Bifurcation of a Cohen-Grossberg Neural Network with Discrete Delays

and Applied Analysis 3 in which a a1 0 b′ 1 0 a2 0 b ′ 2 0 , b a1 0 a2 0 , c b ′ 1 0 b ′ 2 0 , d d1d2f ′ 1 0 f ′ 2 0 . 2.4 If τ 0, it is easy to know that 2.3 has two roots with negative real parts when d < c. Equation 2.10 has at least one positive root when d > c. If τ / 0, let us assume that λ iω ω > 0 is a root of 2.3 ; we have bc −ω2 bd cosωτ, aω −bd sinωτ. 2.5 Adding up the squares of the corresponding sides of the above equation 2.5 , we obtain ω4 kω2 b2 ( c2 − d2 ) 0, 2.6 where k a1 0 b′ 1 0 2 a2 0 b′ 2 0 . Denote z ω2; then 2.6 becomes z2 kz b2 ( c2 − d2 ) 0. 2.7 Since k > 0, 2.7 has no positive real roots when |d| ≤ c. Equation 2.7 has a positive real roots when |d| > c, which means 2.3 has a pair of purely imaginary root for every |d| > c. We have from 2.5 that aω ω2 − bc tanωτ. 2.8 It has roots as follows 4, 12 : ω j ∈ [ 2 ( j − 1π τ , ( 2j − 1π τ ] , ω− j ∈ [( 2j − 1π τ , ( 4j − 1π 2τ ] , j 1, 2, . . . . 2.9 From 2.5 and the discussion of 2.7 , we obtain that d− j − aω j b sinω j τ < −b′ 1 0 b′ 2 0 < 0, d j − aω− j b sinω− j τ > b′ 1 0 b ′ 2 0 > 0, 2.10 that is, 2.3 has a pair of purely imaginary roots ± iω0 when d d± j . On the other hand, we have λ′ d −1 2λ a be−λτ τd ( 2λ a λ2 aλ bc τ ) d, 2.11 4 Abstract and Applied Analysis thus we have Re ( λ′ d −1 ) | d d± j ⎛ ⎜ ⎝ abc a ( ω± j )2 ( bc −ω± j )2 a2 ( ω± j )2 τ ⎞ ⎟ ⎠d± j ; 2.12


Introduction
In 1983, Cohen and Grossberg 1 proposed a kind of neural networks, which are now called Cohen-Grossberg neural networks. The networks have been successfully applied to signal processing, pattern recognition, optimization, and associative memories.
It is well known that the analysis of the dynamical behaviors is a necessary step for practical design of neural networks since their applications heavily depend on the dynamical behaviors. The research on dynamical behaviors of neural networks involves not only the dynamic analysis of equilibrium but also that of periodic solution, bifurcation, and chaos, especially, the periodic oscillatory behavior of the neural networks is of great interest in many applications 2, 3 . Since periodic oscillatory can arise through the Hopf bifurcation in different system with or without time delays, it is very important to discuss the Hopf bifurcation of neural networks. Up to now, to the best of the author's knowledge, bifurcation of Hopfield neural networks has been discussed by many researchers 4-12 , but only a few results on the bifurcation of Cohen-Grossberg neural networks have been obtained. Zhao discussed the bifurcation of a discrete-time Cohen-Grossberg neural network in 13 2 Abstract and Applied Analysis and the bifurcation of a continuous-time Cohen-Grossberg neural network with distributed delays in which kernel function is αe −αs in 14 ; Liu discussed the local Hopf bifurcation of the following Cohen-Grossberg neural network with discrete delays by regarding time delay as the bifurcation parameter in 15 : where x i t denotes the state variable associated with the ith neuron; a i · represents amplification function; f i · denotes the signal function of the ith neuron at time t; b i · is appropriately behaved function; d i · is connection weight of the neural network; discrete delay τ i corresponds to the finite speed of the axonal signal transmission at time t, i 1, 2.
On the other hand, realistic modelling of neural networks inevitably depend on careful design and variation of the connection weight; in 16 , Liu discussed the Neimark-Sacker bifurcation of a discrete-time version of the neural network 1.1 by regarding the connection weight as the bifurcation parameter. The objective of this paper is to discuss Hopf bifurcation of continuous-time system 1.1 by regarding the connection weight as the bifurcation parameter while time delays are fixed. The rest of this paper is organized as follows. Stability property, existence of Hopf bifurcation, and the stability of bifurcating periodic solutions for system 1.1 are obtained in Section 2. Global existence of Hopf bifurcation for system 1.1 is obtained in Section 3. An example is given in Section 4 to demonstrate the main results.

