Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays

Abstract

Time delay is an inevitable factor in neural networks due to the finite propagation velocity and switching speed. Neural system may lose its stability even for very small delay. In this paper, a two-neural network system with the different types of delays involved in self- and neighbor- connection has been investigated. The local asymptotic stability of the equilibrium point is studied by analyzing the corresponding characteristic equation. It is found that the multiple delays can lead the system dynamic behavior to exhibit stability switches. The delay-dependent stability regions are illustrated in the delay-parameter plane, followed which the double Hopf bifurcation points can be obtained from the intersection points of the first and second Hopf bifurcation, i.e., the corresponding characteristic equation has two pairs of imaginary eigenvalues. Taking the delays as the bifurcation parameters, the classification and bifurcation sets are obtained in terms of the central manifold reduction and normal form method. The dynamical behavior of system may exhibit the quasi-periodic solutions due to the Neimark- Sacker bifurcation. Finally, numerical simulations are made to verify the theoretical results.

Introduction

Time delay is an inevitable factor in the signal transmission between biological neurons or electronic-model-neurons due to the finite propagation velocity and switching speed (Campbell 2007; Sun et al. 2010). The neural network with time delays can be described by a class of infinite-dimensional delayed differential equations. To study the effect of time delay on neural systems, Marcus and Westervelt (1989) improved the Hopfield neural network by introducing delay-coupling into the nonlinear activation function. Due to the existence of time delay, some complex dynamic phenomena can be easily obtained near the equilibrium point of neural system.

It is well known that neural system may lose its stability by making the equilibrium point unstable even for very small delays. However, effect of time delay is an interesting problem. For example, in field of machine tool industry, the disturbance of the pure regeneration (one single delay) by e.g. variable angle distribution between milling edges, time variation of the delay can increase the stability of the cutting process (Budak 2003; Zatarain et al. 2008). A small delay is sufficient to destabilize a system, but a large delay may stabilize a system (Liao and Lu 2011; Ye and Cui 2012). This implies that time delays can induce system to exhibit the multi-stable regions, which is called the delay-dependent stability regions. The system behavior switches from being stable to unstable and then back to stable as delay increases. As early as 1980s, Cooke and Grossman (1982) constructed a delayed system with at least N times switching. Hale and Huang (1993) investigated a linear differential equation and obtained some global geometrical characteristics of the stability regions located in two-delayed plane. For the general linear delayed systems consisting of multiple delays, the stability regions have been shown in two-delayed plane (Gu et al. 2005, 2007) and three-delayed space (Sipahi and Delice 2009; Gu and Naghnaeian 2011; Sipahi et al. 2011). Multiple delays can induce the stability switching of linear dynamical systems.

Bifurcation is one of the most important dynamical phenomena for the nonlinear neural systems (Jia et al. 2012; Liebovitch et al. 2011). Different bifurcations can lead to different kinds of dynamic behaviors. For example, Hopf bifurcation gives rise to the periodical activity, while double Hopf bifurcation results in two types of oscillations with different frequencies and a quasi-periodic behavior. Hopf bifurcation and periodical activity for neural network models with one single delay have been investigated extensively, see, e.g., Liao et al. (2007), Yuan and Li (2010), Guo and Huang (2003), Yan (2006) and the references cited therein. Due to the complexity of multiple delays, the analysis of stability and Hopf bifurcation for the multi-delayed differential equations is far from complete. Many researchers tried to fill in some “piece of the puzzle” for the multiple delays problem (Li et al. 1999). Considered the average/sum of time delays as the variable parameter and used the time-scale transformation, some multiple-delayed neural network systems with simple architecture can be transformed to the equivalent models with one single delay (Cao and Xiao 2007; Mao and Hu 2008; Wei and Zhang 2008).

