synchronized stability in a reaction–diffusion neural network model
TRANSCRIPT
JID:PLA AID:22881 /SCO Doctopic: Nonlinear science [m5G; v1.142; Prn:20/10/2014; 14:50] P.1 (1-14)
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Contents lists available at ScienceDirect
Physics Letters A
www.elsevier.com/locate/pla
Synchronized stability in a reaction–diffusion neural network model
Ling Wang, Hongyong Zhao
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
a r t i c l e i n f o a b s t r a c t
Article history:Received 8 August 2014Received in revised form 10 October 2014Accepted 14 October 2014Available online xxxxCommunicated by F. Porcelli
Keywords:Delayed neural networkSynchronizationReaction–diffusionHopf bifurcation
The reaction–diffusion neural network consisting of a pair of identical tri-neuron loops is considered. We present detailed discussions about the synchronized stability and Hopf bifurcation, deducing the non-trivial role that delay plays in different locations. The corresponding numerical simulations are used to illustrate the effectiveness of the obtained results. In addition, the numerical results about the effects of diffusion reveal that diffusion may speed up the tendency to synchronization and induce the synchronized equilibrium point to be stable. Furthermore, if the parameters are located in appropriate regions, multiple unstability and bistability or unstability and bistability may coexist.
© 2014 Published by Elsevier B.V.
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1. Introduction
The study of the dynamics of the neural networks is an inter-disciplinary matter, which has concentrated the interest of many researchers for the last decades [1] (e.g., mathematicians, physi-cists, computer scientists and so on). Since Marcus and Westervelet [2] incorporated a single time lag into the connection term of Hopfield’s model, delays have been inserted into various simple neural networks, many authors have also investigated the dynam-ics of the neural networks of two or more neurons with delays, and have shown various types of dynamical behaviors (see, for ex-ample [3–8] and references therein). However, most of these work only considered the individual neural network but did not investi-gate the interactions between different neural networks.
As a matter of fact, neural networks consist of many nonlinear components which are interdependent and form a complex system with new emergent properties that are not held by each individ-ual item in the system alone. Coupled networks, which are com-bined by subnetworks and each subnetwork has its own dynamical property, are ubiquitous and also common in many branches of science [9]. For instance, in order to describe the complicated in-teraction between billions of neurons in large neural networks, the neurons are often lumped into highly connected subnetworks and the brain organization can be viewed in gross sense as a number of local subnetworks coupled by long distance connections [10].
Recently, Shayer and Campbell [11] considered the following two coupled units
E-mail address: [email protected] (H. Zhao).
http://dx.doi.org/10.1016/j.physleta.2014.10.0190375-9601/© 2014 Published by Elsevier B.V.
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
dx1
dt= −kx1(t) + β tanh
(x1(t − τs)
)+ a12 tanh
(x2(t − τ2)
),
dx2
dt= −kx2(t) + β tanh
(x2(t − τs)
)+ a21 tanh
(x1(t − τ1)
).
They were interested in studying how time delays can affect not only the stability of fixed points of the network but also the bi-furcation of new solutions when stability is lost. The authors [12]provided the stability and bifurcation of periodic solutions for a neural network with n elements where delays between adjacent units and external inputs were included, the particular cases n = 2and n = 3 were discussed in detail.
The subnetwork of the coupling models both in [11] and [12]were single neuron. Song et al. [13] considered a neural network coupled by two sub-networks, each consisting of two neurons as follows⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
du1
dt= −u1(t) + a12 f
(u2(t − τ )
) + α f(u4(t − τ )
),
du2
dt= −u2(t) + a21 f
(u1
(t − τ
)),
du3
dt= −u3(t) + a12 f
(u4(t − τ )
) + α f(u2(t − τ )
),
du4
dt= −u4(t) + a21 f
(u3(t − τ )
),
the conditions ensuring the stability and direction of the Hopf bi-furcation being determined. In [14], Campbell et al. studied the delayed neural network model coupled by a pair of Hopfield-like tri-neuron loops
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⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
dx1
dt= −x1(t) + tanh
(bx3(t)
),
dx2
dt= −x2(t) + tanh
(bx1(t)
),
dx3
dt= −x3(t) + tanh
(bx2(t)
) + c1 tanh(bx6(t − τ )
),
dx4
dt= −x4(t) + tanh
(bx6(t)
),
dx5
dt= −x5(t) + tanh
(bx4(t)
),
dx6
dt= −x6(t) + tanh
(bx5(t)
) + c2 tanh(bx3(t − τ )
).
They analyzed the roots of characteristic equation explicitly and specially investigated the local stability and bifurcation and ob-served some in-phase or anti-phase oscillations in numerical sim-ulations. Recently, Hsu et al. [15] extended the results of Camp-bell et al. [14] to a delayed model comprised of a pair of loops each with n neurons. Based on [14] and [15], Peng and Song [16]studied a delayed neural network consisting of a pair of identical tri-neuron network loops with bidirectional coupling of all neurons between loops, while Yuan and Li [17] gave the explicit conditions ensuring the stability and direction of the Hopf bifurcation of the model in [16].
The rich dynamics arising from the interaction of simple units have been a source of interest for scientists modeling the collective behavior of real-life systems. Inspired by the above, a coupled net-work of dynamical systems can exhibit a range of interesting be-havior, qualitatively very different from their behavior in isolation, such as synchronization [18,19], phase trapping, phase locking, and amplitude death.
Among them, synchronization, which is the phenomenon where systems, due to some kind of interaction, adjust their individual behavior in such a way that their behaviors become identical, has been causing researchers’ wide focus [8,30,38] since the works by Pecora and Carroll [20]. Experiment and theoretical analysis have revealed that a mammalian brain not only displays in its storage of associative memories, but also modulates oscillatory neuronal synchronization by selective perceive attention [21,22]. The dif-ference of using the benefits between synchronized stability and synchronized bifurcation is that memorized images correspond to equilibrium point attractors in the former and limit cycle attractors in the latter. In the theory and applications of content address-able memories, a stable solution can be used as coded informa-tion of a memory of the system to be stored and retrieved [7], pattern recognition by coupled neural networks consisting in con-vergence to the corresponding limit cycle attractor, which storesand retrieves complex oscillatory patterns in the synchronization states [23]. Periodic oscillation in neural networks is an interest-ing phenomenon, like many biological and cognitive activities [1]. So, how to understand the synchronized stability and synchronized bifurcation is very useful. In [24], Wei and Yuan considered the synchronized periodic oscillation in a ring neural network model with two different delays.