Stability Analysis and Existence of Local Hopf Bifurcation
Throughout this paper, we assume that H 2 there exist constants a i , a i such that 0 < a i ≤ a i · ≤ a i for i 1, 2; Let u 1 t x 1 t − τ 1 and u 2 t x 2 t ; we transform system 1.1 into the following systemu in which τ τ 1 τ 2 .
Obviously, system 2.1 has an equilibrium 0, 0 under conditions H 1 and H 2 .
The linearized system of system 2.1 at 0, 0 iṡ The associated characteristic equation of system 2.2 is Abstract and Applied Analysis 3 in which If τ 0, it is easy to know that 2.3 has two roots with negative real parts when d < c. Equation 2.10 has at least one positive root when d > c.

2.5
Adding up the squares of the corresponding sides of the above equation 2.5 , we obtain where k a 1 0 b 1 0 2 a 2 0 b 2 0 2 . Denote z ω 2 ; then 2.6 becomes Since k > 0, 2.7 has no positive real roots when |d| ≤ c. Equation 2.7 has a positive real roots when |d| > c, which means 2.3 has a pair of purely imaginary root for every |d| > c.

2.8
It has roots as follows 4, 12 : From 2.5 and the discussion of 2.7 , we obtain that that is, 2.3 has a pair of purely imaginary roots ± iω 0 when d d ± j . On the other hand, we have Abstract and Applied Analysis thus we have and note that at 3 has a root λ c 0, and λ c −1 a/b τ c > 0 according to 2.11 .
From the above discusses and Hopf bifurcation theorem in 17 for functional differential equations, we have the following results.
From discussion above, we obtain some conditions under which system 1.1 undergoes local Hopf bifurcation near d d ± j j 1, 2, . . . . In addition, we can obtain explicit formulas for determining the properties of the Hopf bifurcation at critical d d ± j . Based on the normal form theory and the center manifold theorem 18 , similar to discussion in 15 , we can compute the following quantities: 2.14 where the explicit formulas for g ij in 2.14 can be obtained when we replace τ j and ω 0 in g ij by τ and ω ∓ j in 15 , respectively. It is well known that μ 2 determines the direction of the Hopf bifurcation and β 2 determines the stability of the bifurcating periodic solutions 18 . Since Re{λ d j } > 0, Abstract and Applied Analysis 5 we know if μ 2 > 0 μ 2 < 0 , then the Hopf bifurcation is supercritical subcritical , the bifurcating periodic solutions exist for d > d j d < d j , and the bifurcating periodic solutions are stable unstable , and due to Re{λ d − j } < 0, we know if μ 2 < 0 μ 2 > 0 , then the Hopf bifurcation is supercritical subcritical , the bifurcating periodic solutions exist for d < d − j d > d − j , and the bifurcating periodic solutions are stable unstable . T 2 determines the period of the bifurcating periodic solutions: the period increases decreases if T 2 > 0 T 2 < 0 .
Remark 2.2. The main results above coincide with the main results in 16 .