In order to get a deep and clear understanding for multiple delays on dynamical behaviors, many researchers focused on the neural network systems with two delays, which are relatively simple. The stability and Hopf bifurcation for some special system with two delays have been investigated, such as neural system (Xu et al. 2006; Xu 2008), Lotka-Volterra predator–prey system (Nakaoka et al. 2006; Song et al. 2004), epidemic model (Cooke et al. 1996), tumor growth (Shi et al. 2011; Xu 2009), plastic deformation instability (Hilout et al. 2010), Mackey–Glass system (Li and Jiang 2011; Wan and Wei 2009), and so on. In 1994, Bélair and Campbell (1994) considered a simple motor control system with two negative delayed feedback loops. The stability regions of the equilibrium point are analyzed in the parameter plane for the fixed delay τ₁. In addition, the criticality of the Hopf bifurcation was considered in detail in terms of some numerical simulations. The result shows that the first order differential equation containing two delays can exhibit the double Hopf bifurcation. Similar analysis for the neural network with multiple delays can be referred in (Shayer and Campbell 2000; Yuan and Campbell 2004; Campbell et al. 2006). This approach can obtain a “piecewise global” perception for the bifurcation diagram in the multi-delayed system. However, to the best of our knowledge, in the existing relative literatures, there are few researchers chosen the multiple delays as the independent parameters for the dynamics analysis. The mutual effect of multiple delays on system dynamical behavior has been not considered yet. It is our research motivation in the present paper. In terms of the stability analysis of the corresponding linear part, the bifurcation points with high co-dimensional singularity and the bifurcation diagram are presented in the two-delayed plane for the neural network system.

To study the two-delayed effects on neural system in detail, in this paper, a codimension-two singularity, namely the double Hopf bifurcation, is analyzed employing the multiple delays as the bifurcation parameters. Various dynamical behaviors are classified in the neighborhood of the singularity point. The bifurcation sets consisting of the oscillation behaviors with the different frequencies and quasi-periodic state are obtained in the delay parameter plane in terms of the central manifold reduction and normal form method (Orosz and Stepan 2004; Dombovari et al. 2008), which was introduced firstly by Faria and Magalhaes (1995a, b). The double Hopf bifurcation analysis for the differential equation with one single delay can be found in some existing works, such as (Yu et al. 2002; Xu et al. 2007; Xu and Pei 2008). This approach provides a convenient tool to compute a relatively simple form of the original differential equation, which can be used to analyze the system dynamic behaviors.

Neural system considered in this paper is described by the following differential equation:

1

where x(t) and y(t) represent the neural behaviors at time t, a₁, a₂, a₃ and a₄ denote the coupled weights, τ₁ and τ₂ are the signal transmission delays for the self- and neighbor-connection, respectively, P and Q are the external inputs, the neuron activation function in S(u) is given by .

For neural network (1), when the self-connection delay τ₁ = 0, our previous works have illustrated that system exhibits the complex dynamical behaviors for the different values of time delay and external input (Song and Xu 2009). The types of equilibrium points are studied analytically in details in terms of the characteristic equation and static bifurcation. The central manifold reduction and normal form method are employed to determine the Hopf bifurcation and its stability. In addition, time delay can affect the existence of the Bogdanov-Takens bifurcation (refer to Song and Xu 2012a). For some delayed interval, system may exhibit the saddle-node homoclinic bifurcation. Furthermore, when the variation slope ratio is proposed to the sigmoid activation function S, the dynamical behaviors of system (1) with τ₁ = τ₂ = τ may exhibit the chaotic behavior (Song and Xu 2012b).

The paper is organized as follows. Firstly, the local asymptotic stability of equilibrium point for system (1) is investigated by analyzing the corresponding characteristic equation of the linearized system. General stability criteria involving the time delays are obtained. The results show that stability switches may occur for some interval of delay τ₁ when τ₂ increases. In section “Existence of double Hopf bifurcation”, the stability regions and Hopf bifurcation curves are illustrated in the delay (τ₁, τ₂) parameter plane. The double Hopf bifurcation points can be obtained from the intersection points of the first and second Hopf bifurcation, i.e., the characteristic equation has double imaginary eigenvalue with multiplicity two. In section “Central manifold reduction”, regarding two delays as the bifurcation parameters, the typical dynamic behaviors near the double Hopf bifurcation are investigated in detail in terms of the central manifold reduction and normal form method. The system may exhibit the quasi-periodic solutions due to the Neimark-Sacker bifurcation. Finally, numerical simulations are made to verify the theoretical results. Conclusions are given in the last section.