Reaction–diffusion (RD) mechanisms can describe many biolog-ical phenomena such as neuron firing in the brain, the heartbeat, cellular organization activities or even biological disorders such as fibrillation [37]. It is known that the foundations of neural pro-cessing refer to a phenomenon which takes place both in space and in time and involves an ensemble of neurons mutually con-nected, their dynamics is governed by the law of diffusion [35]. In signal transmission, the signal will become weak due to diffusion [36]. In addition, inspired by [25,26], we know that not only the evolution time of each variable and its position (space) but also the interactions deriving from the space-distributed structure of the whole networks determine the whole structure and dynamic
behavior of multi-layer cellular neural networks seriously and in-tensively. Therefore, it is essential to consider the state variables that are varying not only with time but also with space [27–31]and reaction–diffusion effects cannot be neglected in both biologi-cal and man-made neural networks.
The simplest model to display features of neural interaction comprised of two coupled neural systems. Starting from this sim-plest network motif, larger networks can be built, and their effects may be studied. So, we focus on the simplest example in which each network copy is capable of oscillation, namely, a pair of sim-ple loops of three neurons. With these in mind, based on the models in [12] and [14], we consider two kinds neural networks coupled by two sub-loop networks, each including three neurons: one way with delay in coupling; the other way with delay in sub-networks, both shown as follows (Fig. 1).
1) Coupled loops with delayConsider a pair of loops with delayed coupling connection⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂u1(t, x)
∂t= d∇2u1(t, x) + bf
(u2(t, x)
) − u1(t, x),
∂u2(t, x)
∂t= d∇2u2(t, x) + bf
(u3(t, x)
) − u2(t, x),
∂u3(t, x)
∂t= d∇2u3(t, x) + bf
(u1(t, x)
) − u3(t, x)
+ cf(
v3(t − τ , x)),
∂v1(t, x)
∂t= d∇2 v1(t, x) + bf
(v2(t, x)
) − v1(t, x),
∂v2(t, x)
∂t= d∇2 v2(t, x) + bf
(v3(t, x)
) − v2(t, x),
∂v3(t, x)
∂t= d∇2 v3(t, x) + bf
(v1(t, x)
) − v3(t, x)
+ cf(u3(t − τ , x)
),
(1)
the Neumann boundary and initial conditions are given by
⎧⎪⎪⎨⎪⎪⎩
∂ui
∂n:= ∂ui
∂x= 0,
∂vi
∂n:= ∂vi
∂x= 0,
t ≥ 0, x = 0,π, i = 1,2,3.
(2)
{ ui(s, x) = ηi(s, x),vi(s, x) = ζi(s, x),(s, x) ∈ [−τ ,0] × [0,π ], i = 1,2,3.
(3)
2) Ring Structure with delayConsider a pair of loops with delays in sub-network⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂u1(t, x)
∂t= d∇2u1(t, x) + bf
(u2(t, x)
) − u1(t, x),
∂u2(t, x)
∂t= d∇2u2(t, x) + bf
(u3(t, x)
) − u2(t, x),
∂u3(t, x)
∂t= d∇2u3(t, x) + bf
(u1(t − τ , x)
)− u3(t, x) + cf
(v3(t, x)
),
∂v1(t, x)
∂t= d∇2 v1(t, x) + bf
(v2(t, x)
) − v1(t, x),
∂v2(t, x)
∂t= d∇2 v2(t, x) + bf
(v3(t, x)
) − v2(t, x),
∂v3(t, x)
∂t= d∇2 v3(t, x) + bf
(v1(t − τ , x)
)− v3(t, x) + cf
(u3(t, x)
),
(4)
with Neumann boundary conditions (2), initial conditions is given by
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Fig. 1. Architecture of the model described by (a) system (1); (b) system (4).⎧⎨⎩
ui(s, x) = φi(s, x),vi(s, x) = ρi(s, x),(s, x) ∈ [−τ ,0] × [0,π ], i = 1,2,3.
(5)
Here ui(t, x), vi(t, x) (i = 1, 2, 3) represent the state variable associated with the ith neurons at time t and in space x. τ in system (1) and system (4) is the transmission delay, respectively, for coupling connections and internal connections. ∇2 = ∂2
∂x2 , d > 0is diffusion coefficient. The synaptic strength of neighborhood-interaction b is the factor of inhibitory (b < 0) or excitatory (b > 0). The coupling strength between two individual loops is given by c. We will talk about coupling inhibitory or excitatory if c < 0 or c > 0.
In initial conditions (3) and (5), we assume that
ηi(s, x) ∈ C = C([−τ ,0], X
),
ζi(s, x) ∈ C = C([−τ ,0], X
),
φi(s, x) ∈ C = C([−τ ,0], X
),
ρi(s, x) ∈ C = C([−τ ,0], X
),
where i = 1, 2, 3 and X is defined by
X = {θ ∈ W 2,2(0,π) : θx(0) = θx(π) = 0
}(6)
with the inner product 〈·,·〉, where W 2,2(0, π) is the standard Sobolev space.
In this paper, our main contributions are as follows.
• In order to point to a possible non-trivial role of delay τ which is in different positions of the both coupled networks, we re-gard τ as the bifurcation parameter to aim to shed some light on the synchronous dynamics of system (1) and (4).
• We reveal the effects of diffusion terms on the convergence of synchronization and stability of the synchronized equilibrium point. Our results yield new insight: diffusion may speed up
the tendency to synchronization and induce the synchronized equilibrium point to be stable.
• We investigate that the synchronized system has different numbers of equilibrium points with the variation of coupling strength c. If the parameters are located in appropriate re-gions, multiple unstability and bistability or unstability and bistability may coexist.
Accordingly, our focus in the remaining of this paper is organized as follows: In Sections 2 and 3, we establish the conditions of synchronized stability and Hopf bifurcation of the two systems, respectively. To demonstrate the effectiveness of the theoretical results obtained in the previous sections, some numerical simu-lations and discussions are performed in Section 4. Finally, conclu-sions are drawn in Section 5.
To begin with, we define N be the nonnegative integer set and N+ be the positive integer set. Throughout the following two sec-tions, we give the following hypotheses
(H1) 3 − 2βc > 0(H2) 1 − βc − αβγ b3 > 0(H3) 8 − 8βc + 2β2c2 + αβγ b3 > 0.
