Neural Networks 49 (2014) 87–95
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Neural Networks journal homepage: www.elsevier.com/locate/neunet
Projective synchronization for fractional neural networks Juan Yu a , Cheng Hu a , Haijun Jiang a,∗ , Xiaolin Fan a,b a
College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, PR China
b
Foundation Department, Xinjiang Institute of Engineering, Urumqi, Xinjiang 830091, PR China
article
info
Article history: Received 27 December 2012 Received in revised form 8 October 2013 Accepted 10 October 2013 Keywords: Fractional-order Neural network Projective synchronization Fractional adaptive control
abstract In this paper, the global projective synchronization of fractional-order neural networks is investigated. First, a sufficient condition in the sense of Caputo’s fractional derivation to ensure the monotonicity of the continuous and differential functions and a new fractional-order differential inequality are derived, which play central roles in the investigation of the fractional adaptive control. Based on the preparation and some analysis techniques, some novel criteria are obtained to realize projective synchronization of fractional-order neural networks via combining open loop control and adaptive control. As some special cases, several control strategies are given to ensure the realization of complete synchronization, anti-synchronization and the stabilization of the addressed neural networks. Finally, an example with numerical simulations is given to show the effectiveness of the obtained results. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Although fractional calculus dates from 17th century, fractional differential systems have recently been used in diffusion, turbulence, electromagnetism, signal processing, and quantum evolution of complex systems. In the past few decades, the study of dynamics of fractional-order differential systems has attracted the interest of many scholars and many interesting and important results have been reported including the existence of chaos in factional-order differential systems such as fractional order Chua circuit (Hartley & Lorenzo, 1995), fractional order Chen system (Lu & Chen, 2006) and fractional order Lü system (Deng & Li, 2005), stability analysis (Baleanu, & Sadati, 2012; Li, Chen, & Podlubny, 2010) and synchronization (Deng, Li, Wang, & Li, 2009; Ge & Ou, 2008; Odibat, 2010). Since synchronization of chaotic systems was first introduced by Perora and Carroll (1990), it has become an active research subject in nonlinear science and has been widely investigated in many fields such as image processing, ecological systems and secure communications. At present, synchronization of chaotic fractional order differential systems becomes a challenging and interesting problem due to its potential applications in secure communication and control processing. So far, various types of chaos synchronization have been revealed to investigate fractional-order chaos synchronization which include complete synchronization (Yan &
∗
Corresponding author. Tel.: +86 13079900716. E-mail addresses: [email protected] (J. Yu), [email protected] (H. Jiang). 0893-6080/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.neunet.2013.10.002
Li, 2007), phase synchronization (Erjaee & Momani, 2008; Taghvafard & Erjaee, 2011), generalized synchronization (Wu, Lai, & Lu, 2012), lag synchronization (Zhu, He, & Zhou, 2011) and projective synchronization (Si, Sun, Zhang, & Chen, 2012; Wang & He, 2008; Xin, Chen, & Liu, 2011). Amongst all kinds of fractional-order chaos synchronization, projective synchronization has been extensively investigated in recent years because it can obtain faster communication with its proportional feature. In the literature (Wang & He, 2008), the authors pointed out that projective synchronization can be used to extend binary digital to M-nary digital communication for achieving fast communication in application to secure communications, so the research on projective synchronization has important theory significance and application value. Recently, fractional-order neural networks were presented and designed to distinguish the classical integer-order models (Arena, Caponetto, Fortuna, & Porto, 1998; Arena, Fortuna, & Porto, 2000; Boroomand & Menhaj, 2010; Petráš, 2006). In Petráš (2006), the authors pointed out that fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various processes. Actually, fractional-order systems have infinite memory. Taking into account these facts, the incorporation of a memory term into a neural network model is an extremely important improvement. In the literature (Lundstrom, Higgs, Spain, & Fairhall, 2008), the authors emphasized the utility of developing and studying fractional-order mathematical models of neural networks because fractional differentiation provides neurons with a fundamental and general computation ability that can contribute to efficient information processing, stimulus anticipation and frequency-independent phase shifts of oscillatory neuronal firing. In addition, fractional-order recurrent neural
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networks are expected to be very effective in applications such as parameter estimations due to the fact that they are characterized by infinite memory. Therefore, it is necessary and interesting to study fractional-order neural networks both in theory and in applications. Currently, some excellent results about fractional-order neural networks have been investigated, such as Kaslik and Sivasundaram (2011, 2012); Zhang, Qi, and Wang (2010); Zhou, Lin, Zhang, and Li (2010); Zhu, Zhou, and Zhang (2008). In Zhang et al. (2010), a fractional order three-dimensional Hopfield neural network was proposed and it was pointed out that chaotic behaviors can be emerged in fractional network. In Kaslik and Sivasundaram (2011, 2012), the local stability analysis for fractional-order neural networks was discussed by applying the linear stability theory of a fractional-order system Besides, the chaos control and synchronization for some simple fractional-order neural networks were discussed in Zhou et al. (2010), Zhu et al. (2008), Zhou, Li, and Zhu (2008) by mainly using Laplace transformation theory and numerical simulations. In Yu, Hu, and Jiang (2012), a class of fractional-order neural networks were investigated, α -exponential stability was introduced and some criteria were derived for such kind of stability of fractional-order neural networks. To the best of our knowledge, there are few results on the projective synchronization of fractional-order neural networks. Motivated by this, the globally asymptotically projective synchronization of fractional-order neural networks is concerned in this paper. The main contribution of this paper lies in the following aspects. First, a sufficient condition in the sense of Caputo’s fractional derivation is given to ensure the monotonicity of the continuous and differential functions. Besides, a new fractional-order differential inequality is derived, which plays a central role in the investigation of the fractional adaptive control. Based on those preparations and some analysis techniques, some novel criteria are derived to realize projective synchronization of fractional-order neural networks via combining open loop control and adaptive control. Especially, when the projective coefficient is appropriately chosen, the derived results can ensure the realization of globally asymptotically complete synchronization, globally asymptotically anti-synchronization and the globally asymptotically stabilization of chaotic fractional-order neural networks. It is noted that the Lyapunov-like method and analysis techniques used in this paper are totally different from the existing results such as Si et al. (2012), Xin et al. (2011), Wang and He (2008), Wang and Song (2009), Wang and Wang (2007), Wang, Zhang, and Ma (2012) in which the stability theorems of a fractional-order linear system and sliding mode control were utilized. It is believed that all of those are new and useful for the design and applications of fractional-order neural networks. This paper is organized as follows. In Section 2, some definitions and new fractional differential inequalities are introduced. The proposed model and some assumptions are given in Section 3. Some criteria for globally asymptotically projective synchronization of fractional-order neural networks are obtained in Section 4. The effectiveness and feasibility of the theoretical results are shown by some examples in Section 5.