Linear stability switches

The equilibrium point (x₀, y₀) for system (1) is the solution of the following equations:

2

To translate (x₀, y₀) to the trivial equilibrium point, letting x → x−x₀ nd y → y−y₀ one has the linearized equation of system (1) given by

3

where , . The corresponding characteristic equation of system (3) is

4

namely

5

It is well known that the equilibrium point is locally asymptotically stable if and only if each of eigenvalues has negative real parts. It follows that the boundary of the stability region can be determined by equations λ = 0 and λ = iω (ω > 0). However, from the dynamical bifurcation theory, the typical static bifurcation is exhibited as the parameters varying near λ = 0. That is to say, system exhibits the different number of equilibrium point when the eigenvalue passes through the imaginary axis along the real axis (Song and Xu 2012a, b). However, in this paper, we just investigate the stability switches induced by multiple delays. Since time delay can not change the number of the equilibrium point, in the following section, the purely imaginary eigenvalues λ = iω (ω > 0) for the characteristic equation (5) are only studied. For simplify, the investigation begins with the case τ₂ = 0 in (5) as follows

6

Without loss of generality, letting τ₁ = 0 in (6) implies

7

It follows from the Routh-Hurwitz criterion that the necessary and sufficient conditions for all roots of (7) having negative real parts is given by

8

and

9

This implies that the equilibrium point (x₀, y₀) is locally asymptotically stable if the parameters are satisfied with the conditions of (8) and (9). With the time delay τ₁ varying, system (1) will lose the stability. To obtain such critical values of time delay, supposing λ = iv (v > 0) is a pair of purely imaginary roots of (6), one has

10

Separating (10) into the real and imaginary parts yields

11

Taking the square, adding the equations and performing some simplification processes, we have

12

where

Without loss of generality, letting (12) has a number of positive and simple roots v_i = 1, 2, …, one has the following critical delays

13

where and satisfied with

14

In order to determine whether the real part of eigenvalue increases or decreases when delay τ₁ crosses the critical values .

Differentiating λ with respect to τ₁ in (10) reaches

15

Moreover, as shown in Hu and Wang (2002), one obtains

16

Therefore, based on the conditions of (8) and (9), one obtains the effect of delay τ₁ on the eigenvalue of characteristic equation (6) by using the transversality condition (16) and the Hopf bifurcation theorem of functional differential equation. When the polynomial L(v) has no positive real root, all eigenvalues of the characteristic equation (6) have negative real parts for the arbitrary τ₁. If L(v) has only one positive and simple root (not multiple root) v₀ satisfied with , there exists the critical delayed value such that all the eigenvalues of the characteristic equation (6) have negative real parts for and at least one root has a positive real part for . Furthermore, if L(v) has two positive and simple roots 0 < v₁ < v₂ satisfied with and , there exist a finite number of the delayed τ₁ intervals in which all eigenvalues of the characteristic equation (6) have negative real parts. However, when L(v) has at least three positive and simple roots 0 < v₁ < v₂ < v₃ < …, the distribution of eigenvalues in the complex plane is very complexity.

In order to investigate the effect of multiple delays on the local stability of equilibrium point, we regard τ₂ as the varying parameter for any fixed delay τ₁. Letting λ = iω (ω > 0) is the simple root of characteristic equation (5), one obtains

Separating the real and imaginary parts gives

17

Eliminating τ₂ from (17), one has

18

where

If (18) has a number of positive and simple roots ω_i, i = 1, 2, …, then (5) has a series of critical delays

19

where and satisfied with

20

To make sure the occurrence of the Hopf bifurcation, it is needed to check the transversality condition. Without loss of generality, the time delay τ₂ is chosen as the bifurcation parameter. The necessary condition for the existence of the Hopf bifurcation is that the critical eigenvalues cross the imaginary axis with non-zero velocity. Differentiating λ with respect to τ₂ in (5), one reaches

