Stability switches and fold-Hopf bifurcations in an inertial four-neuron network model with coupling delay

Neurocomputing 110 (2013) 70–79

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Stability switches and fold-Hopf bifurcations in an inertial four-neuron network model with coupling delay Juhong Ge a,b, Jian Xu b,n a b

Department of Mathematics and Information Science, He’nan University of Economics and Law, Zhengzhou 450002, PR China School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 February 2012 Received in revised form 4 June 2012 Accepted 26 August 2012 Communicated by Long Cheng Available online 8 January 2013

The dynamics of a four-neuron delayed bidirectional associative memory (BAM) model with inertia are investigated. Local stability for the trivial equilibrium is analyzed for various system parameters. Stability switches and fold-Hopf bifurcations are found to occur in this model as progressive increasing of coupling delay values. Fold-Hopf bifurcations are completely analyzed in the parameter space of the coupling delay and the connection weight by employing the extended perturbation-incremental scheme. Various dynamical behaviors are qualitatively classified in the neighbor of fold-Hopf bifurcation point and bifurcating periodic solutions are expressed analytically in an approximate form. The validity of the results is shown by their consistency with the numerical simulation. & 2013 Elsevier B.V. All rights reserved.

Keywords: Inertial neural network Time delay Stability switches Fold-Hopf bifurcations

1. Introduction In the last few decades, modeling of biological neural systems has received increasing attention due to their wide and important applications including classification [1], optimization [2–4], pattern recognition [5,6] and so on. And these applications depend heavily on the network’s dynamics. Some researchers have used chaotic dynamics to overcome the drawback of being trapped in local minima for standard neural networks [7]. And the inertia can be considered as an effective tool to help in the generation of chaos in neural systems. Babcock and Westervelt [8,9] introduced inertial terms into simple deterministic models involving one or two nonlinear threshold switching elements, which they modeled as coupled Hopfield neurons and explored chaotic behaviors. In the paper by Wheeler and Schieve [10], a two-neuron system with one or two inertial terms added is shown to exhibit chaos. There are also strong biological supports for the addition of an inertial term to a standard neural equation. For example, Koch put forth the notion that under certain conditions neurons exhibit a quasi-active membrane, which can be modeled by a phenomenological inductance that allows the membrane to behave like a bandpass filter, enabling electrical tuning, or temporal differentiation, or spatio-temporal filtering [11]. The membrane of a hair cell in the semicircular canals of some animals can be described by equivalent circuits that contain an inductance [12,13]. It is

*

Corresponding author. Tel.: þ 86 21 6598 5364; fax: þ 86 21 6598 3267. E-mail addresses: [email protected] (J. Ge), [email protected] (J. Xu).

0925-2312/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.08.048

known that the squid axon can be modeled as having a phenomenological inductance [14], which is the second derivative of voltage with respect to time. The constructed circuit network will include an inertial term, i.e., inductance. Such network may describe biological neural network to be more accurate. Therefore, it is very useful and significant to introduce an inertial term (the influence of inductance) into the standard neural system. In real systems, time delay is ubiquitous and common due to finite propagation speeds of signals and finite reaction times. Unavoidably, neural networks also incorporate time delays in the signal transmissions among neurons because of finite propagation velocity of action potentials and non-negligible time of a signal from a neuron to the receiving site of a postsynaptic neuron [15]. It is well known that time delay usually plays a very significant role in the network dynamics. Time delay is often regarded as a source of instability and oscillations of the network and can lead to rich and complicated behavior, such as periodic oscillations, multi-stability, quasi-periodic oscillations, and even chaotic phenomena [16–29]. More recently, dynamics of delayed neural networks with inertia have begun to receive initial research interests. Li et al. [30] considered the single inertial neuron model with time delay. Liu et al. [31] investigated the stability of bifurcating periodic solutions for a single delayed inertial neuron model under periodic excitation. The complete behaviors for an inertial one or two-neuron system with time delay were worked out by Liu et al. [32]. In most of the existing literature on theoretical studies of delayed artificial neural networks, it is easy to see that the works

J. Ge, J. Xu / Neurocomputing 110 (2013) 70–79

focused on stability and Hopf bifurcation analysis. Investigation on inertial neural models only considered lower dimensional systems with or without time delay. Actually, the dimensions of inertial neural systems are no more than two. However, the relationship between dynamics and system parameters has not been clarified yet, which makes it difficult to apply them to engineering problems. Particularly, studies on codimension-2 bifurcations of delayed systems are very seldom although they are very important in practical applications. Moreover, by now the most popular approach that has been used to perform such studies is the normal form method and the center manifold theorem. But this approach is difficult to analyze codimension-2 bifurcation of the four-dimension delayed system with inertia due to its tedious calculation with normal form. All of these mentioned above construct our present research motivation. To the best of our knowledge, stability switches and foldHopf bifurcation of delayed neural systems with inertia have not been investigated elsewhere. In this paper, we try to find out some critical conditions under which the stability switches and fold-Hopf bifurcations occur when the parameters of the network model pass through some critical values. The results obtained in this paper give an explicit view of the dynamics of the four-neuron delayed neural network with inertia based on bifurcation analysis from a simple and effective method, rather than the normal form method and center manifold theorem and clarify the relationship between two system parameters and dynamical behaviors for parameter values near the degenerate bifurcation point, which is more convenient to be applied in practice. In this paper, similar to the model in Ref. [33] but with the inclusion of inertial terms, the inertial four-neuron model is considered, which can be modeled by 8 v€ 1 ¼ v_ 1 m1 v1 þ a1 f ðv3 ðttÞÞ þ a2 f ðv4 ðttÞÞ, > > > > < v€ 2 ¼ v_ 2 m v2 þ b1 f ðv3 ðttÞÞ þb2 f ðv4 ðttÞÞ, 2 v€ 3 ¼ v_ 3 m3 v3 þ c1 f ðv1 ðttÞÞ þ c2 f ðv2 ðttÞÞ, > > > > : v€ 4 ¼ v_ 4 m v4 þ d1 f ðv1 ðttÞÞ þ d2 f ðv2 ðttÞÞ:

ð1Þ

4

where the real constants aj , bj and cj , dj ðj ¼ 1,2Þ are the connection weights through neurons in two layers: the I-layer and J-layer; mi ð 40Þði ¼ 1,2,3,4Þ describe the stability of internal neuron processes on the I-layer and J-layer, respectively; vi ði ¼ 1,2,3,4Þ denote the states of the neurons on the I-layer and J-layer; tð 4 0Þ is the time delay; and f ðUÞ the nonlinear activation function. The rest of the paper is organized as follows. In Section 2, a linear stability analysis of (1) is performed. In Section 3, stability switches are verified by the results of numerical simulation. In Section 4, fold-Hopf bifurcations are completely analyzed and illustrated qualitatively and quantitatively by a simple and effective method. Conclusions are drawn and further research directions are outlined in Section 5.