Definition 1 (Kilbas, Srivastava, & Trujillo, 2006). The fractional integral of order α for a function f is defined as I f (t ) =
1
Γ (α)
D f (t ) =
Γ (n − α)
f (n) (τ )
t
1
(t − τ )α−n+1
t0
dτ ,
where t ≥ t0 and n is a positive integer such that n − 1 < α < n. Particularly, when 0 < α < 1, Dα f (t ) =
Γ (1 − α)
f ′ (τ )
t
1
(t − τ )α
t0
dτ .
Lemma 1 (Kilbas et al., 2006). Let Ω = [a, b] be an interval on the real axis R, let n = [α] + 1 for α ̸∈ N or n = α for α ∈ N. If y(t ) ∈ C n [a, b], then I α Dα y(t ) = y(t ) −
n−1 (k) y ( a)
(x − a)k .
k!
k=0
In particular, if 0 < α ≤ 1 and y(t ) ∈ C 1 [a, b], then I α Dα y(t ) = y(t ) − y(a). Lemma 2. Assume that x(t ) ∈ C 1 [a, b] and satisfies Dα x(t ) = f (t , x(t )) ≥ 0
(1)
for all t ∈ [a, b], then x(t ) is monotonously non-decreasing for 0 < α < 1. If Dα x(t ) = f (t , x(t )) ≤ 0, then, x(t ) is monotonously non-increasing for 0 < α < 1. Proof. Without loss of generality, we only prove that x(t ) is monotonously non-decreasing for 0 < α < 1 under condition (1). If the above statement is not true, then there exist a ≤ t2 < t1 ≤ b such that x(t1 ) < x(t2 ).
(2)
On the other hand, integrating both sides of Eq. (1) from t2 to t1 , according to Definition 1 and Lemma 1, we have x(t1 ) − x(t2 ) =
1
Γ (α)
t1
(t1 − τ )α−1 f (τ , x(τ ))dτ ≥ 0,
t2
which contradicts with inequality (2). Therefore, x(t ) is monotonously non-decreasing in this case. Remark 1. It is well known that x(t ) is monotonously nondecreasing if its derivative is equal to or greater than zero in the interval [a, b]. From Lemma 2, the property can be extended to Caputo’s fractional derivative with order 0 < α < 1. Lemma 3. Suppose function g (t ) is nondecreasing and differentiable on t ∈ [0, ∞), then for any constant h and t ∈ [0, ∞),
where 0 < α < 1.
In this section, we will recall some definitions of fractional calculation and introduce some useful lemmas.
α
α
Dα (g (t ) − h)2 ≤ 2(g (t ) − h)Dα g (t ),
2. Preliminaries
Definition 2 (Kilbas et al., 2006). Caputo’s fractional derivative of order α for a function f ∈ C n ([t0 , +∞), R) is defined by
t
(t − τ )
α−1
f (τ )dτ ,
Proof. According to Definition 2, we obtain Dα (g (t ) − h)2 =
=
Γ (1 − α) Γ (1 − α)
−
t
(t − τ )−α [(g (τ ) − h)2 ]′ dτ
0
2
t0
where t ≥ t0 and α > 0.
1
0
2h
Γ (1 − α)
t
(t − τ )−α g (τ )g ′ (τ )dτ t (t − τ )−α g ′ (τ )dτ 0
(3)
J. Yu et al. / Neural Networks 49 (2014) 87–95
2g (t )
t
Remark 3. If β = 0, the inequality in Definition 3 is reduced to the following form
(t − τ )−α g ′ (τ )dτ − 2hDα g (t ) Γ (1 − α) 0 = 2(g (t ) − h)Dα g (t ). ≤
The proof of Lemma 3 is completed.
89
lim ∥y(t )∥ = 0.
t →∞
Remark 2. Evidently, the equality
In this case, system (5) is said to be globally asymptotically stabilized to the origin.