21

It follows that

22

Based on the transversality condition (22) and the Hopf bifurcation theorem, one has the following assertions when each eigenvalue of the characteristic equation (6) has negative real part. If (18) has no positive root, the equilibrium point of system (1) is locally asymptotically stable for the arbitrary delay τ₂, which is called the delay-independent stability. On the other hand, if (18) has one simple positive root ω₀ satisfied with , there exists only a critical delay value denoted by τ_c. All eigenvalues of system (1) have strictly negative real parts for , while at least one of eigenvalues with positive real part for . The equilibrium point of system loses its stability via a Hopf bifurcation. The direction and stability of the Hopf bifurcation can be studied by means of the central manifold reduction and normal form method (Song and Xu 2012c). Furthermore, if G(ω, τ₁) has two positive and simple roots 0 < ω₁ < ω₂ satisfied with and , there exist a finite number of the delayed τ₂ intervals in which all eigenvalues of the characteristic equation (5) have negative real parts. It implies that there exist a finite number of delayed intervals. If time delay is fixed into these intervals, the equilibrium point is locally asymptotically stable, while instable for the outside of the delayed ranges. That is to say, the system dynamic switches from stable to unstable, and then back to stable when delay increases and crosses the critical delayed values.

For example, we choose the coupled weights and external inputs as a₁ = −6, a₂ = 2.5, a₃ = 2.5, a₄ = −6, P = 0.4 and Q = 0.4, respectively. The time delays of self- and neighbor-connection, namely τ₁ and τ₂ are considered as the variable parameters. The figures of function G (18) and eigenvalue real parts of (5) are displayed in Fig. 1 for the different self-connection delay τ₁, where the eigenvalue real parts are obtained with the aid of the numerical tool DDE BIFTOOL (Engelborghs et al. 2002). Figure 1a shows that the curve determined by function G and the line G = 0 have not intersection point when ω > 0 holds. It implies that function G (18) exhibits no positive real root for the fixed delay τ₁ = 1. The real parts of all eigenvalues have the negative part, as shown in Fig. 1b. The equilibrium point of system (1) is locally asymptotically stable for the arbitrary delay τ₂. With the self-connection delay τ₁ increasing to 1.5, the curve determined by function G moves down and crosses the line G = 0. Function G (18) has two positive roots 1.2827 and 1.523, as shown in Fig. 1c. It follows from Fig. 1d that there exist the delay τ₂ intervals in which the eigenvalues have the negative parts. The system dynamics switch from stable to unstable, and then back to stable state with delay τ₂ increasing. It implies that if self-connection delay τ₁ is fixed, we can adjust the neighbor-connection delay τ₂ into the stability regions to suppress system vibration. When delay τ₁ is increased to 8, function G has two pairs of positive roots, as shown in Fig. 1e. However, at this time, the maximum eigenvalue of (5) for the fixed delay τ₁ = 0, i.e. the characteristic equation (6) has positive real part . The equilibrium point of system (1) is unstable for any delay τ₂, as shown in Fig. 1f.

Fig. 1.

Roofs of function G (left column) and eigenvalue real parts with τ ₂ varying (right column) for the fixed self-connection delay a–b

, c–d and e–f , respectively. The other parameters are chosen as a ₁ = −6, a ₂ = 2.5, a ₃ = 2.5, a ₄ = −6, P = 0.4, Q = 0.4

To get a deep investigation for the stability switches of equilibrium point, we fix the self-connection delay τ₁ = 1.5. The partial eigenvalues and phase portraits are shown in Figs. 2 and 3 respectively for the different delay τ₂. Figure 2a shows the maximum eigenvalues are a conjugate pair with negative part given by −0.0295 ± 1.5705i, where the neighbor-connection delay is fixed as τ₂ = 3. It implies that the equilibrium point of system (1) is locally asymptotically stable, as shown in Fig. 3a. Letting τ₁ = 1.5 and varying τ₂ yield that the conjugate pair start to pass through the imaginary axis and go into the right-half in the complex plane. For τ₂ = 4, the pair become to be 0.0256 ± 1.4012i shown in Fig. 2b, which suggests that the equilibrium point loses its stability (Fig. 3b). Continuing to increase τ₂, the conjugate pair return and pass through the imaginary axis again. The maximum eigenvalues with negative real parts occurs in the left-hand of complex plane. Figure 2c shows the eigenvalues are −0.00936 ± 1.2591i for the fixed delay τ₂ = 5. The equilibrium point of system is stabilized again by the neighbor-connection delay τ₂, as shown in Fig. 3c. If the delay τ₂ is fixed as τ₂ = 6, the maximum eigenvalues are conjugate pair with positive part 0.0199 ± 1.4311i, as shown in Fig. 2d. The equilibrium point loses its stability again (Fig. 3d). In a word, time delay can lead system to exhibit multi-stable regions. The dynamics of equilibrium point can multi-switch from stable to unstable, and then back to stable state.