2. Stability analysis of linearized system In this section, the linear stability of the trivial equilibrium point is presented by analyzing the corresponding transcendental characteristic equation of the linearized system of (1). Also, we try to find out some critical conditions under which the stability switches and fold-Hopf bifurcation occur when the parameters of the network model pass through some critical values. For notational convenience, let v1 ¼ x1 , v_ 1 ¼ x2 , v2 ¼ x3 , v_ 2 ¼ x4 , v3 ¼ x5 , v_ 3 ¼ x6 , v4 ¼ x7 , and v_ 4 ¼ x8 . Then (1) is equivalent to the

71

following system: 8 x_ 1 ¼ x2 , > > > > > _ 2 ¼ m1 x1 x2 þa1 f ðx5 ðttÞÞ þ a2 f ðx7 ðttÞÞ, x > > > > > _ x > 3 ¼ x4 , > > > < x_ 4 ¼ m x3 x4 þb1 f ðx5 ðttÞÞ þ b2 f ðx7 ðttÞÞ, 2 x_ 5 ¼ x6 , > > > > > x_ 6 ¼ m3 x5 x6 þc1 f ðx1 ðttÞÞ þ c2 f ðx3 ðttÞÞ, > > > > > x_ 7 ¼ x , > 8 > > > : x_ ¼ m x x þd f ðx ðttÞÞ þ d f ðx ðttÞÞ: 8

4 7

8

1

1

2

ð2Þ

3

To simplify the analysis, the activation function f ðUÞ satisfies the following conditions: 0 ðH1 Þf A C, f ð0Þ ¼ f ð0Þ ¼ 1 and there exists L 40 such that f ðxÞ9 rL for all x A R. A special case, which will be used for numerical simulations here, is f ðUÞ ¼ tanhðUÞ. Under the hypothesis ðH1 Þ, ð0,0,0,0,0,0,0,0Þ is always an equilibrium point of (2). This section focuses on the local stability analysis of the trivial equilibrium point. The analysis for the nontrivial equilibrium point, if it exists, is similar after a simple transform. The characteristic equation associated with (2) is X8 X4 a li þ e2lt i ¼ 0 bi þ 1 li þ b0 e4lt ¼ 0, ð3Þ i¼0 i where ai ði ¼ 0,1,. . .,8Þ and bj ðj ¼ 0,1,. . .,5Þ are given in Appendix A. It is well known that changes in stability may occur when the system has a zero eigenvalue or a purely imaginary pair. This means that it is necessary to investigate the distribution of roots of (3). It is easy to verify that l ¼ 0 is a root of (3) if and only if a0 þ b0 þ b1 ¼ 0: Regarding t as the parameter, we determine when (3) has a pair of purely imaginary roots. Letting l ¼ 7io ðo 4 0Þ be a root in (3) and separating the real and imaginary parts leads to the pair of equations cosð2otÞ ¼

P , D

sinð2otÞ ¼

Q , D

ð4Þ

where P, Q and D are given in Appendix A. Taking square on the both sides of the equations in (4) and summing them up, we get X16 o32 þ i ¼ 1 gi o322i ¼ 0, ð5Þ where gi ði ¼ 1,2,. . .,16Þ are expressed by aj ðj ¼ 0,1,2,. . .,8Þ and

bk ðk ¼ 0,1,2,. . .,5Þ. The expressions for gi are not presented due to their long formulae. Let z ¼ o2 . And (5) becomes X16 g z16i ¼ 0 z16 þ i¼1 i

ð6Þ

Lemma 1. If g16 o0, then (6) has at lease one positive real root. The proof of the lemma is given in Appendix B. Suppose that (6) has positive real roots. Without loss of generality, we assume that it has sixteen positive real roots, defined by zk ðk ¼ 1,2,. . .,16Þ. Thus (5) has sixteen positive roots pffiffiffiffiffi ok ¼ zk ðk ¼ 1,2,. . .,16Þ. Correspondingly, time delays can be solved in (4)     1 P þ 2jk , k ¼ 1,2,. . .,16; j ¼ 0,1,. . ., tkðjÞ ¼ arccos 2ok Q then 7 ok is a pair of purely imaginary roots of (3) with tkðjÞ . The critical delay is defined by

t0 ¼

min

k A f1,2,...,16g

ftð0Þ g, k

o0 ¼ ok

ð7Þ

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J. Ge, J. Xu / Neurocomputing 110 (2013) 70–79

Appling Lemma 1, according to the Hopf bifurcation theorem for functional differential equations [34], the following theorem follows immediately.

When t 4 0, l ¼ ioðo 4 0Þ may be one of roots in (11) if and only if o2 satisfies the following equation:  2  z 2mz þ z þ m2 b z2 2mz þ z þ m2 þb ¼ 0, ð12Þ

Theorem 2. Suppose that ðH  1 Þ and  a0 þ b0 þ b1 a 0 hold. If the condition of Lemma 1 and Re dl=dt 9t ¼ t0 4 0 hold, then the trivial equilibrium is asymptotically stable for t A ½0, t0  and unstable for t 4 t0 when all characteristic roots of (3) have negative real parts at t ¼ 0. System (1) undergoes a Hopf bifurcation at the trivial solution when t ¼ t0 where t0 is defined by (7).

where z ¼ o2 : From (12), the following results hold for (10).