((g (t ) − h)2 )′ = 2(g (t ) − h)g ′ (t )
4. Projective synchronization
is true for any differentiable function g (t ) and constant h. However, from Lemma 3 we can see that it is not fit for Caputo’s fractional derivative with order 0 < α < 1 and is replaced by the inequality (3). For example, the function g (t ) = t is nondecreasing for t ∈ [0, +∞) and for any constant h, we have Dα (t − h)2 =
2t 1−α
Γ (2 − α)
2(t − h)Dα t =
2t 1−α
Γ (2 − α)
t 2−α
−h ,
(t − h),
Choose the control input ui in slave system (5) as the following form
ui (t ) = vi (t ) + wi (t ), n v ( t ) = aij β fj (xj (t )) − fj (β xj (t )) + (β − 1)Ii , i j =1 w ( t ) = −di (t ) yi (t ) − β xi (t ) , i α D di (t ) = ki |yi (t ) − β xi (t )|, i ∈ I,
(6)
where each ki > 0 and β is the projective coefficient.
Obviously, when 0 < α < 1, Dα (t − h)2 < 2(t − h)Dα t .
Remark 4. In fact, control scheme (6) is a hybrid control, vi (t ) is an open loop control and wi (t ) is an adaptive feedback control.
3. Model description
Let ei (t ) = yi (t ) − β xi (t ) (i ∈ I) be the synchronization errors. From system (4) and system (5), the error system can be described by
In this section, we consider a class of fractional-order neural networks as the master system described by Dα xi (t ) = −ci xi (t ) +
n
(4)
where i ∈ I = {1, 2, . . . , n}, 0 < α < 1, n corresponds to the number of units in a neural network; xi (t ) corresponds to the state of the ith unit at time t; fj (·) denotes the activation function of the jth neuron; aij denotes the constant connection weight of the jth neuron on the ith neuron; ci > 0 represents the rate with which the ith neuron will reset its potential to the resting state when disconnected from the network and each Ii denotes an external input. To ensure the existence and uniqueness of the solutions of system (4) (see, Kilbas et al., 2006), the following assumption is given. (H1 ) Functions fj are Lipschitz-continuous on R with Lipschitz constants Lj > 0, i.e.,
|fj (u) − fj (v)| ≤ Lj |u − v| for all u, v ∈ R and j ∈ I. The slave system is given by n
n
aij fj (yj (t )) − fj (β xj (t )) − di (t )ei (t ). (7)
j =1
aij fj (xj (t )) + Ii ,
j =1
Dα yi (t ) = −ci yi (t ) +
Dα ei (t ) = −ci ei (t ) +
aij fj (yj (t )) + Ii + ui (t ),
Evidently, ei (t ) = 0 is a trivial solution of the error system (7), and the problem of globally asymptotically projective synchronization between system (4) and system (5) is converted to the globally asymptotically stability problem for the zero solution. Theorem 1. Under the assumption (H1 ), system (4) and system (5) are globally asymptotically projective synchronized based on the control scheme (6). Proof. Assume that x(t ) = (x1 (t ), . . . , xn (t ))T and y(t ) = (y1 (t ), . . . , yn (t ))T are any two solutions of system (4) and system (5) with initial values x(0) = (x1 (0), . . . , xn (0))T and y(0) = (y1 (0), . . . , yn (0))T satisfying ei (0) ̸= 0 for i ∈ I, respectively. Evidently, ei (t ) = 0 is a solution of the error system (7). From the existence and uniqueness theorem of fractional order differential functions (Yu et al., 2012), ei (0)ei (t ) > 0 for t ≥ 0. If ei (0) > 0, then ei (t ) > 0 and Dα |ei (t )| =
(5)
j =1
for all i ∈ I, where each ui (t ) is a control input. Definition 3. If there exists nonzero constants β ∈ R, such that for any two solutions x(t ) and y(t ) of system (4) and system (5) with different initial values denoted by x0 and y0 , one has lim ∥y(t ) − β x(t )∥ = 0,
Γ (1 − α)
0
dτ = Dα ei (t ),
(t − τ )α
and if ei (0) < 0, then ei (t ) < 0 and Dα |ei (t )| = −
Γ (1 − α)
e′i (τ )
t
1
0
(t − τ )α
dτ = −Dα ei (t ).
Hence, Dα |ei (t )| = sgn(ei (t ))Dα ei (t )
n = sgn(ei (t )) −ci ei (t ) + aij [fj (yj (t )) j =1
t →∞
then, master system (4) and slave system (5) are said to be globally asymptotically projective synchronized, where ∥ · ∥ denotes the Euclidean norm, and β denotes the projective coefficient. Especially, master system (4) and slave system (5) are said to be globally asymptotically complete synchronized if β = 1 and are said to be globally asymptotically anti-synchronized if β = −1.
e′i (τ )
t
1
− fj (β xj (t ))] − di (t )ei (t ) . Construct the following auxiliary function V (t ) = U (t ) +
n 1 i=1
2ki
2
di (t ) − di ,
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J. Yu et al. / Neural Networks 49 (2014) 87–95
Similarly, for tk − T ≤ t ≤ tk and k = 1, 2, . . . , it follows from (9) and (10) that
where U (t ) =
n
|ei |,
U (tk ) − U (t ) ≤
i=1
each di is an adaptive constant which is determined in the later analysis. It follows from (6) and Lemma 2 that each di (t ) is nondecreasing, which together with Lemma 3, the fractional-order derivatives of V (t ) along the solutions of system (7) can be derived as Dα V (t ) = Dα U (t ) +
n 1
2ki
i=1
≤
n
sgn(ei (t )) −ci ei (t ) +
n
i=1
aij fj (yj (t )) − fj (β xj (t ))
ki
n ≤ −ci |ei (t )| + |aij |Lj |ej (t )| − di |ei (t )|
n
2
ϵ 2
,
tk − T ≤ t ≤ tk ,
n
n
i=1
j =1
ci −
|aji |Li − ci ,
Therefore,
ϵ 2
,
tk − T ≤ t ≤ tk + T ,
(11)
(12)
With inequality (8) and (11), when tk − T ≤ t ≤ tk + T , we get
|aji |Li + di |ei (t )|.