Fig. 2.

Distribution of eigenvalues with time delay τ ₂ varying a , b , c and d for the fixed parameters a ₁ = −6, a ₂ = 2.5, a ₃ = 2.5, a ₄ = −6, P = 0.4, Q = 0.4, , where the asterisk in green represents the eigenvalues with negative real part and one with positive real part is in red color

Fig. 3.

Phase portraits with time delay τ ₂ varying a , b , c and d for the fixed parameters a ₁ = −6, a ₂ = 2.5, a ₃ = 2.5, a ₄ = −6, P = 0.4, Q = 0.4,

Existence of double Hopf bifurcation

It follows from above section that the multiple roots of equation (18) can lead system (1) exhibit the phenomenon of stability switches. The dynamical behavior of equilibrium point undergoes the multiple transitions between stability and instability when delays increase and cross the critical delayed values. In this section, we will illustrate these critical values and find the stability regions located in the delayed plane. To this end, the Hopf bifurcation curves represented by delay parameters are investigated. The coordinate values of the intersection points of these curves are presented, which are the points of the double Hopf bifurcation.

As mentioned above, two pairs of purely imaginary solutions of equation (5) are denoted by , with ω₂ > ω₁ > 0. Then two series of the critical delays corresponding to such pairs of purely imaginary eigenvalues are given by

23

where ,

and

24

where ,

It follows from (21) that the sign of at , and , is difficult to determine analytically in terms of the expression of ω_1,2. But it can be easily computed numerically. In fact, numerical results show that the quantity of is negative and is positive. This implies that the crossing of the imaginary axis is from left to right as τ₂ increases to a certain critical value corresponding to ω₂, and crossing from right to left occurs at a certain values corresponding to ω₂.

For instance, if holds for some certain intervals of τ₁, then the unstable equilibrium point becomes stable as τ₂ is increased to cross the critical value . When τ₂ is further increased to pass through the critical value , the equilibrium point loses its stability again, where a pair of eigenvalues cross the imaginary axis and arrive to the right half plane. The occurrence and termination of stability switches for the equilibrium point are located at the points of the first and second Hopf bifurcation corresponding ω₁ and ω₂ frequencies, respectively. Similarly, higher order stability regions may be located by for 1, 2, …, resulting in a possibility of multiple stability regions.

Furthermore, the double Hopf bifurcation may occur when the two critical delays are identical, which correspond to the different frequencies ω₁ and ω₂. Geometrically, the double Hopf bifurcation points are the intersections of the first and second Hopf bifurcation curves. Owing to the multiple delays of system (1) considered in the present paper, more complicated expressions have been derived for the critical delays. The possible points of the intersection for the Hopf bifurcation curves cannot be given in a theoretical form. But for a given set of the system parameters, the double Hopf bifurcation points can be solved by the numerical procedure.

As an illustrative example, consider a specific system with the fixed parameters a₁ = −6, a₂ = 2.5, a₃ = 2.5, a₄ = −6, P = 0.4 and Q = 0.4, which are analyzed in section “Linear stability switches”. The delays τ₁ and τ₂ are taken as the variable parameters. It follows from (2) that system (1) has only unique equilibrium point given by . The frequencies of the first and second Hopf bifurcation can be obtained in terms of (18) for each fixed value of delay τ₁. For example, chosen time delay as τ₁ = 1.8, one has the frequencies and , respectively. The equilibrium point switches to stable state τ₂ when increases to corresponding to the first Hopf bifurcation, and remains its stability until τ₂ reaches relating to the second Hopf bifurcation. Similarly, the equilibrium point is locally asymptotically stable if the delay is fixed into the range of , which is the second order stability region. In such a way, the equilibrium point switches its stability and instability a finite number of times. For each fixed delay τ₁, based on (23) and (24), a series of critical curves of the stability regions can be exhibited in the (τ₁, τ₂) parameter plane, as shown in Fig. 4. These curves divide the parameter plane into several stable and unstable regions. The stability regions are indicated by , where j stands for the order of the stability regions.