Global Existence of Hopf Bifurcation for the Model
In this section, we mainly prove that the local Hopf bifurcation of 1.1 can be extended for large values by applying the global Hopf bifurcation theorem in 19 .   Proof. First, we can prove that system 1.1 has a unique equilibrium 0, 0 under conditions H 1 -H 4 when d < d − . The equilibria of system 1.1 satisfy the following equation according to condition From the first equation of 3.1 , we have u 1 b − 1 d 1 f 2 u 2 ; substitute it into the second equation of 3.1 ; we obtain and we have When d 1 d 2 < 0, obviously, g u 2 > 0 due to f i · > 0 according to H 4 . When d 1 d 2 > 0, we obtain that f i 0 max u i ∈R f i u i , i 1, 2, due to u i f i u i < 0 according to H 4 , combined 1 < 0; consequently, g u 2 > 0. Hence g u 2 is an increasing bijective function, since g 2 0 0, u 2 0 is a unique solution of 3.3 , and 3.1 has a unique solution 0, 0 , that is, 0, 0 is a unique equilibrium of system 1.1 when d < d − .
Second, let us prove the global existence of Hopf bifurcation of system 1.1 . Following the work of Wu 19 , we make the following definitions. Denote the equivalent system 2.1 of system 1.1 as follows: and ω j are given by 2.9 and 2.10 , respectively. From Theorem 2.1, we know that 0, d − j , 2π/ω j is nonempty, and all centers are isolated center. At fact, sets of centers are The characteristic function is continuous in d, p, λ ∈ R×R×R×C. Theorem 2.1 above ensures that there exist ε > 0, > 0 and a smooth curve λ : and we can compute the crossing number of every isolated center 0, d − j , 2π/ ω j as follows: 2π ω j , Ω ε −1, 3.10 hence we can obtain from Theorem 3.3 in 19 that the connected component 0, d − j , 2π/ ω j in Σ is unbounded.

Abstract and Applied Analysis 7
On the other hand, we can prove that all nontrivial periodic solutions of system 2.1 are uniformly bounded.

3.11
where k 1 max{a 1 b 1 , a 2 b 2 } and k 2 max{|d 1 |M 1 , |d 2 |M 2 }. It follows that Hence, all nontrivial periodic solutions of system 2.1 are uniformly bounded. At the same time, we can prove that the period p of a periodic solution of system 2.1 with d < d − on 0, d − j , 2π/ω j is also uniformly bounded. In fact, system has no nontrivial τ-periodic solution. For a contradiction, suppose that system 2.1 has a nontrivial τ-periodic solution, then the following differential equation also has a nontrivial τ-periodic solutioṅ On the other hand, let 3.14 Thus, we conclude that system 2.6 has no τ-periodic solution according to Bendixson-Dulac criterion. Consequently, system 2.1 has no τ/n-periodic solution for positive integer n ≥ 1. From definition of ω j , we have where m is a positive integer. Therefore, the projection of 0, d − j , 2π/ω j on p-space is bounded.

Abstract and Applied Analysis
Hence, we obtain that the projection 0, d − j , 2π/ω j on d-space must be unbounded. As a result, for any d < d − j , system 2.1 has at least a periodic solution with a period in τ/m, 2π/ω 1 and the proof of Theorem 3.2 is complete.
Remark 3.3. The discussions of bifurcations of system 1.1 at d b 1 0 b 2 0 are a very cockamamie case. It is difficult to obtain good conditions to guarantee the global existence of periodic solutions of system 1.1 , but if a 1 x 1 t a 2 x 2 t 1, b 1 t, x 1 t μ 1 x 1 t , and b 2 t, x 2 t μ 2 t x 2 t , system 1.1 reduces to the following Hopfield neural network:

3.16
A supercritical pitchfork of system 3.16 can occur at d μ 1 μ 2 under assumptions H 1 -H 4 11 , and the periodic solutions of system 2.2 are also globally existent 12 . Of course, Theorems 2.2 and Theorem 3.1 for 3.16 in 12 are special cases of Theorems 2.1 and 3.2 in this paper.