therefore, it is necessary to investigative the dynamical behaviors of this kind of networks.
In general, in the investigating on a delay model, linearization of the system at its steady state gives us a transcendental characteristic equation or called an exponential polynomial equation.It is well known that the steady state is stable if all eigenvalues of the corresponding transcendental characteristic equation have negative real parts, and unstable if at least one root has positive real part.Thus, a Hopf bifurcation occurs when the real part of a certain eigenvalue changes from negative to zero and to positive (i.e., the steady state changes from stability to instability).This is usually caused by the delays.However, there is a strong possibility that if the coefficients of the exponential polynomial satisfy certain assumptions, then the real parts of all eigenvalues remain negative for all values of the delay; that is, independent of the delay.The corresponding delay system is called absolutely stable (see, e.g., Hale et al. [9]).A general result in Hale et al. [9] says that a delay system is absolutely stable if and only if the corresponding ODE system is asymptotically stable and the characteristic equation has no purely imaginary roots.Therefore, a first important step in the study of the dynamics of a delayed model is to analyze in detail the distribution of zeros of the associated characteristic equation.However, most aforementioned works on the neural network focused mainly on the second-dimensional case and hence the associated characteristic equation is also second-degree exponential polynomial equation.Thus, the second-degree exponential polynomial equation has been investigated widely and exhaustively (see [20] and the cited reference therein).As far as the study of third and fourth-degree exponential polynomial equation, there are also a few literature (see, e.g., [17,18] for the case of the third degree and [4,12] for the case of the fourth degree).
Among a great deal of artificial neural networks, there exist a class of important twolayer heteroassociative networks, called bidirectional associative memory (BAM) neural networks with axonal signal transmission delays, has been proposed and applied in many fields such as pattern recognition and automatic control (see [6]).If there is only one neuron on the I-layer and three neurons on the J-layer, the time delay from the I-layer to another J-layer is τ 1 while the delay from the J-layer back to the I-layer is τ 2 , and the activation functions are f l (l = 2,3,4) (see Figure 1.1).Then the network model can be described by the following delayed differential equations, namely, functional differential X.-P.Yan and W.-T. Li 3 where x k (k = 1,2,3,4) denote the state of the kth neuron; μ k > 0 (k = 1,2,3,4) describe respectively the stability of internal neuron processes on the I-layer and the J-layer; the real constants c l1 (l = 2,3,4) and c 1k (k = 2,3,4) denote the connected weights through the neurons in two layers: the I-layer and the J-layer.
The linearization of the system (1.1) at its steady state leads to a transcendental characteristic equation in the form of which is a fourth-degree exponential polynomial equation.
If b 1 = b 2 = 0, then (1.1) reduces to which was investigated by Li and Wei [12], and applied the obtained results to a ring of neural network model consisting of four different neurons with instantaneous selfconnection and multiple delays, see Figure 1.2.Clearly, if b 2 1 + b 2 2 = 0, then the results obtained by Li and Wei [12] fail to (1.2).Therefore, in order to study the dynamics of the neural network (1.1), it is necessary to investigate further the fourth-degree exponential polynomial equation (1.2) with b 2 1 + b 2 2 = 0.In this paper, we first study the distribution of the roots of (1.2) and find that there are the following two possibilities.
(a) Under certain assumptions on the coefficients, all roots of (1.2) have negative real parts for all delay value τ ≥ 0.
(b) If the assumptions in (a) are not satisfied, then there is a critical value τ 0 .When the delay τ < τ 0 , the real parts of all roots of (1.2) are still negative; when τ = τ 0 , there is A four-neuron ring with delays and self-connection.
a pair of purely imaginary roots and all other roots have negative real parts; when τ > τ 0 , there is at least one eigenvalue which has positive real part.By regarding the sum τ of the two delays τ 1 and τ 2 as a bifurcation parameter, applying the obtained results to the BAM neural network (1.2), we show that under a set of assumptions on the coefficients, the steady state is absolutely stable (i.e., asymptotically stable independent of the delay τ).Under another set of conditions, the steady state is conditionally stable; that is, there is a sequence of critical delay values τ 0 < τ 1 < ..., and the steady state is asymptotically stable when τ < τ 0 , loses its stability when τ = τ 0 , and becomes unstable when τ > τ 0 .Thus, a Hopf bifurcation occurs at the steady state when τ passes through the critical values τ j ( j = 0,1,...).
This paper is organized as follows.In the next section, we will analyze in detail the distribution of roots of the fourth exponential polynomial equation (1.1).In Section 3, we apply the results obtained in Section 2 to (1.2), the absolute stability, conditional stability of zero equilibrium and the existence of Hopf bifurcation are studied.In Section 4, based on the normal form theory and the center manifold argument introduced by Hassard et al. [10], we derive the formulae determining the direction, stability and the period of the bifurcating periodic solution.In Section 5, by applying the results obtained in Section 4, we give a result for a special case of (1.1) determining the direction, stability and the period of the bifurcating periodic solution.Finally, to verify the theoretic analysis, a conclusion is also drawn in the end.