Fig. 4.

Variation of the critical delay values τ ₂ with τ ₁ in the (τ ₁, τ ₂) parameter plane for the fixed parameters as a ₁ = −6, a ₂ = 2.5, a ₃ = 2.5, a ₄ = −6, P = 0.4 and Q = 0.4, respectively, where denote the stability regions of equilibrium point (x ₀, y ₀) and P_j, j = 0, 1, 2, … are the double Hopf bifurcation points

It follows from Fig. 4 that the curves of the first and second Hopf bifurcation intersect at the point . The frequencies corresponding to the Hopf bifurcation are respectively and , as shown in Fig. 5. It implies that the intersection point is a non-resonate double Hopf bifurcation, which is usually referred as the codimension-two bifurcation. This singularity point is a source of more complicated dynamics, such as the stable quasi-periodic behavior and unstable solution. Similarly, the higher order double Hopf bifurcation points and the corresponding frequencies can be obtained in the (τ₁, τ₂) parameter plane, which are shown in Fig. 4 and Table 1.

Fig. 5.

a Roots of the characteristic equation in the complex plane for and b enlargement near (0, 0) corresponding to a, where the other parameters are fixed as a ₁ = −6, a ₂ = 2.5, a ₃ = 2.5, a ₄ = −6, P = 0.4, Q = 0.4

Table 1.

Coordinate values of the double Hopf bifurcation points and the corresponding frequencies

	ω 1	ω 2
(2.8467, 1.1767)	0.7314	0.9735
(1.9473, 3.5226)	1.0052	1.2985
(1.6481, 5.4804)	1.1682	1.4474
(1.5225, 7.4198)	1.2630	1.5121
(1.4584, 9.3768)	1.3222	1.5422

Central manifold reduction

It follows from the previous section that multiple delays of system (1) induce two different values of frequencies corresponding to the first and second Hopf bifurcation. The existence of a double Hopf bifurcation can be detected by the local stability analysis. The intersections of these Hopf bifurcation curves are the double Hopf bifurcation points. In this section, we represent the typical dynamical behaviors and the bifurcation sets in the neighborhood of the double Hopf bifurcation point by means of the central manifold reduction and normal form method (Buono and Belair 2003; Faria and Magalhaes 1995a, b). The periodic solutions corresponding to the different frequencies and the stable quasi-periodic solutions resulting from the double Hopf bifurcations point are investigated in the system under consideration. Introducing the transformation yields that system (1) becomes

25

where . For the rescale system defined by system (25), the two Hopf bifurcation frequencies, namely and , are now given by and , respectively. Setting and yields that system (25) can be rewritten as system (26).

26

Regarding the time delays τ₁ and r as the bifurcation parameters, we let

where and are the unfolding parameters. It follows that

27

Then system (25) becomes

28

To apply the central manifold reduction, it is necessary to changes system (28) into a functional differential equation (Hale 1977). Let as the Banach space of continuous functions from to R² with the supremum norm, where . For any , we define

29

where is a real-valued function of bounded variation in with

30

where , , and δ(θ) is Dirac function. Further, we define

31

where

32

and

For , the linear operator defined by system (28) generates an infinitesimal generator of the semi-flow of bounded linear operators with

33

and

34

Set

35

where

36

and

Thus, system (28) becomes

37

where ,. For , the adjoint operators and of and , respectively, are given by

38

and

39

For and , we introduce the bilinear form

40

From the above discussion, we know that system (25) has two pairs of purely imaginary eigenvalues and the other eigenvalues with negative real parts at the double Hopf bifurcation point. Therefore, the Banach space C can be split into two subspaces as , where is the four-dimensional center subspace spanned by the basic vectors of the linear operator associated with the imaginary characteristic roots, and is the complement subspace of (Faria and Magalhaes 1995a, b). Now, We suppose and are the eigenvectors of and , respectively, corresponding to eigenvalue and , , i.e. , and . By the direct computation, we have

where

Furthermore, it follows that the real bases for and its dual space can be expressed as (Chen and Yu 2006)