ϵ
Dα V (t ) ≤ − λ. 2 As a result, for any k = 1, 2, . . . , we have
i ∈ I.
V tk + T − V tk − T
Denote
≤− =−
di + ci −
1≤i≤n
k = 1, 2, . . . .
tk−1 + T < tk − T < tk + T < tk+1 − T .
j =1
λ = min
k = 1, 2, . . . .
j=1
Let the adaptive constants satisfy di >
(tk − t )α
Without loss of generality, we assume these interval disjoint and t1 − T > 0, then for any k = 1, 2, . . . , we have
n
=−
Γ (α + 1) ϵ ≤ ,
U (t ) ≥
n 1 − di (t )ei (t ) + (di (t ) − di )Dα di (t )
i=1
(tk − τ )α−1 dτ
t
M
=
U (t ) ≥
j =1
i=1
Γ (α)
which implies that
Dα (di (t ) − di )2
tk
M
n
|aji |Li
> 0.
Then t ≥ 0.
(8)
t ≥ 0,
which implies that ei (t ) and di (t ) are bounded on t ≥ 0. Then, there exists a constant M > 0 such that
|Dα U (t )| ≤ M ,
2α−1 ϵλT α
Γ (α + 1)
tk + T − τ
α−1
dτ
tk − T
.
(13)
V (tk−1 + T ) ≥ V (tk − T ),
k = 1, 2, . . . .
(14)
By (13) and (14),
According to Lemma 2, we have V (t ) ≤ V (0),
tk + T
On the other hand, according to Lemma 2 and inequality (8), V (t ) is monotonously non-increasing, which together with (12), we obtain
j =1
Dα V (t ) ≤ −λU (t ),
ϵ λ 2Γ (α)
t ≥ 0.
(9)
V (tk + T ) − V (0) ≤ −
2α−1 ϵλT α
Γ (α + 1)
k,
which reveals that V (tk + T ) → −∞ as k → +∞, which contradict with V (t ) ≥ 0. Hence
In the following, we will prove
lim U (t ) = 0,
lim U (t ) = 0.
t →∞
t →∞
Otherwise, there exist a constant ϵ > 0 and time series {tk } satisfying 0 < t1 < t2 < · · · < tk < tk+1 < · · · and limk→∞ tk = ∞ such that
that is, master system (4) and slave system (5) are globally asymptotically projective synchronized based on the control scheme (6).
U (tk ) ≥ ϵ,
Remark 5. Obviously, the adaptive control gains di (t ) tend to some positive constants when system (4) and system (5) achieve projective synchronization.
k = 1, 2, . . . .
(10)
1 ( Γ (α+1)ϵ ) α
> 0. When tk ≤ t ≤ tk + T for any Denote T = 2M k = 1, 2, . . . , it follows from (9) and (10) that U (t ) − U (tk ) ≥ −
M
Γ (α)
t
(t − τ )
dτ
tk
M
=−
α−1
Γ (α + 1) ϵ ≥− ,
(t − tk )α
2
ϵ
2
,
tk ≤ t ≤ tk + T ,
ui (t ) = −di (t )(yi (t ) − xi (t )), Dα di (t ) = ki |yi (t ) − xi (t )|.
(15)
In this case, the following result is directly derived from Theorem 1.
which shows that U (t ) ≥
When β = 1, the control input (6) is reduced to the following form
k = 1, 2, . . . .
Corollary 1. Assume that (H1 ) holds. System (4) and system (5) are globally asymptotically complete synchronized under the adaptive control (15).
J. Yu et al. / Neural Networks 49 (2014) 87–95
Evidently, when β rewritten as
=
91
−1, the control input (6) can be
n aij −fj (xj (t )) − fj (−xj (t )) ui (t ) = j =1 − 2Ii − di (t ) yi (t ) + xi (t ) , α D di (t ) = ki |yi (t ) + xi (t )|.
(16)
As a special case of Theorem 1, we have the following result. Corollary 2. Assume that (H1 ) holds. System (4) and system (5) are globally asymptotically anti-synchronized under the control strategy (16). Remark 6. If each activation function fj is an odd function in model (4), the controller (16) can be rewritten in the following form
Fig. 1. Chaotic behavior of system (4) with initial value (0.2, −0.5, 0.8).
ui (t ) = −2Ii − di (t ) yi (t ) + xi (t ) , Dα di (t ) = ki |yi (t ) + xi (t )|.
It follows from Corollary 2 that system (4) and system (5) are globally asymptotically anti-synchronized in this case. If β = 0, the control scheme (6) is degenerated to n u (t ) = − a f (0) − I − d (t )y (t ), i
ij j
j=1 Dα d (t ) = k |y (t )|. i
i
i
i
i
(17)
i
The following statements are directly obtained from Theorem 1 in this case. Corollary 3. Suppose that (H1 ) holds. System (5) is globally asymptotically stabilized to the origin under the control scheme (17). Fig. 2. Synchronization errors with β = 3.