2. Coupled loops with delay
In what follows, we investigate the influence of delay τ on the stability of the synchronization system. Under the point of synchronization, the dynamics of system (1) is completely char-acterized by the following equation⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂u1(t, x)
∂t= d∇2u1(t, x) + bf
(u2(t, x)
) − u1(t, x),
∂u2(t, x)
∂t= d∇2u2(t, x) + bf
(u3(t, x)
) − u2(t, x),
∂u3(t, x)
∂t= d∇2u3(t, x) + bf
(u1(t, x)
) − u3(t, x)
+ cf(u3(t − τ , x)
),
(7)
where ui is the common component of a synchronous solution of system (1).
In the reminder of this section, we consider the synchronized system (7). System (7) admits u∗ = (u∗
1, u∗2, u
∗3)
T be the equilibrium point, letting
ui → ui − u∗i , i = 1,2,3,
one has the linearized equation of (7)⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂u1(t, x)
∂t= d∇2u1(t, x) + αbu2(t, x) − u1(t, x),
∂u2(t, x)
∂t= d∇2u2(t, x) + βbu3(t, x) − u2(t, x),
∂u3(t, x)
∂t= d∇2u3(t, x) + γ bu1(t, x) − u3(t, x)
+ βcu3(t − τ , x),
(8)
where α = f ′(u∗2), β = f ′(u∗
3), γ = f ′(u∗1).
Let
U (t) = (u1(t, ·), u2(t, ·), u3(t, ·)
)T
= (u1(t), u2(t), u3(t)
)T,
then (8) can be rewritten as an abstract differential equation in the phase space C = C([−τ , 0], X) of the form
dU (t) = D U (t) + L(Ut), (9)
dt
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where
D = diag{d,d,d}, = diag
{∂2/∂x2, ∂2/∂x2, ∂2/∂x2},
L(ϕ) =⎛⎝ αbϕ2(0) − ϕ1(0)
βbϕ3(0) − ϕ2(0)
γ bϕ1(0) − ϕ3(0) + βcϕ3(−τ )
⎞⎠ (10)
for ϕ = (ϕ1, ϕ2, ϕ3)T ∈ C . The characteristic equation of (9) is
λy − D y − L(eλ· y
) = 0, y ∈ dom( )\{0}. (11)
From the properties of the second order differential operator de-fined on the interval (0, π) with homogeneous Neumann boundary conditions, we know that the linear operator ∂2
∂x2 has the eigenval-
ues −n2 (n ∈ N) with the corresponding eigenfunctions on X are
ξ1n = (γn,0,0)T , ξ2
n = (0, γn,0)T , ξ3n = (0,0, γn)T
where γn = cos(nx) and {ξ1n , ξ2
n , ξ3n }∞n=0 construct a basis of the
phase space X . Therefore, any element y in X can be expanded as a Fourier series in the form
y =∞∑
n=0
Y Tn
⎛⎝ ξ1
nξ2
nξ3
n
⎞⎠ , Yn =
⎛⎝ 〈y, ξ1
n 〉〈y, ξ2
n 〉〈y, ξ3
n 〉
⎞⎠ (12)
and
L
⎛⎝ϕT
⎛⎝ ξ1
nξ2
nξ3
n
⎞⎠
⎞⎠ = (Lϕ)T
⎛⎝ ξ1
nξ2
nξ3
n
⎞⎠ , n ∈ N, (13)
for ϕ = (ϕ1, ϕ2, ϕ3)T ∈ C . Using this decomposition, we note that
(11) is equivalent to
∞∑n=0
Y Tn
(λI + Dn2 − M1
)⎛⎝ ξ1
nξ2
nξ3
n
⎞⎠ = 0, (14)
where
M1 =(−1 αb 0
0 −1 βbγ b 0 −1 + βce−λτ
).
Hence, we conclude that the characteristic equation (11) is equiv-alent to the sequence of characteristic equations
1(λ) := λ3 + 3λ2(dn2 + 1) + 3λ
(dn2 + 1
)2
− βc(λ + dn2 + 1
)2e−λτ + (
dn2 + 1)3
− αβγ b3 = 0, n ∈ N. (15)
In the following, we emphasize the importance of delay τ , regard-ing time delay as the bifurcation parameter.
Theorem 1. If delay τ is absent, that is, τ = 0, then the equilibrium point of the synchronized system (7) is locally asymptotically stable if and only if (H1)–(H3) hold.
Proof. When τ = 0, the characteristic equation (15) becomes
λ3 + P1λ2 + P2λ + P3 = 0, n ∈ N, (16)
where
P1 = 3dn2 + 3 − βc,
P2 = 3(dn2 + 1
)2 − 2βc(dn2 + 1
),
P3 = (dn2 + 1
)3 − βc(dn2 + 1
)2 − αβγ b3.
We first prove the sufficiency. It follows from (H1)–(H3) and n ≥ 0that, for any n ∈ N , P1 > 0, P2 > 0, P3 > 0 and P1 P2 − P3 > 0. By Routh–Hurwitz criteria, all the roots of (16) have negative real parts. Therefore, the equilibrium point of the synchronized system (7) is locally asymptotically stable when τ = 0.
The necessity is obvious. Otherwise, we assume that either (H1), (H2), or (H3) cannot be fulfilled, then it follows from Routh–Hurwitz criteria that, not all the roots of (16) have negative real parts, that is, the equilibrium point of the synchronized system (7)is not locally asymptotically stable. This completes the proof. �
Next, we shall carry out the linear stability analysis of the synchronized system (7). We suppose that τ �= 0, pure imaginary eigenvalues λ = iω with ω > 0 make 1(λ) = 0 when their real and imaginary parts are zero, namely,⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
βc[−ω2 + (
dn2 + 1)2]
sinωτ
− 2ωβc(dn2 + 1
)cosωτ = ω3 − 3ω
(dn2 + 1
)2,
βc[−ω2 + (
dn2 + 1)2]
cosωτ
+ 2ωβc(dn2 + 1
)sinωτ = −3ω2(dn2 + 1
)+ (
dn2 + 1)3 − αβγ b3,
(17)
taking the square, adding the equations and performing some sim-plification processes, we have
ω6 + Q 1ω4 + Q 2ω
2 + Q 3 = 0, n ∈ N, (18)
where
Q 1 = 3(dn2 + 1
)2 − β2c2,
Q 2 = 3(dn2 + 1
)4 − 2β2c2(dn2 + 1)2 + 6αβγ b3(dn2 + 1
),
Q 3 = (dn2 + 1
)6 − β2c2(dn2 + 1)4 − 2αβγ b3(dn2 + 1
)3
+ α2β2γ 2b6.