It should be noted that, in general, the computation of o in (5) is very tedious. This is why we first consider the following cases to present and understand rich dynamical behaviors in (1) for readers’ convenience. Case 1. mi ¼ m

ði ¼ 1,2,3,4Þ

In this case, Eq. (5) can be decomposed as follows:



X3 X3 o8 þ i ¼ 0 xi o2i o8 þ i ¼ 0 Zi o2i ¼ 0,

ð8Þ

where

(b) If m2 4b 4 0 and m 41=2 hold, then (10) has two pairs of pure imaginary roots l ¼ 7io þ and l ¼ 7 io ðo þ 4 o Þ where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  u t 2m1 7 2m1 4 m2 þ b : ð14Þ o7 ¼ 2 If

4

2

2

3

x0 ¼ m þ bm þ d, x1 ¼ b2bm þ2m 4m , x2 ¼ 1 þb4m þ 6m2 , x3 ¼ 24m ¼ Z3 , Z0 ¼ m4 bm2 þ d,

Z1 ¼ bþ 2bm þ 2m2 4m3 , Z2 ¼ 1b4m þ 6m2 , b ¼ a1 c1 þb1 c2 þ a2 d1 þb2 d2 ,

d ¼ ða2 b1 a1 b2 Þðc2 d1 c1 d2 Þ: P In the next, we only consider o8 þ 3i ¼ 0 xi o2i ¼ 0. For P o8 þ 3i ¼ 0 Zi o2i ¼ 0, we may obtain similar results. Let z ¼ o2 . P The equation o8 þ 3i ¼ 0 xi o2i ¼ 0 becomes z4 þ x3 z3 þ x2 z2 þ x1 z þ x0 ¼ 0: 4

3

ð9Þ 2

Denote hðzÞ ¼ z þ x3 z þ x2 z þ x1 z þ x0 : From the standard Lodovicio Ferrari’s method in Ref. [35], the following Lemma is obtained. Lemma 3. For the polynomial (9), we have the following results. (1) If x0 o 0, then (9) has at least one positive real root. (2) If D Z0, then (9) has positive roots if and only if zn Z0 and hðzn Þ o 0. (3) If D o0, then (9) has positive root if and only  if there  exists at least one zn Z 0 and hðzn Þ o0 where D ¼ q2 =4 þ p3 =27 , 2 3 p ¼ 8x2 3x3 =16, q ¼ x3 4x2 x3 þ 8x1 =32, and zn is the 0 maximum root of the equation h ðzÞ ¼ 0. From Lemma 3, it is known that in some cases, it is possible to show the presence of points at which the original characteristic equation (3) has two pairs of purely imaginary roots or a zero and a pair of purely imaginary roots. In what follows, we try to find out some critical conditions under which the stability switches and fold-Hopf bifurcations occur when the parameters of the network model pass through some critical values. Case 2. mi ¼ m, a2 ¼ a1 , b2 ¼ b1 , c2 ¼ c1 , d2 ¼ d1 ði ¼ 1,2,3,4Þ

  (a) If 1=4 o m 1=4 ob o m2 holds, then (10) has two pairs of pure imaginary roots l ¼ 7io þ and l ¼ 7io ðo þ 4 o Þ where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  u t 2m1 7 2m1 4 m2 b : ð13Þ o7 ¼ 2

In this case, Eq. (3) can be rewritten as 

2 

2 l2 þ l þ m l2 þ l þ m be2lt ¼ 0

m2 ¼ b and m 41=2 hold, then (10) has a simple zero root and a p pair of purely imaginary roots l ¼ 7io þ ffiffiffiffiffiffiffiffiffiffiffiffiffi o þ ¼ 2m1 .

Correspondingly, the values of critical time delay can be expressed by (see Appendix C) 

  2o m þ o2 tl ðoÞ ¼ 21o arctan ð 2 2 Þ2 þ lp , ðmo Þ o 8  2 < t2k ðoÞ, mo2 o2 4 0 ð15Þ tl ðoÞ ¼  2 : t2k þ 1 ðoÞ, mo2 o2 o 0 with kA f0,1,2,3,. . .g. Let tl ðoÞ ¼ tlþ ðo þ Þ if o ¼ o þ and tl ðoÞ ¼ tl ðo Þ if o ¼ o . For further analysis, we need the following results on transversality condition.  Lemma 4. Let lðtÞ be the root of (10) satisfying l tl7 ðo 7 ÞÞ ¼ 7 io 7 . We have     dl dl o 0: 9l ¼ 7 io þ , t ¼ t þ ðo þ Þ 4 0, Re 9 Re  l dt dt l ¼ 7 io , t ¼ tl ðo Þ The proof of the lemma is given in Appendix D. Summarizing the above results, using Lemma 4, together with Cooke and Grossman’s work [36], the following theorem follows immediately. Theorem 5. Assume that ðH1 Þ and mi ¼ m, a2 ¼ a1 , b2 ¼ b1 , c2 ¼ c1 , d2 ¼ d1 ði ¼ 1,2,3,4Þ in (1) hold. The following conclusions are true:  (a) If 1=4 o m 1=4 o b o m2 or m2 4b 4 0 and m 4 1=2 hold, then (1) undergoes Hopf bifurcation at tl7 ðo 7 Þ and the stability of the trivial solution changes a finite number of times, at most, as t is increased, and eventually it becomes unstable. tl7 ðo 7 Þ are defined by (15) where o 7 are given by , respectively. (b) If m2 ¼ b and m 4 1=2 hold, then (1) undergoes an interaction of a steady state bifurcation and a Hopf p bifurcation ffiffiffiffiffiffiffiffiffiffiffiffiffi around the trivial equilibrium at t ¼ t0þ ðo þ Þ o þ ¼ 2m1 , where t0þ ðo þ Þis given by (15).

ð10Þ 3. Stability switches

Since m 4 0, we often consider the equation about l in (10)

2

l2 þ l þ m

be2lt ¼ 0:

ð11Þ

In this section, some results of numerical simulating of (2) are presented to justify the stability switches in (a) of Theorem 5 in Section 2.

J. Ge, J. Xu / Neurocomputing 110 (2013) 70–79

73

Fig. 1. Hopf bifurcation curves in ðb, tÞ plane for (2) where m ¼ 2. The trivial equilibrium point is asymptotically stable where b A ð0,1:75: Solid line indicates Hopf þ bifurcation curves corresponding to t l ðo Þ while dashed line denotes Hopf bifurcation curves corresponding to tl ðo þ Þ:

The curves of Hopf bifurcations for different parameters’ values (b versus t) can be shown in Fig. 1 when mi ¼ 2ði ¼ 1,2,3,4Þ. If the parameters in (2) are taken as (b ¼ 1:8, see Fig. 1 (left)) a1 ¼ a2 ¼ 0:5,

b1 ¼ b2 ¼ 0:5,

c1 ¼ c2 ¼ 0:4, and d1 ¼ d2 ¼ 1:4, ð16Þ

then one can obtain at t ¼ 0

l1,2 ¼ 0:5 þ 1:32288i, l3,4 ¼ 0:51:32288i

then (12) yields two positive and simple roots, i.e.,o þ ¼ 1:44624, o ¼ 0:953096: And, when t A ½0, 0:895016Þ [ ð2:12851, 4:19121Þ [ ð4:30076, 1Þ, at least one root of (10) has a positive real part. However, when t A ð0:895016, 2:12851Þ [ ð4:19121, 4:30076Þ, all roots of (10) have negative real parts. Two switches from instability to stability to instability occur as t is increased, and eventually it becomes unstable. The results are in good agreement with conclusion (a) of Theorem 2.