The fourth-degree transcendental polynomial equation
In this section, we will study in detail the distribution of zeros of the fourth-degree transcendental polynomial equation (1.2).
Consider the following fourth-degree transcendental polynomial equation where a k ,b l ∈ R (k = 0,1,2,3; l = 0,1,2) are all real constants and 2 l=0 b 2 l = 0. Clearly, iω (ω > 0) is a root of (2.1) if and only if ω satisfies the following equation (2.2) X.-P.Yan and W.-T. Li 5 Separating the real and imaginary parts of the above equation yields the following equations Adding up the squares of the corresponding sides of the above equations leads to (2.4) (2.5) Then, (2.4) can be denoted simply as the following equation: Noting that lim z→+∞ h(z) = +∞, we conclude that if d < 0, then (2.6) has at least one positive root.From (2.7), we have where f (z Suppose that D 0 > 0, then from the Cardano's formulae for the third-degree algebra equation, we know that the equation f (z) = 0 has only one real root Noticing that lim z→±∞ h(z) = +∞, thus we know that z * 1 is a unique minimum value point of h(z) on R. Therefore, if d ≥ 0 and z * 1 ≤ 0, then (2.6) has no positive roots; if d ≥ 0, z * 1 > 0 and h(z * 1 ) < 0, then (2.6) has at least one positive roots.Assume that D 0 = 0, then in this case the equation f (z) = 0 has three real roots it is easy to know that h(z) is strictly monotonously increasing when z > z * 2 .Therefore, if d ≥ 0 and z * 2 ≤ 0, then (2.6) has no positive roots; if d ≥ 0, z * 2 > 0 and h(z * 2 ) < 0, then (2.6) has at least one positive root.
If D 0 < 0, then we know that the equation f (z) = 0 has three real roots where α is one of cubic roots of the complex number −q/2 + D 0 and 3 > 0 and h(z * 3 ) < 0, then (2.6) has at least one positive root.Summarizing these remarks, we have the following result.(ii) Suppose that d ≥ 0, then (2.6) has no positive root if one of the following conditions holds: (1)

6) has at least a positive root if one of the following conditions holds:
(1) 3 > 0 and h(z * 3 ) < 0. Suppose now that (2.6) has positive roots.Without loss of generality, we may assume that (2.6) has four positive roots denoted respectively as z 1 , z 2 , z 3 and z 4 .Then (2.4) has four positive roots where (2.15) Therefore, if we define then (2.1) with τ = τ (k) j has a pair of purely imaginary roots ±iω k .Let (2.17) X.-P.Yan and W.-T. Li 7 In what follows, we use a result due to Ruan and Wei [17] to analyze (2.1), which is stated as follows.

.,e −λτm ) on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.
Note that when τ = 0, (2.1) becomes the following fourth-degree algebra equation It is easily to get the following result regarding the distribution of roots of the exponential polynomial equation (2.1) by using Lemmas 2.1 and 2.2. 3 < 0, then the order of all roots with positive real parts of (2.1) has the same sum as that of all roots with positive real parts of (2.19) for all τ ≥ 0. (ii) If d < 0 or d ≥ 0 and one of the following conditions holds, (1) 3 > 0 and h(z * 3 ) < 0, then the order of all roots with positive real parts of (2.19) has the same sum as that of all roots with positive real parts of (2.19) be the root of (2.1) near τ = τ (k) j satisfying Lemma 2.4.Suppose that z k = ω 2 k and dh(z k )/dz = 0. Then the following transversality conditions hold: and the sign of Re[dλ k (τ)/dτ| τ=τ (k) j ] is the same as that of dh(z k )/dz.Proof.Differentiating the two sides of (2.1) with respect to τ and noticing that λ is the function of τ, one can obtain (2.23) Thus, we have Notice that λ = ±iω k when τ = τ (k) j (k = 1,2,3,4, j = 0,1,2,...) and we only need to consider the case that λ = iω k because the case λ = −iω k can be obtained similarly.Accordingly, when τ = τ (k)  j (k = 1,2,3,4, j = 0,1,2,...), we have (2.25) (2.26) X.-P.Yan and W.-T. Li 9 Then it follows from (2.3) and (2.25) that (2.27) Therefore, from (2.5) and (2.27), we can get (2.28) Thus, Re[dλ k (τ)/dτ| τ=τ (k) j ] = 0.In addition, since z k > 0, we conclude that Re[dλ k (τ)/ dτ| τ=τ (k)  j ] and dh(z k )/dz have the same sign.This completes the proof.