Particularly, when n = 0, we have
Q 10 = 3 − β2c2,
Q 20 = 3 − 2β2c2 + 6αβγ b3,
Q 30 = 1 − β2c2 − 2αβγ b3 + α2β2γ 2b6
= (1 − αβγ b3 + βc
)(1 − αβγ b3 − βc
).
Let z = ω2, then (18) can be rewritten into the following form
z3 + Q 1z2 + Q 2z + Q 3 = 0, n ∈ N. (19)
Then, we get the following result.
Theorem 2. Under the assumption of Theorem 1, furthermore (H4)–(H7)
are satisfied. Then the following statements are true.
(i) When τ ∈ [0, τ (0)10 ), the equilibrium point u∗ = (u∗
1, u∗2, u
∗3)
T of the synchronized system (7) is asymptotically stable.
(ii) The synchronized system (7) undergoes a Hopf bifurcation at u∗ =(u∗
1, u∗2, u
∗3)
T when τ = τ(0)10 . That is, system (7) has a branch of
periodic solutions bifurcating from u∗ near τ = τ(0)10 . Here, τ (0)
10 can be found in the proof of this theorem.(H4) 1 + βc − αβγ b3 < 0(H5) 3 − β2c2 > 0(H6) 2β2c2 − 3 < 6αβγ b3 < 6 − 4β2c2
(H7) d > |βc|.
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Proof. It is easy to know that there is no zero root of (15). In fact, substituting λ = 0 into (15), we have(dn2 + 1
)3 − βc(dn2 + 1
)2 − αβγ b3 = 0,
which yields that
d3n6 + d2n4(3 − βc) + dn2(3 − 2βc) + 1 − βc − αβγ b3 = 0.
(20)
According to (H1)–(H2), we have 3 − βc > 0, 3 − 2βc > 0 and 1 −βc − αβγ b3 > 0. Furthermore, noting that d > 0, n ≥ 0, then the equality (20) does not hold.
When n = 0, (19) can be written as
z3 + Q 10z2 + Q 20z + Q 30 = 0. (21)
It follows from (H2), (H4), (H5) and (H6) that Q 30 < 0, Q 10 > 0, Q 20 > 0. According to Descartes’s rule of signs [32], (21) has a unique positive real root, denoted by z10, and thus, (18) has a unique positive root ω10 = √
z10. By (17), we have
cosω10τ = ω410 + (2 + αβγ b3)ω2
10 + 1 − αβγ b3
βc(−ω210 + 1)2 + 4ω2
10βc. (22)
Thus, if we denote
τ( j)10 = 1
ω10
{cos−1
{ω4
10 + (2 + αβγ b3)ω210
βc(−ω210 + 1)2 + 4ω2
10βc
+ 1 − αβγ b3
βc(−ω210 + 1)2 + 4ω2
10βc
}}+ 2 jπ
ω10,
j = 0,1, · · · , (23)
then ±iω10 is a pair of purely imaginary roots of (15) at τ = τ( j)10 .
Clearly, τ (0)10 is the first value of τ ( j)
10 > 0, such that (15) has roots appearing on the imaginary axis. Furthermore, it follows from (H6)
and (H7) that, for n ∈ N+ , Q 1 > 0, Q 2 > 0, Q 3 > 0. That is, when n ∈ N+ , (18) has no positive real roots and hence (15) has no purely imaginary roots. Therefore, all the roots of (15) have neg-ative real parts for τ ∈ [0, τ (0)
10 ), all the roots of (15) except ±iω10
have negative real parts for τ = τ(0)10 . As n = 0, differentiating (15)
with respect to τ yields(dλ
dτ
)−1
= 3λ2 + 3 + 6λ + (τ (λ + 1) − 2)(λ + 1)βce−λτ
−λβc(λ + 1)2e−λτ. (24)
Eq. (24) with respect to λ, τ at iω10 and τ (0)10 , respectively, and
under (H5)–(H6), yields{d(Reλ(τ ))
dτ
}−1∣∣∣∣τ=τ
(0)10 , λ=iω10
= 3ω610 + 2ω4
10(3 − β2c2) + ω210(6αβγ b3 + 3 − 2β2c2)
4ω410β
2c2 + (ω310βc − ω10βc)2
> 0.
Hence, the transversal condition is satisfied. Furthermore, ω10 is simple. Otherwise, differentiating (15) (n = 0) with respect to λyields
3λ2 + 6λ + 3 + (τ (λ + 1) − 2
)(λ + 1)βce−λτ = 0. (25)
Obviously, the left side of (25) equals to the numerator of the right side of (24). According to transversality above, we can easily know that{
3λ2 + 6λ + 3 + (τ (λ + 1) − 2
)(λ + 1)βce−λτ
}∣∣λ=iω10, τ=τ
(0)10
�= 0.
This completes the proof. �
3. Ring structure with delays
Under the point of synchronization, the dynamics of system (4)is completely characterized by the following equation⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂u1(t, x)
∂t= d∇2u1(t, x) + bf
(u2(t, x)
) − u1(t, x),
∂u2(t, x)
∂t= d∇2u2(t, x) + bf
(u3(t, x)
) − u2(t, x),
∂u3(t, x)
∂t= d∇2u3(t, x) + bf
(u1(t − τ , x)
) − u3(t, x)
+ cf(u3(t, x)
),
(26)
where ui is the common component of a synchronous solution of system (4).