41

and

42

respectively. Note that the factor is required for normalizing the linear part of transformed equation. It is easily verified that and , where and

43

Define

44

which actually represents the local coordinate system on the four-dimensional center manifold, induced by the basis Ψ. It follows (41) and (42) that one can decompose v_i into two parts given by

45

which implies that Φu is the projection of v_i on the center manifold. Substituting (45) into (37) and applying the bilinear operator (37) with Ψ given by (37) to the resulting equation yields

46

With the aid of (45), one obtains

47

where B is determined by (43). It should be noted that and are very small in the neighborhood of . This implies that

Executing the standard normalization technique as (Xu and Pei 2008; Buono and Belair 2003), one obtains the amplitude equation coming from the polar normal form of the central manifold system given by (Guckhenheimerm and Holmes 1983)

48

As mentioned above, chosen the system parameters as , the double Hopf bifurcation points are illustrated in Fig. 4 and Table 1. For convenience, we only investigate the first order critical delayed value of the double Hopf bifurcation given by . The same procedure can be applied to the other points. Substituting the parameter values into system (48), one obtains the corresponding coefficients given by , respectively. From the dynamical theory (Guckhenheimerm and Holmes 1983), the curves of the secondary bifurcation are represented as and in the (μ₁, μ₂) plane. To analyze the dynamical behaviors of system (1) in parameter (τ₁, τ₂) plane, the bifurcation curves in terms of τ₁ and τ₂ can be obtained by virtue of (27). The dynamical classification near the first order point of double Hopf bifurcation is shown in Fig. 6, which is called the two-parameter bifurcation diagram. It should be noticed that the fixed point with r₁ = r₂ = 0 in system (48) corresponds to the equilibrium point of the origin system. Possible fixed point in the invariant coordinate axes for (48) with r₁ = 0 or r₂ = 0 correspond to limit cycle of the origin system, while a nontrivial fixed point with r₁ > 0 and r₂ > 0 generates a two-dimensional tours (quasi-periodic solution with two incommensurate frequencies). The stability of all these solutions in the original system is clearly detectable from that of the corresponding object in (48), as shown in Fig. 6.

Fig. 6.

Classification and bifurcation sets for system (1) near the first order point of the double Hopf bifurcations in the parameter (τ ₁, τ ₂) plane. The other parameters are chosen at the same as those in Fig. 4

Numerical simulations

To illustrate the theoretical works of determining the double Hopf bifurcation properties, in this section, we present some numerical simulation to verify the system solutions. As mentioned above, we kept on choosing the parameters as a₁ = −6, a₂ = 2.5, a₃ = 2.5, a₄ = −6, P = 0.4 and Q = 0.4, respectively. The time delays τ₁ and τ₂ are considered as the variable parameters. Fig. 7 shows the one-parameter bifurcation diagrams with varying delay τ₂ for fixed delay τ₁. Phase portraits of the typical behaviors near the double Hopf bifurcation point are illustrated in Fig. 8, respectively.

Fig. 7.

One-parameter bifurcation diagram with varying the delay τ ₂ for a τ ₁ = 2.5, τ ₂ ∈ [0.7,1], b τ ₁ = 2.5, τ ₂ ∈ [1.05, 1.3], c τ ₁ = 3.5, τ ₂ ∈ [1.1, 1.8], and d the partial enlargement of the c box, respectively. The other parameters are chosen at the same as those in Fig. 4

Fig. 8.

Phase portraits of the numerical simulation for the dynamical behavior in system (1) near the first order point of the double Hopf bifurcation, where (τ ₁, τ ₂) is fixed as a (2.0, 0.8), b (2.5, 1.6), c (3.5, 1.6), d (3.5, 1.64), e (3.5, 1.4), and f (3.5, 0.8), respectively. The other parameters are chosen at the same as those in Fig. 4