Remark 7. The projective synchronization for a class of fractional neural networks is proposed via combining open loop control with adaptive control in this paper. The derived criteria are novel and effective. In fact, the designed control scheme and analysis technique can be extended to the investigation of synchronization for fractional chaotic systems. Remark 8. In this paper, a fractional adaptive control is introduced to realize projective synchronization. It is easy to see that the adaptive control is related to the differential order of systems, i.e., the different orders of systems affect the synchronized adaptive feedback gains. Hence, the synchronization of fractional neural network is influenced by the order of fractional system. Remark 9. Recently, projective synchronization of the chaotic and hyperchaotic systems with fractional-order has been extensively investigated based on the stability theorems of a fractional-order linear system (Si et al., 2012; Wang & He, 2008; Wang & Song, 2009; Wang & Wang, 2007; Wang et al., 2012; Xin et al., 2011), in which the eigenvalues of the coefficient matrix or Jacobian matrix were required to be calculated, this leads to a large amount of calculation and the obtained synchronization may be local such as Wang and Wang (2007). Different from these results, in this paper, the globally projective synchronization of fractional-order neural networks is proposed for the first time by introducing a new fractional-order differential inequality and by means of the Lyapunov-like method, some effective analysis techniques such as reduction to absurdity, and the fractional-order adaptive control. Evidently, the proposed methods in this paper are meaningful and new and can be extended to the study of the projective synchronization of the chaotic and hyperchaotic systems with fractional-order.
Fig. 3. Evolutions of master–slave systems with β = 3.
5. Numerical simulations In this section, several numerical examples are given to verify the effectiveness of the theoretical results. In system (4), select x = (x1 , x2 , x3 )T , α = 0.98, fj (xj ) = tanh(xj ) for j = 1, 2, 3, c1 = c2 = c3 = 1, I1 = I2 = I3 = 0 and 2 1.8 −4.75
A = (aij )3×3 =
−1.2 1.71 0
0 1.15 . 1.1
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J. Yu et al. / Neural Networks 49 (2014) 87–95
Fig. 4. Trajectories of master–slave systems with β = 3.
Fig. 7. Evolutions of master–slave systems with β = −2.
Fig. 5. Time responses of di (t ) with β = 3.
Fig. 6. Synchronization errors with β = −2.
Under those parameters, system (4) has a chaotic attractor, which is shown in Fig. 1. In control scheme (6), choose k1 = 0.05, k2 = 0.06, k3 = 0.08, d1 (0) = d2 (0) = d3 (0) = 0.05. Select the projective coefficient β = 3. According to Theorem 1, master system (4) and response system (5) are globally asymptotically projective synchronized, which is verified in Figs. 2–5. In Fig. 2, the projective synchronization errors with different initial values converge to zero, which show that master system (4) and slave system (5) are globally asymptotically projective
Fig. 8. Trajectories of master–slave systems with β = −2.
Fig. 9. Time responses of di (t ) with β = −2.
synchronized with projective coefficients β = 3 and they are verified in Figs. 3 and 4. In addition, the adaptive gains di (t ) (i = 1, 2, 3) converge to some positive constants in this case, see Fig. 5. Similarly, the projective synchronization with projective coefficient β = −2 is simulated in Figs. 6–9. In the following, we consider the globally asymptotically complete synchronization of system (4) and system (5). In this case, β = 1. Choose k1 = k2 = k3 = 0.03, d1 (0) = d2 (0) = d3 (0) = 0.01. It follows from Corollary 1 that master system (4) and slave system (5) are globally asymptotically complete synchronized.
J. Yu et al. / Neural Networks 49 (2014) 87–95
Fig. 10. Synchronization errors with β = 1.
Fig. 11. Evolutions of master–slave systems with β = 1.
93
Fig. 13. Time responses of di (t ) with β = 1.
Fig. 14. Synchronization errors with β = −1.
Fig. 12. Trajectories of master–slave systems with β = 1.
The synchronization errors and synchronization evolutions are presented in Figs. 10–12, respectively. The adaptive gains are also shown in Fig. 13, from which we can see that di (t ) (i = 1, 2, 3) approach to some positive constant. Similarly, when we choose β = −1, and select k1 = k2 = k3 = 0.1, d1 (0) = d2 (0) = d3 (0) = 0.01, Figs. 14–17 imply that system (4) and system (5) are globally asymptotically anti-synchronized. When β = 0, according to Corollary 3, system (5) is globally asymptotically stabilized to the origin, which is verified in Figs. 18–20 with control parameters k1 = k2 = k3 = 0.2 and d1 (0) = d2 (0) = d3 (0) = 0.01.
Fig. 15. Evolutions of master–slave systems with β = −1.
6. Conclusion The globally asymptotically projective synchronization of fractional-order neural networks is concerned by designing a novel control input in this paper. To obtain the main results, a sufficient condition in the sense of Caputo’s fractional derivation is given to ensure the monotonicity of the continuous and differential functions. Besides, a new fractional-order differential inequality
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J. Yu et al. / Neural Networks 49 (2014) 87–95
Fig. 16. Trajectories of master–slave systems with β = −1.
Fig. 19. Trajectory of system (5) with β = 0.
Fig. 17. Time responses of di (t ) with β = −1.