In the reminder of this section, we consider the synchronized system (26). System (26) admits u∗ = (u∗
1, u∗2, u
∗3)
T be the equilib-rium point, letting
ui → ui − u∗i , i = 1,2,3,
one has the linearized equation of (26)⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂u1(t, x)
∂t= d∇2u1(t, x) + αbu2(t, x) − u1(t, x),
∂u2(t, x)
∂t= d∇2u2(t, x) + βbu3(t, x) − u2(t, x),
∂u3(t, x)
∂t= d∇2u3(t, x) + γ bu1(t − τ , x) − u3(t, x)
+ βcu3(t, x),
(27)
where α = f ′(u∗2), β = f ′(u∗
3), γ = f ′(u∗1). Similar to the analysis
of the synchronized system (7), we conclude that the characteris-tic equation of (27) is equivalent to the sequence of characteristic equations
∞∑n=0
Y Tn
(λI + Dn2 − M2
)⎛⎝ ξ1
nξ2
nξ3
n
⎞⎠ = 0, (28)
where
M2 =( −1 αb 0
0 −1 βbγ be−λτ 0 −1 + βc
),
that is
2(λ) := λ3 + λ2[3(dn2 + 1
) − βc] + λ
[3(dn2 + 1
)2
− 2βc(dn2 + 1
)] + (dn2 + 1
)3 − βc(dn2 + 1
)2
− αβγ b3e−λτ = 0, n ∈ N. (29)
Theorem 3. If delay τ is absent, that is, τ = 0, then the equilibrium point of the synchronized system (26) is locally asymptotically stable if and only if (H1)–(H3) hold.
Proof. When τ = 0, (29) becomes
λ3 + P1λ2 + P2λ + P3 = 0, n ∈ N, (30)
where Pi (i = 1, 2, 3) are the same as the corresponding items of (16). The proof is the same as that of Theorem 1 and hence we shall omit it.
Next, we shall carry out the linear stability analysis of the syn-chronized system (26). We suppose that τ �= 0, pure imaginary eigenvalues λ = iω with ω > 0 make 2(λ) = 0 when their real and imaginary parts are zero, namely,
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⎧⎪⎨⎪⎩
αβγ b3 sinωτ = ω3 − ω[3(dn2 + 1
)2 − 2βc(dn2 + 1
)]αβγ b3 cosωτ = −ω2[3
(dn2 + 1
) − βc]
+ (dn2 + 1
)3 − βc(dn2 + 1
)2,
(31)
taking the square, adding the equations and performing some sim-plification processes, we have
ω6 + R1ω4 + R2ω
2 + R3 = 0, n ∈ N, (32)
where
R1 = 3(dn2 + 1
)2 − 2βc(dn2 + 1
) + β2c2,
R2 = 3(dn2 + 1
)4 − 4βc(dn2 + 1
)3 + 2β2c2(dn2 + 1)2
,
R3 = (dn2 + 1
)6 − 2βc(dn2 + 1
)5 + β2c2(dn2 + 1)4
− α2β2γ 2b6.
Particularly, when n = 0, we have
R10 = 3 − 2βc + β2c2,
R20 = 3 − 4βc + 2β2c2,
R30 = 1 − 2βc + β2c2 − α2β2γ 2b6
= (1 − βc + αβγ b3)(1 − βc − αβγ b3).
Let z = ω2, then (32) can be rewritten into the following form
z3 + R1z2 + R2z + R3 = 0, n ∈ N. (33)
Make the following assumes
(H8) 1 − βc + αβγ b3 < 0(H9) 5 − 5βc + β2c2 > 0(H10) β2c2d4 − α2β2γ 2b6 > 0 �Theorem 4. Under the assumption of Theorem 3, if (H8)–(H10) hold, then the following statements are true.
(i) When τ ∈ [0, τ (0)20 ), the equilibrium point u∗ = (u∗
1, u∗2, u
∗3)
T of the synchronized system (26) is asymptotically stable.
(ii) The synchronized system (26) undergoes a Hopf bifurcation at u∗ =(u∗
1, u∗2, u
∗3)
T when τ = τ(0)20 . That is, system (26) has a branch of
periodic solutions bifurcating from u∗ near τ = τ(0)20 . Here, τ (0)
20 can be found in the proof of this theorem.
Proof. It is easy to know that there is no zero root of (29). In fact, substituting λ = 0 into (29), we have(dn2 + 1
)3 − βc(dn2 + 1
)2 − αβγ b3,
which yields that
d3n6 + d2n4(3 − βc) + dn2(3 − 2βc) + 1 − βc − αβγ b3
= 0. (34)
The proof is similar to the corresponding part of Theorem 2, hence we shall omit it.
When n = 0, (33) can be written as
z3 + R10z2 + R20z + R30 = 0. (35)
It follows from (H1), (H2) and (H8) that R10 > 0, R20 > 0, R30 < 0. According to Descartes’s rule of signs [32], (35) has a unique posi-tive real root, denoted by z20, and thus (32) has a unique positive root ω20 = √
z20. By (31), we have
cosω20τ = −ω220(3 − βc) + 1 − βc
αβγ b3. (36)
Thus, if we denote
τ( j)20 = 1
ω20
{cos−1 −ω2
20(3 − βc) + 1 − βc
αβγ b3+ 2 jπ
}j = 0,1,2, · · · , (37)
then ±iω20 is a pair of purely imaginary roots of (29) at τ = τ( j)20 .
Clearly, τ (0)20 is the first value of τ ( j)
20 > 0, such that (29) has roots appearing on the imaginary axis. Furthermore, it follows from (H9)
and (H10) that, for n ∈ N+ , R1 > 0, R2 > 0, R3 > 0. That is, when n ∈ N+ , Eq. (32) has no positive real roots and hence (29) has no purely imaginary roots. Therefore, all the roots of (29) have nega-tive real parts for τ ∈ [0, τ (0)
20 ), all the roots of (29) except ±iω20
have negative real parts for τ = τ(0)20 . As n = 0, differentiating (29)
with respect to τ yields(dλ
dτ
)−1
= 3λ2 + 2λ(3 − βc) + 3 − 2βc
−λαβγ b3e−λτ+ αβγ b3τe−λτ
−λαβγ b3e−λτ.
(38)
Eq. (38) with respect to λ, τ at iω20 and τ (0)20 , respectively, yields{
d(Reλ(τ ))
dτ
}−1∣∣∣∣τ=τ
(0)20 , λ=iω20
= 3ω620 + (2β2c2 − 4βc + 6)ω4
20
ω220α
2β2γ 2b6+ (2β2c2 − 4βc + 3)ω2
20
ω220α
2β2γ 2b6
> 0.
Therefore, the transversal condition is satisfied. Furthermore, ω20is simple. Otherwise, differentiating equation (29) (n = 0) with re-spect to λ yields
3λ2 + 2λ(3 − βc) + 3 − 2βc + αβγ b3τe−λτ = 0. (39)
Obviously, the left side of (39) equals to the numerator of the right side of (38). According to transversality above, we can easily know that{
3λ2 + 2λ(3 − βc) + 3 − 2βc + αβγ b3τe−λτ}∣∣
λ=iω20, τ=τ(0)20
�= 0.