l5,6 ¼ 0:5 7 0:63903i, l7,8 ¼ 0:5 71:75831i Therefore, all roots of (10) have negative real parts at t ¼ 0, which implies that the trivial equilibrium point is stable at t ¼ 0: Substituting (16) into (12) yields two positive and simple roots, i.e., o þ ¼ 1:31286, o ¼ 1:12978. From (15), one can obtain

tlþ ðo þ Þ ¼ 1:35452,3:74745, 6:14039,8:53332, 10:9263, 13:3192, 15:7121, 18:1051,. . .

tl ðo Þ ¼ 1:89458, 4:67531, 7:45603, 10:2368, 13:0175, 15:7982, 18:5789, 21:3596,. . . These critical time delays can be ranked as þ   0 o t0þ ðo þ Þ o t 0 ðo Þ o t1 ðo þ Þ o t1 ðo Þ o    o t4 ðo Þ

o t5þ ðo þ Þ o t6þ ðo þ Þ o      Furthermore, Refdl=dtg9l ¼ 7 io þ , t ¼ t þ ðo þ Þ 4 0, and Re dl=dt l 9l ¼ 7 io , t ¼ t ðo Þ o0. l Hence, all roots of (10) have negative real parts when    t A 0, t0þ ðo þ Þ [ t0 ðo Þ, t1þ ðo þ Þ [ t1 ðo Þ, t2þ ðo þ Þ  þ [ t2 ðo Þ, t3 ðo þ Þ    þ þ [ t 3 ðo Þ, t4 ðo þ Þ [ t4 ðo Þ, t5 ðo þ Þ , which implies that the trivial solution is asymptotically stable. However, at least one root of (10) has a positive real part when    t A t0þ ðo þ Þ, t0 ðo Þ [ t1þ ðo þ Þ, t1 ðo Þ [ t2þ ðo þ Þ, t2 ðo þ Þ [ t3þ ðo þ Þ, t 3 ðo Þ  þ   [ t4 ðo þ Þ, t4 ðo Þ [ t5þ ðo þ Þ, 1 , which implies that the trivial equilibrium point is unstable. Six switches from stability to instability occur as time delay t is increased, and eventually it becomes unstable. It is in good agreement with conclusion (a) of Theorem 2. If the parameters in (2) are taken as (b ¼ 2:1, see Fig. 1 (right)) a1 ¼ a2 ¼ 0:5,

b1 ¼ b2 ¼ 0:5,

c1 ¼ c2 ¼ 1:1,

d1 ¼ d2 ¼ 1; ð17Þ

4. Fold-Hopf bifurcations Our goal of this section is to investigate fold-Hopf bifurcations by using the extended perturbation-incremental scheme (PIS) when choosing the coupling delay and the connection weight as two controlling parameters. Some results of numerical simulating of (2) are given for justifying the theoretical results. In what follows, it is necessary to assume that f A C 3 : The PIS [37] can be extended to investigate fold-Hopf bifurcations in a system of first-order delayed differential equations. The scheme is described in two steps, namely, the perturbation step (noted as step one) for bifurcation parameters close to the bifurcation point and the incremental step (noted as step two) for those far away from the bifurcation point. In this paper, we only discuss a small-amplitude harmonic solution near a fold-Hopf bifurcation point obtained by the linear analysis of (2). Therefore, here we only introduce the extended step one as follows. A class of delayed models can be written as _ ¼ CZðtÞ þ DZðttÞ þ eFðZðtÞ, ZðtÞ

ZðttÞÞ,

ð18Þ

where ZðtÞ ¼ ðz1 ðtÞ, z2 ðtÞ,:::,zn ðtÞÞT A Rn , C and D are n  n real constant matrices, FðUÞ is a continuous nonlinear function in its variables with Fð0, 0Þ ¼ 0, e is a parameter representing the strength of nonlinear coupling, t is the time delay, and n is a positive integer. We assume Dc and tc to be the critical values of a fold-Hopf bifurcation, namely, (18) has only a simple zero root l0 ¼ 0 and a simple purely imaginary pair l 7 ¼ 7 io a0 at D ¼ Dc and t ¼ tc : Furthermore, the crossing speed of the purely imaginary pair roots is assumed to be nonzero, and all other characteristic roots are neither zero nor purely imaginary pairs. In the next, we derive the analytical expression of solutions arising from a fold-Hopf bifurcation in (18) when D and t are close to Dc and tc : A perturbation to one of the critical values, D ¼ Dc þ eDe and t ¼ tc þ ed2 in (18) yields _ ¼ CZðtÞ þ Dc Zðttc Þ þ F~ ðZðtÞ, Zðttc Þ, ZðttÞ, eÞ, ZðtÞ

ð19Þ

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J. Ge, J. Xu / Neurocomputing 110 (2013) 70–79

Substituting (28) into (26) and using the harmonic balance, one may obtain that

where F~ ðZðtÞ, Zðttc Þ, ZðttÞ, eÞ ¼ Dc ½ZðttÞZðttc Þ þ e½De ZðttÞ þ FðZðtÞ, ZðttÞÞ For e ¼ 0, it can be seen from (19) that D ¼ Dc , t ¼ tc , and F~ ¼ 0: Assume that (19) has a periodic solution with a period 2p=o at e ¼ 0: Then the periodic solution may be expressed as ZðtÞ ¼ acosðotÞ þbsinðotÞ þ z,

ð20Þ T

where a ¼ ða1 ,a2 ,:::,an ÞT , b ¼ ðb1 ,b2 ,:::,bn Þ , z ¼ ðz1 ,z2 ,:::,zn ÞT A Rn , and o and tc are determined by the characteristic equation det lICDc elt ¼ 0. Substituting (20) into (19) at e ¼ 0 and using the harmonic balance [38], one may obtain that Mb ¼ Na,