Firstly, two delays are fixed as located in region 1 of Fig. 6. The numerical solution of system (1) is shown in Fig. 8a. The trajectory asymptotically converges to the equilibrium point . It implies the point is locally asymptotically stable. With the increasing of delay τ₂, the equilibrium point loses its stability in term of the supercritical Hopf bifurcation, where the parameters encounter the second Hopf bifurcation curve and enter region 2 of Fig. 6. The system periodically oscillates with the second frequency, as shown in Fig. 8b. The one-parameter bifurcation diagram with increasing τ₂ is shown in Fig. 7b. The oscillation amplitude of system is increasing with delay τ₂ varying. However, when delay τ₂ is decreased from region 1 of Fig. 6, the parameters will encounter the first Hopf bifurcation curve related to ω₁ frequency. This implies that the equilibrium point loses its stability through a reverse Hopf bifurcation, as shown in Fig. 7a. It follows that this bifurcation is also supercritical Hopf bifurcation. The system is exhibited the oscillation solution with ω₁ frequency as it enters into region 6, as shown in Fig. 8f.

Furthermore, when we choose delay , the one-parameter bifurcation diagram with varying τ₂ is shown in Fig. 7c, where Fig. 7d is the partial enlargement of the box in Fig. 7c. Figure 8c and e respectively show the stable state of periodical solutions with the different frequencies. With delay τ₂ varying from region 3–4, the stable periodic solution loses its stability through the Neimark-Sacker bifurcation. It suggests that the system behavior moves to a two-frequency quasi-periodic state, as shown in Fig. 8d. The phase portrait clearly shows the modulation of the peak intensities, which is also called as 2-torus. Keeping delay τ₂ vary, when parameters encounter and cross the other secondary bifurcation, the system behavior transforms into the periodic activity again from the 2-torus solution. The analytical predictions of system solutions are good representatives of the numerical results.

Conclusion

In neural system, action potential plays a crucial role in many information communications. To understand the information representation, many mathematical models are proposed and the mechanism of information processing is investigated by using the analytical method of nonlinear dynamics. The quiescent state, periodic spiking, quasi-periodic behavior and bursting activity are all the important biological behavior with the different neuro-computational properties (Song and Xu 2012a; 2012b). The action potential exhibits the quasi-periodic behavior when the number of incommensurable frequencies is finite, which is called the multi-frequency rhythmic activity. Their power spectra are discrete with peaks corresponding to the composed frequencies (Izhikevich 1999). In fact, EEG recordings of brain can demonstrate the rhythmic activity with a few pronounced frequencies. For example, the most prominent are gamma (30–100 Hz) oscillations in the cortex and theta (4–8 Hz) oscillations in the hippocampus (Llinas 1988). Recently, Marichal et al. (2010) considered the dynamic behavior of a quasi-periodic orbit obtained by the Neimark-Sacker bifurcation in a simple discrete recurrent neural network model.

It is well known that time delay is an inevitable factor in the signal transmission between biological neurons or electronic-model-neurons. Neural systems with time delays have very rich dynamical behaviors. In this paper, a neural network with two different types of delays involved in self- and neighbor-connection has been studied. The occurrence of the multiple delays greatly complicates the analysis of stability and bifurcation. The stability regions are strongly relied on time delays. For the different values of delays, the system may exhibit the different parameter regions involved in the delay-independence stability and delay-dependence stability. In special, when the characteristic equation of the system has multiple roots, there exists the delay window, where the equilibrium point is either locally asymptotically stable. To illustrate these critical values and find the stability regions, the Hopf bifurcation curves are illustrated. The multiple stability regions are obtained in the delayed parameter plane. The transverse direction of the imaginary eigenvalues located at the critical values is determined by means of the numerical computation.

Furthermore, the system exhibits the double Hopf bifurcation points due to the two pairs of imaginary eigenvalues appearing on the margin of stability regions simultaneously. Considering the multiple delays as the bifurcation parameters, the classification of the various dynamic behaviors in the neighborhood of the double Hopf bifurcation point is obtained in detail in term of the central manifold reduction and normal form method. The system may exhibit the equilibrium solution, periodic solutions with the different frequencies of the first and second Hopf bifurcations, and quasi-periodic solutions. In addition, the quasi-periodic solution is obtained by the Neimark- Sacker bifurcation of the periodic solutions. Numerical results are given to illustrate that the double Hopf bifurcation is an interaction of the supercritical–supercritical Hopf bifurcations. Finally, the one-parameter bifurcation diagrams and phase portraits verify the agreement between the theoretical analysis and numerical simulations.