This completes the proof. �4. Numerical solutions and discussions
To supplement our theoretical work, in this section, we give some numerical simulations.
The activation function, determining the input–output relation denotes the intrinsic properties of neural systems. The history of the activation function starts from the original paper by Mccul-loch and Pitts [33]. In [34], the activation function is defined as a monotonically increasing function with a lower asymptote of 0 and an upper asymptote of 1. These activation functions are all called sigmoid function and have some “well-behaviors” (boundedness, differentiability, and monotonically increases). Basically, one of the most popular choice for activation function is
f (·) = tanh(k·) = ek· − e−k·
ek· + e−k· ,
k ∈R is the gain of the response function.The task is now to analyze the numbers of equilibrium points.
Equilibrium points of (7) or (26) satisfy
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Fig. 2. (a) The synchronization trajectory of system (1); (b) The equilibrium point (0, 0, 0)T of the synchronized system (7) is stable when τ = 6.2 < τ
(0)10 .
{ u1 = b tanh(ku2),
u2 = b tanh(ku3),
u3 = b tanh(ku1) + c tanh(ku3).
(40)
The above equations give
u3 = bG(u3) + c tanh(ku3),
where G(u3) = tanh[kb tanh(kb tanh(ku3))].Let
h(u3) := u3 − bG(u3) − c tanh(ku3), (41)
differentiating (41) gives
h′(u3) = 1 − ks23
(b3k2s2
1s22 + c
),
where⎧⎪⎨⎪⎩
s21 = sech2[kb tanh
(kb tanh(ku3)
)],
s22 = sech2[kb tanh(ku3)
],
s23 = sech2(ku3).
It is easy to know that 0 < s2i ≤ 1, i = 1, 2, 3. We distinguish two
cases.
1) 1 − ck − b3k3 < 0
We have h′(0) < 0, thus h(u3) is strictly decreasing at the ori-gin. Noting that h(0) = 0 and limu3→∞ h(u3) = ∞, by continuity, there is at least one positive value of u3, namely, u+
3 > 0, such that h(u+
3 ) = 0. By symmetry, h(−u+3 ) = 0 and these values determine
the other variables at a nontrivial equilibrium point. Therefore, sys-tem (7) or (26) has nontrivial equilibrium points.
2) 1 − ck − b3k3 ≥ 0
Fig. 3. (a) The synchronization trajectory of system (1); (b) Hopf bifurcation oc-curs from the equilibrium point (0, 0, 0)T of the synchronized system (7) when τ = 7.2 > τ
(0)10 .
(i) bk > 0, ck > 0, 1 − ck − b3k3 ≥ 0, then we have h′(u3) ≥ 1 −ck − b3k3 ≥ 0, thus h(u3) is increasing for u3. Noting that h(0) =0 and limu3→∞ h(u3) = ∞, therefore system (7) or (26) only has trivial equilibrium point.
(ii) bk < 0, ck < 0, then we have h′(u3) > 1, thus h(u3) is strictly increasing for u3, which shows that there is no nontriv-ial equilibrium point of system (7) or (26).
(iii) bk > 0, ck < 0, 1 − b3k3 ≥ 0, then we have h′(u3) > 1 −b3k3 ≥ 0, thus h(u3) is strictly increasing for u3. System (7) or (26)only has trivial equilibrium point.
(iv) bk < 0, ck > 0, 1 − ck ≥ 0, then we have h′(u3) > 1 − ck ≥ 0, thus h(u3) is strictly increasing for u3. System (7) or (26) only has trivial equilibrium point.
4.1. Impact of delay
Consider system (1) choosing excitatory synaptic strength b =0.75 and inhibitory coupling strength c = −1.06 with d = 1.1, k =0.9. Since ck < 0, bk > 0 and 1 − b3k3 = 0.6925 > 0, according to the analysis about the numbers of equilibrium points, we can ob-tain that the synchronized system (7) only has trivial equilibrium point. It is easy to check that (H1)–(H7) hold, using Theorem 2, the synchronized system (7) has the critical value τ 0
10 = 7.0660. The equilibrium point of synchronized system (7) is asymptoti-cally stable as τ = 6.2 < τ 0
10; unstable and the spatially homo-geneous periodic solutions emerge from the equilibrium point for τ = 7.2 > τ 0
10, that is, the asymptotically stable equilibrium point leads to periodic oscillations. The numerical simulations in Figs. 2and 3 show the good agreements with the results given in Theo-rem 2.
To observe the influence of delay on synchronous behavior of system (1), we illustrate synchronous phenomena of u3(t, x) −v3(t, x), fixing c, b, d and k as above. Different values of τ give no evident influence on the synchronized speed of ui(t, x) − vi(t, x)
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Fig. 4. Trajectory of (u3(t, x) − v3(t, x)) of system (1), the synchronization speed with different delays.
Fig. 5. (a) The synchronization trajectory of system (4); (b) The equilibrium point (0, 0, 0)T of the synchronized system (26) is stable when τ = 10 < τ
(0)20 .
Fig. 6. (a) The synchronization trajectory of system (4); (b) Hopf bifurcation occurs from the equilibrium point (0, 0, 0)T of the synchronized system (26) when τ =11 > τ
(0)20 .
Fig. 7. Trajectory of (u3(t, x)–v3(t, x)) of system (4), the synchronization speed with different delays.
(i = 1, 2) and hence we shall omit it. It is worth that, in Fig. 4(a), the increase of τ slow down the tendency to converge to the synchronization situation when τ < τ
(0)10 . On the contrary, when
τ > τ(0)10 , the increase of τ speed the synchronization tendency up
shown as Fig. 4(b). Obviously, the delay is effective to the syn-chronous speed.
We choose the parameter values of system (4) as c = −1, b =0.5, d = 0.8 and k = −1, i.e., the synaptic strength of neighborhood interaction is excitatory, meanwhile coupling strength is inhibitory. Since bk < 0, ck > 0 and 1 −ck = 0, according to the analysis above
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Fig. 8. The amplitude of ui(t, x) (i = 1, 2, 3) varies with delay τ increasing from 8 to 20.