Direction and stability of the Hopf local bifurcation
In the above section, we have already obtained some sufficient conditions ensuring system (3.1)undergoes a Hopf bifurcation at the equilibrium (0,0,0,0) when τ takes some certain critical values.In this section, we suppose that a Hopf bifurcation for system (3.1) will occur at the zero equilibrium when τ = τ j ( j = 0,1,2,...), that is, a family of periodic solutions bifurcate from the zero equilibrium and will establish the explicit formulae determining the direction, stability, and period of these periodic solutions bifurcating from the zero equilibrium at these critical values τ j ( j = 0,1,2,...) of τ by using the normal theory and the center manifold argument developed by Hassard et al. [10].Without loss of generality, we denote any one of these critical values τ = τ j ( j = 0,1,2,...) by τ, at which (3.4) has a pair of purely imaginary roots ±iω and system (3.1)undergoes a Hopf bifurcation from the zero equilibrium.
For φ ∈ C 1 ([−1,0],R 4 ), define the operator A(μ) as where If we further define the operator R(μ) as then system (4.1) is equivalent to and a bilinear inner product where η(θ) = η(θ,0).Then A(0) and A * are adjoint operators.From the discussion in Section 2, we know that ±iω τ are eigenvalues of A(0) and therefore they are also eigenvalues of A * .It is not difficult to verify the vector is the eigenvector of A(0) corresponding to the eigenvalue iω τ, and is the eigenvector of A * corresponding to the eigenvalue −iω τ.Moreover, q * (s), q(θ) = 1, where (4.17) X.-P.Yan and W.-T. Li 15 Using the same notations as in Hassard et al. [10], we first compute the coordinates to describe the center manifold C 0 at μ = 0. Let y t = (y (1)  t , y (2)  t , y (3)  t , y (4) t ) T be the solution of system (4.1) when μ = 0. Define z(t) = q * (s), y t (θ) , (4.18) On the center manifold C 0 , we have where z and z are local coordinates for center manifold C 0 in the direction of q and q * .Note that W is real if y t is real.We only consider real solution.

Applications and numerical simulations
In this section, we will apply the results obtained in Sections 3 and 4 to investigate a special four-neuron BAM neural network with two delays and give numerical simulations supporting our theoretical analysis.
5.1.Application to a special BAM neural network.Consider the four-neuron BAM neural network with two delays described by the following functional differential equations: with μ > 0, c k1 (k = 2,3,4) > 0, and c 1l (l = 2,3,4) < 0. In addition, the activation function f satisfies the following condition: For instance, the nonlinear activation function commonly used in the studies and applications on neural network given by f (u) = tanh(u) posses the above property.Adopting the same notations as in Sections 3 and 4, for system (5.1),we have (5.2) From (3.5) and (3.6), one can get that (5.5)Thus, dh(z)/dz > 0 for all z > 0, that is, h(z) is strictly monotonous increasing on [0,∞).Noticing that d < 0, it follows that h(z) has only a positive root, say z 0 .Let ω 0 = √ z 0 , τ = τ 1 + τ 2 .From (2.16) and (5.3), we have By Theorem 3.1, we can get the following result.
(i) When τ ∈ [0,τ 0 ), all roots of the corresponding characteristic equation of (5.1) have negative real parts.Meanwhile, when τ = τ 0 , the corresponding characteristic equation of (5.1) has only a pair of simple purely imaginary roots ±iω 0 , and all other roots have negative real parts.However, when τ > τ 0 , the corresponding characteristic equation of (5.1) has at least one root with positive real part.