ð21Þ

Ma ¼ Nb,

ð22Þ

Tz ¼ 0,

ð23Þ

where M ¼ oIþ Dc sinðotc Þ, N ¼ C þ Dc cosðotc Þ and T ¼ C þDc . M and N are the real and imaginary parts of the characteristic matrix  lICDc elt at l ¼ io respectively, where a simple purely imaginary root occurs at t ¼ tc . Therefore, there are only ð2n2Þ independent equations to determine a1 ,a2 ,. . .,an and b1 ,b2 ,. . .,bn in (21) and (22). If a1 and b1 are chosen to be independent, then ai and bi ði ¼ 2,3,. . .,nÞ can be determined by (21) and (22) in terms of a1 and b1 : Similarly, zi ði ¼ 2,3,. . .,nÞ can be determined by (23) in terms of z1 : Eq. (20) can be expressed in a polar coordinate as Z i ðtÞ ¼ r i cosðot þ yi Þ þzi ,

ð24Þ T

T

where ZðtÞ ¼ ðz1 ðt Þ,z2 ðtÞ,:::,zn ðtÞÞ , r ¼ ðr 1 ,r 2 ,:::,r n Þ , z ¼ ðz1 ,z2 ,:::,zn ÞT A Rn , y1 , r 1 , and z1 are constants, r i are functions of r 1 , yi are functions of y1 , and zi are functions of z1 ði ¼ 2,3,. . .,nÞ. Based on the expression in (24), we consider the solution of (19) for small e. The harmonic solution of (19) can be considered as a perturbation to that of (24), given by Z i ðtÞ ¼ r i ðeÞcosððo þ sðeÞÞt þ yi Þ þ zi ðeÞ,

r i ð0Þ ¼ r i , sð0Þ ¼ 0 r i ðeÞ ¼ r i ðr 1 ðeÞÞ,

zi ð0Þ ¼ zi



zi ðeÞ ¼ zi ðz1 ðeÞÞ

ð29Þ

MT q ¼ NT p,

ð30Þ

TT s ¼ 0

ð31Þ

If p1 , q1 and s1 are chosen to be independent, then pi , qi and si ði ¼ 2,3,. . .,nÞ can be determined by (29)–(31) in terms of p1 , q1 and s1 . Substituting (25) and (28) into (27), expanding in power of e and neglecting higher-order terms, and noting the independence of p1 , q1 and s1 yield a set of algebraic equations in r 1 ðeÞ, sðeÞ and z1 ðeÞ: Now, we employ the extended method mentioned above to consider the effect of time delay t and the connection weight c1 on (2). If the parameters in (2) are taken as mi ¼ 2, a1 ¼ a2 ¼ b1 ¼ b2 ¼ 1, c1 ¼ c1c ¼ 1, d1 ¼ d2 ¼ 1ði ¼p1,2,3,4 Þ: Using ffiffiffi pffiffiffi (15), a fold-Hopf bifurcation occurs at tc ¼ p=3 3 and o ¼ 3: Rescale xi -exi ði ¼ 1,2,. . .,8Þ and perturb the bifurcation parameters c1 and t, say c1 ¼ c1c þ e2 d1 and t ¼ tc þ e2 d2 where e2 d1 and e2 d2 are very small. Thus (2) is transformed into the form (19) _ ¼ CZðtÞ þ Dc Zðttc Þ þ Dc ½ZðttÞZðttc Þ ZðtÞ þ e½De ZðttÞ þFðZðt Þ, ZðttÞÞ

ð32Þ

where ZðtÞ, C, Dc , De and FðZðt Þ, Z ðttÞÞ are given in Appendix G. The first approximate periodic solution ZðtÞ of (2) can be expressed as  3 2 ez þ ercos o þ e2 s t þ y  7 6 pffiffiffi 6  3ersin o þ e2 s t þ y 7 6 7  6 7 6 ez þ ercos o þ e2 s t þ y 7 6 pffiffiffi 7   6  3ersin o þ e2 s t þ y 7 6 7 6  7 Z tÞ ¼ 6 ð33Þ 2 7: e z e rcos o þ e s y t þ 6 7 6 pffiffiffi 7  6 3ersin o þ e2 s t þ y 7 6  7 6 7 6 ezercos o þ e2 s t þ y 7 4 pffiffiffi  5  3ersin o þ e2 s t þ y

ð25Þ

where 

MT p ¼ NT q,

ði ¼ 2,3,. . .,nÞ

The following lemma provides a new method to determine r 1 ðeÞ, sðeÞ and z1 ðeÞin (25). Lemma 6. If WðtÞ is a periodic solution of the adjoint equation (see Appendix E) corresponding to the linearized equation of (19) _ WðtÞ ¼ CT WðtÞDc T Wðt þ tc Þ, ð26Þ then   T Z 0 2p Dc T Wðt þ tc ÞW t þ tc þ ZðtÞdt o þ sðeÞ tc  

 2p WT ð0Þ Zð0Þ  WT o þ sðeÞ Z 2p o þ sðeÞ þ WT ðtÞF~ ðZðtÞ, Zðttc Þ, ZðttÞ, eÞdt ¼ 0, ð27Þ 0

 where Wðt Þ ¼ W t þ 2op and ZðtÞ ¼ Z t þ o þ2psðeÞ . The proof of the lemma is performed in Appendix F. To apply Lemma 6 for solving r 1 ðeÞ, sðeÞ and z1 ðeÞ, one must obtain the expression of WðtÞ in (26). The periodic solution of (26) can be written as ð28Þ WðtÞ ¼ pcosðotÞ þ qsinðotÞ þ s,  T  T where p ¼ p1 ,p2 ,:::,pn , q ¼ q1 ,q2 ,:::,qn , and s ¼ ðs1 ,s2 ,:::,sn ÞT .