Fig. 9. (a) The curve of h(u3) = 0 in (c, u3) plane with b = 2, k = −0.8. The syn-chronized system has five equilibrium points in Region 1, but a unique equilibrium point in Region 2; (b) The curve of h(u3) = 0 in (c, u3) plane with b = 0.1, k = 2. The synchronized system has three equilibrium points in Region 1, but a unique equilibrium point in Region 2.
we can obtain that the synchronized system (26) only has trivial equilibrium point. By calculating, all conditions of Theorem 4 are satisfied, so one has the critical value τ (0)
20 = 10.7669. The equilib-
rium point of system (26) is asymptotically stable as τ = 10 < τ(0);
20Fig. 10. (a) The synchronization trajectory of system (1); (b) The solution of system (7) tends to constant steady states u∗
2± with different initial values.
unstable and the spatially homogeneous periodic solutions emerge from the equilibrium point for τ = 11 > τ
(0)20 , that is, the asymp-
totically stable equilibrium point leads to periodic oscillations. The corresponding synchronization of system (4) and time history of synchronized system (26) are shown in Figs. 5 and 6.
To observe the impact of delay in the second scheme, we il-lustrate synchronous phenomena by Fig. 7 with different delays. It is worth that, with the increase of τ , the tendency to converge to synchronization is getting slower, which is different from the sit-uation in the first scheme. We show the synchronized speed of u3(t, x) and v3(t, x), since the synchronous phenomena of ui(t, x)and vi(t, x) (i = 1, 2) are similar to Fig. 7, we shall omit them.
In this part we discuss the effect of τ on the amplitude of sys-tem (26) when τ varies from 8 to 20 continuously. Fig. 8 gives the amplitude of ui , i = 1, 2, 3. From it we can see that when τ < τ
(0)20 ,
the amplitudes of system (26) are zeros. That is, the system is lo-cally asymptotical stable. When τ > τ
(0)20 , system (26) occurs period
oscillations, it is observed that when τ increases, the amplitude of the periodic solution is getting larger.
Noting the examples of the synchronization system above all have a unique trivial equilibrium point, in the following, we give the relationship between the numbers of equilibrium points and coupling strength c. Any u3 satisfying h(u3) = 0 determines all the other variables u1 and u2 at an equilibrium point in (40), the curve governed by h(u3) = 0 in the (c, u3) plane is shown in Fig. 9. It is
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Fig. 11. (a) The synchronization trajectory of system (4); (b) The solution of system (26) tends to constant steady states u∗+ and u∗− with different initial values.
seen that the synchronized system has different numbers of equi-librium points with the variation of coupling strength c.
Considering system (1) choose excitatory synaptic strength b = 2 with k = −0.8. In order to display five equilibrium points, c is fixed to be located in Region 1 of Fig. 9(a), for instance c = −4.5, which implies inhibitory coupling strength. According to Fig. 9(a), we know that the synchronized system (7) has five equi-librium points, that is, u∗
0 = (0, 0, 0)T , u∗1± = (±1.3381, ∓1.0113,
±0.6960)T , u∗2± = (±1.8267, ∓1.9341, ±2.5559)T , correspond-
ingly,
α = β = γ = k = −0.8,
α = f ′(∓1.0113) = −0.4419,
β = f ′(±0.6960) = −0.5955,
γ = f ′(±1.3381) = −0.3012,
α = f ′(∓1.9341) = −0.1327,
β = f ′(±2.5559) = −0.0518,
γ = f ′(±1.8267) = −0.1550,
respectively. In the case where u∗ = u∗0 or u∗ = u∗
1± , it is easy to know that (H1) cannot be fulfilled. By Theorem 1, when τ = 0, the equilibrium points u∗
0 and u∗ = u∗1± of synchronized system (7) are
unstable. In the case where u∗ = u∗2± , the parameters satisfy the
conditions of Theorem 1 which implies that when τ = 0, u∗ are
2±Fig. 12. (a) The synchronized trajectory of system (1) with different values of c; (b) The synchronized trajectory of system (4) with different values of c.
Fig. 13. The synchronization trajectory of system (4).
locally asymptotically stable. Furthermore, there is no imaginary roots of synchronized system (7) when τ increasing, that is to say, u∗
2± of synchronized system (7) are locally asymptotically stable for all τ ≥ 0. Taking τ = 0.5, the synchronization of system (1) at u∗
0 is shown in Fig. 10(a). Because of the limitation of the length for this paper, the synchronization of the system at other equilib-rium points shall be omitted. The synchronized states illustrated in Fig. 10(b) indicates that multiple unstability and bistability coexist.
Taking system (4) choose excitatory synaptic strength b = 0.1with k = 2. In order to display three equilibrium points, c is fixed to be located in Region 1 of Fig. 9(b), for instance c = 0.58,
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Fig. 14. (a) The synchronization trajectory of system (4) without the diffusion for (u1(t)–v1(t)). (b) The synchronization trajectory of system (4) without the diffu-sion for (u2(t)–v2(t)). (c) The synchronization trajectory of system (4) without the diffusion for (u3(t)–v3(t)).
which implies excitatory coupling strength. The synchronization system (26) has three equilibrium points u∗
0 = (0, 0, 0)T , u∗+ =(0.0123, 0.0618, 0.3609)T , u∗− = (−0.0123, −0.0618, −0.3609)T
correspondingly, α = β = γ = k = 2, and γ = f ′(±0.0123) =1.9988, α = f ′(±0.0618) = 1.9697, β = f ′(±0.3609) = 1.2360, re-spectively. In the case where u∗ = u∗
0, it is easy to know that (H2)
cannot be fulfilled, according to Theorem 3, when τ = 0, the equi-librium point u∗ = (0, 0, 0)T of system (26) is unstable. In the case
0Fig. 15. (a) The synchronization trajectory of system (42); (b) Trajectory of ui(t, x)(i = 1, 2, 3) of system (43), the equilibrium point (0, 0, 0)T is unstable.
where u∗ = u∗+ or u∗ = u∗− , the parameters satisfy the conditions of Theorem 3 which implies that when τ = 0, u∗+ and u∗− are sta-ble. Furthermore, there is no imaginary roots of system (26) when τ increasing, that is to say, u∗± of system (26) are asymptotically stable for all τ ≥ 0. Taking τ = 2, the synchronization of system (4)at u∗
0 is shown in Fig. 11(a). Given the limited space available, the synchronization of the system at other equilibrium points shall be omitted. The synchronized states illustrated in Fig. 11(b) indicates that unstability and bistability coexist.