Fig. 20. Time responses of di (t ) with β = 0.
Acknowledgments This work was supported by Natural Science Foundation of Xinjiang (Grant No. 2013211B06), China Postdoctoral Science Foundation funded project (2013M540782), Natural Science Foundation of Xinjiang University (Grant No. BS120101), and National Natural Science Foundation of People’s Republic of China (Grant No. 61164004). References
Fig. 18. Time response of system (5) with β = 0.
is derived, which differs from the corresponding integer-order case and plays a central role in the investigation of the fractional adaptive control. Based on those preparations, some criteria are derived to realize projective synchronization of fractional-order neural networks via combining open loop control and adaptive control. It is worthwhile to note that the complete synchronization, anti-synchronization and the stabilization of the addressed neural networks are guaranteed as some special cases of our main results. It is believed that our results should provide some practical guidelines for engineering applications of fractional-order neural networks. The future work will focus on the synchronization of fractional-order delayed neural networks.
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Contents lists available at ScienceDirect
Neural Networks journal homepage: www.elsevier.com/locate/neunet
Projective synchronization for fractional neural networks Juan Yu a , Cheng Hu a , Haijun Jiang a,∗ , Xiaolin Fan a,b a
College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, PR China
b
Foundation Department, Xinjiang Institute of Engineering, Urumqi, Xinjiang 830091, PR China
article
info
Article history: Received 27 December 2012 Received in revised form 8 October 2013 Accepted 10 October 2013 Keywords: Fractional-order Neural network Projective synchronization Fractional adaptive control
abstract In this paper, the global projective synchronization of fractional-order neural networks is investigated. First, a sufficient condition in the sense of Caputo’s fractional derivation to ensure the monotonicity of the continuous and differential functions and a new fractional-order differential inequality are derived, which play central roles in the investigation of the fractional adaptive control. Based on the preparation and some analysis techniques, some novel criteria are obtained to realize projective synchronization of fractional-order neural networks via combining open loop control and adaptive control. As some special cases, several control strategies are given to ensure the realization of complete synchronization, anti-synchronization and the stabilization of the addressed neural networks. Finally, an example with numerical simulations is given to show the effectiveness of the obtained results. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Although fractional calculus dates from 17th century, fractional differential systems have recently been used in diffusion, turbulence, electromagnetism, signal processing, and quantum evolution of complex systems. In the past few decades, the study of dynamics of fractional-order differential systems has attracted the interest of many scholars and many interesting and important results have been reported including the existence of chaos in factional-order differential systems such as fractional order Chua circuit (Hartley & Lorenzo, 1995), fractional order Chen system (Lu & Chen, 2006) and fractional order Lü system (Deng & Li, 2005), stability analysis (Baleanu, & Sadati, 2012; Li, Chen, & Podlubny, 2010) and synchronization (Deng, Li, Wang, & Li, 2009; Ge & Ou, 2008; Odibat, 2010). Since synchronization of chaotic systems was first introduced by Perora and Carroll (1990), it has become an active research subject in nonlinear science and has been widely investigated in many fields such as image processing, ecological systems and secure communications. At present, synchronization of chaotic fractional order differential systems becomes a challenging and interesting problem due to its potential applications in secure communication and control processing. So far, various types of chaos synchronization have been revealed to investigate fractional-order chaos synchronization which include complete synchronization (Yan &
∗
Corresponding author. Tel.: +86 13079900716. E-mail addresses: [email protected] (J. Yu), [email protected] (H. Jiang). 0893-6080/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.neunet.2013.10.002
Li, 2007), phase synchronization (Erjaee & Momani, 2008; Taghvafard & Erjaee, 2011), generalized synchronization (Wu, Lai, & Lu, 2012), lag synchronization (Zhu, He, & Zhou, 2011) and projective synchronization (Si, Sun, Zhang, & Chen, 2012; Wang & He, 2008; Xin, Chen, & Liu, 2011). Amongst all kinds of fractional-order chaos synchronization, projective synchronization has been extensively investigated in recent years because it can obtain faster communication with its proportional feature. In the literature (Wang & He, 2008), the authors pointed out that projective synchronization can be used to extend binary digital to M-nary digital communication for achieving fast communication in application to secure communications, so the research on projective synchronization has important theory significance and application value. Recently, fractional-order neural networks were presented and designed to distinguish the classical integer-order models (Arena, Caponetto, Fortuna, & Porto, 1998; Arena, Fortuna, & Porto, 2000; Boroomand & Menhaj, 2010; Petráš, 2006). In Petráš (2006), the authors pointed out that fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various processes. Actually, fractional-order systems have infinite memory. Taking into account these facts, the incorporation of a memory term into a neural network model is an extremely important improvement. In the literature (Lundstrom, Higgs, Spain, & Fairhall, 2008), the authors emphasized the utility of developing and studying fractional-order mathematical models of neural networks because fractional differentiation provides neurons with a fundamental and general computation ability that can contribute to efficient information processing, stimulus anticipation and frequency-independent phase shifts of oscillatory neuronal firing. In addition, fractional-order recurrent neural