If WðtÞ is assumed to be a periodic solution of the adjoint equation corresponding to the linearized equation of (32), then it can be represented in the following form: 2 3 pcosðotÞ þ qsinðotÞ þs



p ffiffiffi p ffiffiffi 61 7 6 p þ 3q cosðot Þ þ 1  3p þq sinðot Þ þs 7 4 64 7 6 7 6 7 pcosð o tÞ þ qsinð o tÞ þs 6 7



pffiffiffi pffiffiffi 61 7 6 p þ 3q cosðot Þ þ 1  3p þq sinðot Þ þs 7 4 64 7 6 7: ð34Þ WðtÞ ¼ 6 7 pcosð o tÞqsinð o tÞ þ s 6 7



pffiffiffi pffiffiffi 61 7 1 6 7 6 4 p 3q cosðot Þ þ 4 3pq sinðot Þ þ s 7 6 7 6 7 pcosðotÞqsinðotÞ þ s 6 7



pffiffiffi pffiffiffi 41 5 1 ð Þ ð Þ p 3 q cos o t þ o t þ s 3 pq sin 4 4 According to the extended PIS mentioned above, substituting (33) and (34) into (27), expanding in power of e and neglecting higher-order terms, noting the independence of p,  q and s yields a set of algebraic equations in er, ez and e2 s1 at O e2 as follows: 

2perðezÞ2 pðerÞ3 per e2 d1 2p2 er e2 s1 pffiffiffi þ pffiffiffi pffiffiffi  pffiffiffi þ 2 3 4 3 3 3 3 pffiffiffi þ 2 3per e2 d2 þ per e2 s1 ¼ 0,

2pðezÞ2 er þ

1 1 pðerÞ3  per e2 d1 þ2pere2 d2 2 4 7per e2 s1 2 2 2 pffiffiffi þ p er e s1 ¼ 0, þ 9 3

ð35Þ

ð36Þ

J. Ge, J. Xu / Neurocomputing 110 (2013) 70–79

75

Fig. 2. Classification and bifurcation sets of the solutions are derived from fold-Hopf bifurcation from (35)–(37) according to the values of c1 and t (left figure). Secondary bifurcations occur along the line labeled PFLC (pitchfork bifurcation of limit cycles) and SH (secondary Hopf bifurcation from equilibrium points). Phase portraits of the planar system in ðr,zÞ plane are given corresponding to different sectors of the plane of bifurcation parameter space ðc1 , tÞ in top figure and the arrows indicate the direction of time evolution of the system (right figure).

t o 0:6561160:05151667c1 g,  VI ¼ ðc1 , tÞjc1 4 1, t o0:7076330:103033c1 g:

Fig. 3. Time history where ðc1 , tÞ ¼ ð0:5,0:5Þ is located in region I. The trivial equilibrium point is asymptotically stable.



16pðezÞ3 8pezðerÞ2 2peze2 d1 pffiffiffi  pffiffiffi pffiffiffi þ ¼ 0, 3 3 3 3

ð37Þ

2 2 2 where pffiffiffi c1 ¼ pcffiffiffi1c þ e d1 , t ¼ tc þ e d2 , o ¼ oc þ e s1 , c1c ¼ 1, tc ¼ p=3 3, o ¼ 3: It follows from (35)–(37) that the number of equilibrium points depends on the values of c1 and t: Stability of the equilibrium point can be determined by using (35)–(37). All roots can easily be solved from (35)–(37) for a fold-Hopf bifurcation around the bifurcation point, i.e., pffiffiffiffiffiffiffiffiffiffiffiffi

ð0,0Þ, 0,0:612372 c1 1 , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2:20291 0:707633 þ t þ0:103033c1 , 0 ,



pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:98517 0:398533t þ 0:206067c1 , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1:20658 0:656116 þ t þ 0:0515167c1 :

Analysis of (35)–(37) reveals that the parameter plane ðc1 , tÞ is divided into six distinct regions depending on two bifurcation parameters c1 and t bounded by  I ¼ ðc1 , tÞjc1 o 1, t o 0:7076330:103033c1 g,  II ¼ ðc1 , tÞjc1 o1, t 4 0:7076330:103033c1 g,  III ¼ ðc1 , tÞjc1 4 1, t 40:398533 þ 0:206067c1 g,  IV ¼ ðc1 , tÞjc1 41, t 4 0:6561160:05151667c1 , t o 0:398533 þ0:206067c1 g,  V ¼ ðc1 , tÞjc1 4 1, t 40:7076330:103033c1 ,

It can be seen from (35)–(37) that r, z and s1 determine the feature of motions of (1) when a fold-Hopf bifurcation point at ðc1c , tc Þ is perturbed by d1 and d2 . The stationary solutions on the z axis in ðr,zÞ correspond to the equilibrium points in (1), whereas stationary solutions off the axis are associated with periodic solutions in (1). All solutions are shown in Fig. 2, which shows the bifurcation set for (1) in parameter space ðc1 , tÞ. The line labeled Hopf corresponds to t ¼ 0:7076330:103033c1 , the line labeled Pitchfork corresponds to c1 ¼ 1:0, the line labeled SH corresponds to t ¼ 0:398533þ 0:206067c1 , and the line labeled PFLC corresponds to t ¼ 0:6561160:05151667c1 : It is in good agreement with fig. 4.2 produced by Be´lair et al. [39] and Fig. 4 by Olien and Be´lair [40]. Under some conditions, there exist complex dynamical behaviors such as multistability, pitchfork bifurcation of limit cycles, secondary Hopf bifurcation. The classification illustrated in Fig. 2 gives us some insight into the dynamics of (1). The dynamics of each region in Fig. 2 are verified qualitatively and the result of each simulation is displayed as shown in Figs. 3–8. As is shown, one of these periodic solutions is stable wherever one exists, except for region IV. It can be seen that the results of simulating (2) are in good consistence with theoretical analysis.

5. Conclusions ANNs have been evolving toward more powerful and more biologically realistic models. There is some strong biological support for the addition of an inertial term to a neural equation. Time delay is unavoidable and can influence the stability of the entire network by creating oscillatory or unstable phenomena [41]. So we consider a neural network with both inertia and time delay for more realistic considerations. From the point of view of nonlinear dynamics, analyzing these neural networks is useful in solving problems of both theoretical and practical importance. In this paper, a simplified four-neuron delayed neural model with inertia has been investigated. Under some conditions, the rest state of the system can be switched from stable to unstable and back to stable and so on or from unstable to stable and back to unstable just by progressive increasing of coupling delay values. Fold-Hopf bifurcations induced by time delay are detected and their nature was elucidated by using the extended PIS without the CMR, in which the coefficients of the normal form are obtained in terms of the original delayed system directly,

76

J. Ge, J. Xu / Neurocomputing 110 (2013) 70–79

Fig. 4. The periodic oscillation derived from Hopf bifurcation around the trivial equilibrium point where ðc1 , tÞ ¼ ð0:5,0:75Þ is located in region II. (a) Time history and (b) phase portrait.

Fig. 5. The periodic oscillation where ðc1 , tÞ ¼ ð1:5,0:75Þ is located in region III. (a) Time history and (b) phase portrait.