4.2. Impact of the coupling strength c
In the following, we consider the role that the coupling strength c plays in synchronization speed.
Taking the excitatory connections b = 0.7 in system (1) and the inhibitory connections b = −0.7 in system (4), respectively, other parameters are fixed as the same, i.e., d = 0.5, k = 1, τ = 0.8. As the growth of the values of c, the synchronization speed be-comes faster in Fig. 12. The synchronous phenomena of ui(t, x)and vi(t, x) (i = 1, 2) with different c are similar to u3(t, x) and v3(t, x), hence we shall omit them.
4.3. Impact of diffusion
The appearance of diffusion terms adds some complications to the analysis and influences the dynamic behavior of the system. If the parameters in system (4) are taken as d = 0.8, c = 1, b = −0.4, k = 1, τ = 15, i.e., the synaptic strength is inhibitory, meanwhile
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Fig. 16. (a) The synchronization trajectory of system (42); (b) Trajectory of ui(t, x)(i = 1, 2, 3) of system (43), the equilibrium point (0, 0, 0)T is stable.
coupling strength is excitatory. The synchronization of system (4)is shown in Fig. 13. To deeply comprehend the impact of diffusion, we choose parameters as the same as those in Fig. 13 regarding d = 0 instead of d = 0.8, shown in Fig. 14. Though the system is also synchronous, but the synchronization speed is obviously slower than the speed with diffusion.
Next, we discuss the impact of diffusion on the stability of the synchronized equilibrium point without delay. In the absence of delay τ , the system (1) and (4) without diffusion item are both as follows
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
du1(t)
dt= bf
(u2(t)
) − u1(t),
du2(t)
dt= bf
(u3(t)
) − u2(t),
du3(t)
dt= bf
(u1(t)
) − u3(t) + cf(
v3(t)),
dv1(t)
dt= bf
(v2(t)
) − v1(t),
dv2(t)
dt= bf
(v3(t)
) − v2(t),
dv3(t)
dt= bf
(v1(t)
) − v3(t) + cf(u3(t)
),
(42)
the corresponding synchronized system is
Fig. 17. Different curves divide the parameter plane (b, c) into different regions with α = β = γ = k = 1.2.⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
du1(t)
dt= bf
(u2(t)
) − u1(t),
du2(t)
dt= bf
(u3(t)
) − u2(t),
du3(t)
dt= bf
(u1(t)
) − u3(t) + cf(u3(t)
).
(43)
Taking excitatory connections b = 0.68, excitatory couplingstrength c = 0.69, k = 1. By calculating we obtain 1 −ck −b3k3 = −0.0044 <0, it is easy to know that there are three equilibrium points u∗
0 = (0, 0, 0)T , u∗± = (±0.0482, ±0.0710, ±0.1048)T . We take u∗0
into account, therefore α = β = γ = f ′(0) = k = 1, the synchro-nization of system (42) is illustrated in Fig. 15(a). The synchronized equilibrium point u∗
0 is unstable shown as Fig. 15(b), the solu-tion with initial value (0.000000001, 0.000000002, 0.000000003)T
tends to the positive synchronized equilibrium point u∗+ .Consider system (42) and (43) with diffusion d = 1, keeping the
values of c, b, k as above, the synchronization of the diffusion sys-tem and the stable synchronized equilibrium point u∗
0 with τ = 0are shown in Figs. 16(a) and 16(b), respectively. The appearance of diffusion terms influence the stability of the synchronized equilib-rium point.
5. Conclusions
In this paper, reaction–diffusion neural networks with differ-ent arrangements that consists of a pair of loops each with three neurons and two-way coupling are considered. We analyze the synchronous stability and establish conditions under which syn-chronized Hopf bifurcation occurs.
Delay τ is effective to the synchronous tendency, it may slow down the speed to synchronization when τ increases (Figs. 4(a) and 7), it also may speed up the synchronous tendency (Fig. 4(b)).
The analysis and numerical solutions point to a possible non-trivial role of the position of the delay. Furthermore, it follows from Theorems 2 and 4 that the critical value τ (0)
10 and τ (0)20 are
dependent on the parameters c, b, k. This implies that the dif-ferent conditions for synchronized Hopf bifurcation can divide the parameter plane into some different regions.
Without loss of generality, as an example, we take α = β = γ =k = 1.2 into account, the curves yield different regions as shown in Fig. 17. The synchronized systems (7) and (26) may undergo Hopf bifurcation at critical τ for those parameter values located in Region 1 and Region 2, respectively. The position of the de-lay determines the selected parametric space. Inspired by Fig. 17and Theorems 2, 4, it is obvious that, if the parameters are chosen appropriately, then the equilibrium point of synchronized system
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Fig. 18. (a) The synchronization of the system (1); (b) The equilibrium point (0, 0, 0)T of the synchronized system (7) is stable when τ = 11.
(26) can undergo Hopf bifurcation at critical τ , in the meantime, the equilibrium point of synchronized system (7) is asymptotically stable for all τ ≥ 0. In turn, the situation is also true. For exam-ple, recall the numerical simulation about the synchronized system (26) in Figs. 5(b) and 6(b), the spatially homogeneous periodic so-lutions emerge from the equilibrium point for τ = 11 > τ
(0)20 . We
consider the synchronized system (7) with the same parameter values as those in system (26), the equilibrium point of the syn-chronized system (7) is stable for τ = 11, as shown in Fig. 18.
The synchronized stable pattern may be in the form of ei-ther trivial synchronized equilibrium point (Fig. 2(b), Fig. 5(b)) or multiple nontrivial synchronized equilibrium points (Figs. 10(b) and 11(b)). It is seen that the synchronized system has differ-ent numbers of equilibrium points with the variation of coupling strength c. In the synchronized states, multiple unstability and bistability (Fig. 10(b)) or unstability and bistability (Fig. 11(b)) may coexist. In addition, the growth of c may accelerate the synchro-nized speed (Fig. 12). It is worth that the appearance of diffusion terms adds some complications to the analysis and influences the dynamic behavior of the system. Numerical studies have been em-ployed to support and extend the obtained theoretical results.
Acknowledgement
This work was supported by National Natural Science Founda-tion of China under Grant 61174155 and Grant 11032009, Funding
of Jiangsu Innovation Program for Graduate Education KYZZ_0088, the Fundamental Research Funds for the Central Universities.
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