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J. Yu et al. / Neural Networks 49 (2014) 87–95
networks are expected to be very effective in applications such as parameter estimations due to the fact that they are characterized by infinite memory. Therefore, it is necessary and interesting to study fractional-order neural networks both in theory and in applications. Currently, some excellent results about fractional-order neural networks have been investigated, such as Kaslik and Sivasundaram (2011, 2012); Zhang, Qi, and Wang (2010); Zhou, Lin, Zhang, and Li (2010); Zhu, Zhou, and Zhang (2008). In Zhang et al. (2010), a fractional order three-dimensional Hopfield neural network was proposed and it was pointed out that chaotic behaviors can be emerged in fractional network. In Kaslik and Sivasundaram (2011, 2012), the local stability analysis for fractional-order neural networks was discussed by applying the linear stability theory of a fractional-order system Besides, the chaos control and synchronization for some simple fractional-order neural networks were discussed in Zhou et al. (2010), Zhu et al. (2008), Zhou, Li, and Zhu (2008) by mainly using Laplace transformation theory and numerical simulations. In Yu, Hu, and Jiang (2012), a class of fractional-order neural networks were investigated, α -exponential stability was introduced and some criteria were derived for such kind of stability of fractional-order neural networks. To the best of our knowledge, there are few results on the projective synchronization of fractional-order neural networks. Motivated by this, the globally asymptotically projective synchronization of fractional-order neural networks is concerned in this paper. The main contribution of this paper lies in the following aspects. First, a sufficient condition in the sense of Caputo’s fractional derivation is given to ensure the monotonicity of the continuous and differential functions. Besides, a new fractional-order differential inequality is derived, which plays a central role in the investigation of the fractional adaptive control. Based on those preparations and some analysis techniques, some novel criteria are derived to realize projective synchronization of fractional-order neural networks via combining open loop control and adaptive control. Especially, when the projective coefficient is appropriately chosen, the derived results can ensure the realization of globally asymptotically complete synchronization, globally asymptotically anti-synchronization and the globally asymptotically stabilization of chaotic fractional-order neural networks. It is noted that the Lyapunov-like method and analysis techniques used in this paper are totally different from the existing results such as Si et al. (2012), Xin et al. (2011), Wang and He (2008), Wang and Song (2009), Wang and Wang (2007), Wang, Zhang, and Ma (2012) in which the stability theorems of a fractional-order linear system and sliding mode control were utilized. It is believed that all of those are new and useful for the design and applications of fractional-order neural networks. This paper is organized as follows. In Section 2, some definitions and new fractional differential inequalities are introduced. The proposed model and some assumptions are given in Section 3. Some criteria for globally asymptotically projective synchronization of fractional-order neural networks are obtained in Section 4. The effectiveness and feasibility of the theoretical results are shown by some examples in Section 5.
Definition 1 (Kilbas, Srivastava, & Trujillo, 2006). The fractional integral of order α for a function f is defined as I f (t ) =
1
Γ (α)
D f (t ) =
Γ (n − α)
f (n) (τ )
t
1
(t − τ )α−n+1
t0
dτ ,
where t ≥ t0 and n is a positive integer such that n − 1 < α < n. Particularly, when 0 < α < 1, Dα f (t ) =
Γ (1 − α)
f ′ (τ )
t
1
(t − τ )α
t0
dτ .
Lemma 1 (Kilbas et al., 2006). Let Ω = [a, b] be an interval on the real axis R, let n = [α] + 1 for α ̸∈ N or n = α for α ∈ N. If y(t ) ∈ C n [a, b], then I α Dα y(t ) = y(t ) −
n−1 (k) y ( a)
(x − a)k .
k!
k=0
In particular, if 0 < α ≤ 1 and y(t ) ∈ C 1 [a, b], then I α Dα y(t ) = y(t ) − y(a). Lemma 2. Assume that x(t ) ∈ C 1 [a, b] and satisfies Dα x(t ) = f (t , x(t )) ≥ 0
(1)
for all t ∈ [a, b], then x(t ) is monotonously non-decreasing for 0 < α < 1. If Dα x(t ) = f (t , x(t )) ≤ 0, then, x(t ) is monotonously non-increasing for 0 < α < 1. Proof. Without loss of generality, we only prove that x(t ) is monotonously non-decreasing for 0 < α < 1 under condition (1). If the above statement is not true, then there exist a ≤ t2 < t1 ≤ b such that x(t1 ) < x(t2 ).
(2)
On the other hand, integrating both sides of Eq. (1) from t2 to t1 , according to Definition 1 and Lemma 1, we have x(t1 ) − x(t2 ) =
1
Γ (α)
t1
(t1 − τ )α−1 f (τ , x(τ ))dτ ≥ 0,
t2
which contradicts with inequality (2). Therefore, x(t ) is monotonously non-decreasing in this case. Remark 1. It is well known that x(t ) is monotonously nondecreasing if its derivative is equal to or greater than zero in the interval [a, b]. From Lemma 2, the property can be extended to Caputo’s fractional derivative with order 0 < α < 1. Lemma 3. Suppose function g (t ) is nondecreasing and differentiable on t ∈ [0, ∞), then for any constant h and t ∈ [0, ∞),
where 0 < α < 1.
In this section, we will recall some definitions of fractional calculation and introduce some useful lemmas.
α
α
Dα (g (t ) − h)2 ≤ 2(g (t ) − h)Dα g (t ),
2. Preliminaries
Definition 2 (Kilbas et al., 2006). Caputo’s fractional derivative of order α for a function f ∈ C n ([t0 , +∞), R) is defined by
t
(t − τ )
α−1
f (τ )dτ ,
Proof. According to Definition 2, we obtain Dα (g (t ) − h)2 =
=
Γ (1 − α) Γ (1 − α)
−
t
(t − τ )−α [(g (τ ) − h)2 ]′ dτ
0
2
t0
where t ≥ t0 and α > 0.