Fig. 6. Multistability where ðc1 , tÞ ¼ ð1:5,0:65Þ is located in region IV with different initial values (a) ð1,1,0:1,0:1,0:1,0:1,0:1,0:1Þ, (b) ð0:1,0:1,0:1,0:1,0:1,0:1, 0:1,0:1Þ, and (c) ð0:1,0:1,0:1,0:1,0:1,0:1,0:1,0:1Þ.

which simplifies the computation procedure. The classification in the plane of bifurcation parameters gives us some insight into the dynamics of all the equations. The advantage of the classification is that we do not have to do extensive numerical simulation to find the interesting behavior because we already know where it exists. The theoretical results have established some necessary and solid foundations for applications of inertial neural networks. The main novelty of this paper relies not only on firstly introducing inertia into a four-neuron delayed neural system, but also on clarifying the relationship between dynamical behaviors and system parameters for parameter values in the neighbor of the degenerate fold-Hopf bifurcation point by a simple and effective method, which is more convenient to be applied in practice. To the best of our knowledge, no result has been obtained for the

Fig. 7. The time history where ðc1 , tÞ ¼ ð1:5,0:56Þ A region V with different initial values ð1,1,0:1,0:1,0:1,0:1,0:1,0:1Þ and ð1,1,0:1,0:1,0:1,0:1,0:1,0:1Þ.

Fig. 8. The time history where ðc1 , tÞ ¼ ð1:5,0:5Þ A region VI with the following initial values ð1,1,0:1,0:1,0:1,0:1,0:1,0:1Þ and ð1,1,0:1,0:1,0:1,0:1,0:1,0:1Þ.

J. Ge, J. Xu / Neurocomputing 110 (2013) 70–79

stability switches and fold-Hopf bifurcations of such system in the existing literature. The inertial system considered in this paper may also exhibit four different chaotic attractors when numerical simulations were carried out. The interesting problem will remained as our future work.

Appendix C. Computation of Eq. (15) In Case 2, Eq. (4) can be simplified as 

Acknowledgments

Appendix A 







a0 ¼ m1 m2 m3 m4 , a1 ¼ m1 m2 m3 þ m4 þ m1 þ m2 m3 m4 ,

a3 ¼



i¼2

X4 i¼1

a4 ¼ 1 þ 3

mi þ 2m1 i¼1



X4

mi þ m1

X4

a5 ¼ 4 þ 3



mi þ m2 m3 þ m4 þ m3 m4 þ a1 ,

X4



i¼2





mi þ 2m2 m3 þ m4 þ 2m3 m4 ,

X4 i¼2



i¼1

mi , a6 ¼ 6 þ







mi þ m2 m3 þ m4 þ m3 m4 , X4 i¼1

 2o m þ o2 : b

ðC:1Þ









Appendix D. Proof of Lemma 4 Proof. Differentiating both sides of (10) with respect to t gives ( )    o2 þ i o3 mo dl Re 9l ¼ io ¼ Re  dt 1o2 t þ mt þ ið2o þ otÞ  o2 2o2 2m þ 1 ¼ : 2 1o2 t þ mt þ ð2o þ otÞ2  2 Notice that o2 , 1o2 t þ mt þ ð2o þ otÞ2 4 0. So the sign    of Re dl=dt9l ¼ io is determined by that of 2o2 2m þ1 . 2 By inserting the expression for o 7 , it is seen that the sign is positive for o2þ and negative for o2 . This completes the proof.

The inner product of two vectors U,Vð A C½0,mÞ is defined as Z m hU,Vi ¼ VT ðtÞUðtÞdt, ðE:1Þ

mi , a7 ¼ 4, a8 ¼ 1,

0







b2 ¼ b2 d2 m1 þ m3 a2 d1 m2 þ m3 b1 c2 m1 þ m4 a1 c1 m2 þ m4 , b3 ¼ b2 d2 a2 d1 b1 c2 a1 c1 þ b2 , b4 ¼ 2b5 , b5 ¼ ða1 c1 þb1 c2 þ a2 d1 þb2 d2 Þ:  10

sinð2otÞ ¼

Appendix E. Adjoint equation

b0 ¼ ða2 b1 a1 b2 Þðc2 d1 c1 d2 Þ,   b1 ¼  b2 d2 m1 þa2 d1 m2 m3  b1 c2 m1 þa1 c1 m2 m4 , 

b

,

Eq. (15) can be solved by (C.1).

The authors are very grateful to the associate editor and reviewers for their valuable comments and suggestions. This research is supported by the State Key Program of National Natural Science Foundation of China under Grant no. 11032009 and the National Natural Science Foundation of China under Grant no. 11202068.

X4

2

mo2 o2

cosð2otÞ ¼

a2 ¼ m1

77



P ¼ o12 b1 þ o a1 b1 4b2 þ b3  þ o8 a3 b1 þ a2 b2 a1 b3 þ4b4 b5  þ o6 a5 b1 a4 b2 þ a3 b3 a2 b4 þ a1 b5  þ o4 a0 b1 þ b0 b1 þ a6 b2 a5 b3 þ a4 b4 a3 b5  þ o2 a0 b3 b0 b3 a6 b4 þ a5 b5 a0 b5 þ b0 b5 ,   Q ¼ o11 4b1 þ b2 þ o9 a2 b1 a1 b2 þ4b3 b4  þ o7 a4 b1 þ a3 b2 a2 b3 þ a1 b4 4b5  þ o5 a6 b1 a5 b2 þ a4 b3 a3 b4 þ a2 b5   þ o3 a0 b2 þ b0 b2 a6 b3 þ a5 b4 a4 b5 þ o a0 b4 b0 b4 þ a6 b5 ,  D ¼ o16 þ o14 ð162a1 Þ þ o12 a1 2 8a2 þ2a3  þ o10 a2 2 2a1 a3 þ 8a4 2a5  þ o8 2a0 þ a3 2 2a2 a4 þ 2a1 a5 8a6  þ o6 2a0 a1 þ a4 2 2a3 a5 þ 2a2 a6   2 þ o4 2a0 a3 þ a5 2 2a4 a6 þ o2 2a0 a5 þ a6 2 þ a0 2 b0 :

where ðUÞT represents transpose and m is a constant. To derive the adjoint equation corresponding to the linearized equation of (19), we rewrite (19) in the operator form for e ¼ 0 LðZÞ ¼ 0,

_ where LðZÞ ¼ ZðtÞCZðtÞD c Zðttc Þ: The operator L is said to be an adjoint operator corresponding to L, provided L and Ln satisfy     LðUÞ,V ¼ U,Ln ðVÞ , ðE:3Þ for all vectors U and V. _ From (E.1)–(E.3), one can obtain Ln ðWÞ ¼ WðtÞ T T C WðtÞDc Wðt þ tc Þ: So the adjoint equation corresponding to the linearized equation of (19) can be expressed as (26).

Appendix F Proof. Multiplying both sides of (19) by WT ðtÞ and integrating with respect to t from zero to 2p=ðo þ sðeÞÞ, one has Z 2p Z 2p o þ sðeÞ o þ sðeÞ _ WT ðtÞZðtÞdt ¼ WT tÞ½CZðtÞ þ Dc Zðttc Þdt 0

0

Z

2p o þ sðeÞ

W ðtÞF~ ðZðtÞ, Zðttc Þ, ZðttÞ, eÞdt     Noting WðtÞ ¼ W t þ 2p=o and ZðtÞ ¼ Z t þ 2p=o þ sðe Þ yields þ

0

WT ðtÞ½CZðtÞ þ Dc Zðttc Þdt

Z

2p o þ sðeÞ

þ Proof. Denote the left side of (6) as gðzÞ. Clearly, gð0Þ ¼ g16 o0, and limz- þ 1 gðzÞ ¼ þ 1: Hence, there exists at least a z0 A ð0,1Þ, so that hðz0 Þ ¼ 0. This completes the proof.

T

0

R o þ2psðeÞ Appendix B. Proof of Lemma 1

ðE:2Þ n

0



 WT

WT ðtÞF~ ðZðtÞ, Zðttc Þ,ZðttÞ, eÞdt

  Z 2p o þ sðeÞ 2p _ T ðtÞZðtÞdt WT ð0Þ Zð0Þ þ W o þ sðeÞ 0

78

J. Ge, J. Xu / Neurocomputing 110 (2013) 70–79

Z

h iT _ WðtÞ þ CT WðtÞ þ Dc T Wðt þ tc Þ ZðtÞdt

2p o þ sðeÞ

¼

References

0

Z

  Dc T Wðt þ tc ÞW t þ

0

þ t c



 WT Z

T

ZðtÞdt

  2p WT ð0Þ Zð0Þ o þ sðeÞ

2p o þ sðeÞ

þ

2p þt o þ sðeÞ c

WT ðtÞF~ ðZðtÞ, Zðttc Þ,ZðttÞ, eÞdt ¼ 0:

0

Now, Lemma 6 follows from (26).

Appendix G 2

3 x1 ðtÞ 6 x ðtÞ 7 6 2 7 6 7 6 x3 ðtÞ 7 6 7 6 7 6 x4 ðtÞ 7 8 7 ZðtÞ ¼ 6 6 x5 ðtÞ 7 A R , 6 7 6 7 6 x6 ðtÞ 7 6 7 6 x7 ðtÞ 7 4 5 x8 ðtÞ 2

0

6 m 6 6 6 0 6 6 6 0 C¼6 6 0 6 6 6 0 6 6 0 4 0

1

0

0

0

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

m

1

0

0

0

0

0

0

0

1

0

0

0

0

m

1

0

0 0

0 0

0 0

0 0

0 0

0 m

2

0 6 6 0 6 6 0 6 6 6 0 Dc ¼ 6 6 0 6 6 6 c1c 6 6 0 4 d1 2

0 6 0 6 6 6 0 6 6 0 6 De ¼ 6 6 0 6 6 ed 6 1 6 4 0 0

0

0

0

0

0

0

0

0

0

0

a1

0

a2

0

0

0

0

0

0

0 0

0 0

0 0

b1 0

0 0

b2 0

0

c2

0

0

0

0

0

0

0

0

0

0

0

d2

0

0

0

0

0 7 7 7 0 7 7 7 0 7 7, 0 7 7 7 0 7 7 1 7 5 1

3

7 07 7 07 7 7 07 7, 07 7 7 07 7 07 5 0

3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

07 7 7 07 7 07 7 7, 07 7 07 7 7 05

0

0

0

0

0

0

0

2

0

3

3 0 6  e a x3 ðtt ed Þ þ a x3 ðtt ed Þ þOe2 7 c c 6 3 1 5 7 2 2 7 2 6 7 6 7 0 6 7 6  e b x3 ðtt ed Þ þ b x3 ðtt ed Þ þ Oe2 7 6 3 1 5 7 c c 2 2 7 2 7: FðZðtÞ,ZðttÞÞ ¼ 6 6 7 0 6  2 7 6 e 7 3 e 3 6  3 c1 x1 ðttc ed2 Þ 3 c2 x3 ðttc ed2 Þ þO e 7 6 7 6 7 0 4   2 5 e 3 3  3 d1 x1 ðttc ed2 Þ þ d2 x3 ðttc ed2 Þ þO e

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Juhong Ge was born in Henan province, China. She received the M.S. degree from Yunnan Normal University, Kunming, China in 2007, in mathematics/ applied mathematics and Ph.D. degree at the School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, China in 2011. Since 2011, she has been working at the Department of Mathematics and Information Science, He’nan University of Economics and Law, Zhengzhou, China. The research interests are in chaos, bifurcation and synchronization of nonlinear systems, and the study of complex networks.

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Jian Xu was born in Shanghai, China, on September 27, 1961. He received the Ph.D. degree in dynamics and control from the Tianjin University, Tianjin, China, in 1994. As a postdoctoral, he worked in Beijing University of Aeronautics and Astronautics, Beijing, China, from 1994 to 1996 and in Huazhong University of Science and Technology, Wuhan, China from 1996 to 1998, respectively. He joined Tongji University (TJU), Shanghai, China, in 1998 as an Associate Professor. Since 2000, he has been a professor of TJU, where he is now Deputy Dean School of Aerospace Engineering and Applied Mechanics. He was the recipient of several honors in recognition of his research activities, including the National Outstanding Young Funds of China, Program of Shanghai Subject Chief Scientist, the Distinguished Professor in TJU, Technological Progress Second Prize from the Ministry of Education of China, Natural Sciences First Prize from Tianjin City. The primary areas of research are in delay-induced dynamics, nonlinear dynamics and control, stability, bifurcation and chaos in nature and engineering, neural networks with delayed connection, systems biology.