1
0
2h
Γ (1 − α)
t
(t − τ )−α g (τ )g ′ (τ )dτ t (t − τ )−α g ′ (τ )dτ 0
(3)
J. Yu et al. / Neural Networks 49 (2014) 87–95
2g (t )
t
Remark 3. If β = 0, the inequality in Definition 3 is reduced to the following form
(t − τ )−α g ′ (τ )dτ − 2hDα g (t ) Γ (1 − α) 0 = 2(g (t ) − h)Dα g (t ). ≤
The proof of Lemma 3 is completed.
89
lim ∥y(t )∥ = 0.
t →∞
Remark 2. Evidently, the equality
In this case, system (5) is said to be globally asymptotically stabilized to the origin.
((g (t ) − h)2 )′ = 2(g (t ) − h)g ′ (t )
4. Projective synchronization
is true for any differentiable function g (t ) and constant h. However, from Lemma 3 we can see that it is not fit for Caputo’s fractional derivative with order 0 < α < 1 and is replaced by the inequality (3). For example, the function g (t ) = t is nondecreasing for t ∈ [0, +∞) and for any constant h, we have Dα (t − h)2 =
2t 1−α
Γ (2 − α)
2(t − h)Dα t =
2t 1−α
Γ (2 − α)
t 2−α
−h ,
(t − h),
Choose the control input ui in slave system (5) as the following form
ui (t ) = vi (t ) + wi (t ), n v ( t ) = aij β fj (xj (t )) − fj (β xj (t )) + (β − 1)Ii , i j =1 w ( t ) = −di (t ) yi (t ) − β xi (t ) , i α D di (t ) = ki |yi (t ) − β xi (t )|, i ∈ I,
(6)
where each ki > 0 and β is the projective coefficient.
Obviously, when 0 < α < 1, Dα (t − h)2 < 2(t − h)Dα t .
Remark 4. In fact, control scheme (6) is a hybrid control, vi (t ) is an open loop control and wi (t ) is an adaptive feedback control.
3. Model description
Let ei (t ) = yi (t ) − β xi (t ) (i ∈ I) be the synchronization errors. From system (4) and system (5), the error system can be described by
In this section, we consider a class of fractional-order neural networks as the master system described by Dα xi (t ) = −ci xi (t ) +
n
(4)
where i ∈ I = {1, 2, . . . , n}, 0 < α < 1, n corresponds to the number of units in a neural network; xi (t ) corresponds to the state of the ith unit at time t; fj (·) denotes the activation function of the jth neuron; aij denotes the constant connection weight of the jth neuron on the ith neuron; ci > 0 represents the rate with which the ith neuron will reset its potential to the resting state when disconnected from the network and each Ii denotes an external input. To ensure the existence and uniqueness of the solutions of system (4) (see, Kilbas et al., 2006), the following assumption is given. (H1 ) Functions fj are Lipschitz-continuous on R with Lipschitz constants Lj > 0, i.e.,
|fj (u) − fj (v)| ≤ Lj |u − v| for all u, v ∈ R and j ∈ I. The slave system is given by n
n
aij fj (yj (t )) − fj (β xj (t )) − di (t )ei (t ). (7)
j =1
aij fj (xj (t )) + Ii ,
j =1
Dα yi (t ) = −ci yi (t ) +
Dα ei (t ) = −ci ei (t ) +
aij fj (yj (t )) + Ii + ui (t ),
Evidently, ei (t ) = 0 is a trivial solution of the error system (7), and the problem of globally asymptotically projective synchronization between system (4) and system (5) is converted to the globally asymptotically stability problem for the zero solution. Theorem 1. Under the assumption (H1 ), system (4) and system (5) are globally asymptotically projective synchronized based on the control scheme (6). Proof. Assume that x(t ) = (x1 (t ), . . . , xn (t ))T and y(t ) = (y1 (t ), . . . , yn (t ))T are any two solutions of system (4) and system (5) with initial values x(0) = (x1 (0), . . . , xn (0))T and y(0) = (y1 (0), . . . , yn (0))T satisfying ei (0) ̸= 0 for i ∈ I, respectively. Evidently, ei (t ) = 0 is a solution of the error system (7). From the existence and uniqueness theorem of fractional order differential functions (Yu et al., 2012), ei (0)ei (t ) > 0 for t ≥ 0. If ei (0) > 0, then ei (t ) > 0 and Dα |ei (t )| =
(5)
j =1
for all i ∈ I, where each ui (t ) is a control input. Definition 3. If there exists nonzero constants β ∈ R, such that for any two solutions x(t ) and y(t ) of system (4) and system (5) with different initial values denoted by x0 and y0 , one has lim ∥y(t ) − β x(t )∥ = 0,
Γ (1 − α)
0
dτ = Dα ei (t ),
(t − τ )α
and if ei (0) < 0, then ei (t ) < 0 and Dα |ei (t )| = −
Γ (1 − α)
e′i (τ )
t
1
0
(t − τ )α
dτ = −Dα ei (t ).
Hence, Dα |ei (t )| = sgn(ei (t ))Dα ei (t )
n = sgn(ei (t )) −ci ei (t ) + aij [fj (yj (t )) j =1
t →∞
then, master system (4) and slave system (5) are said to be globally asymptotically projective synchronized, where ∥ · ∥ denotes the Euclidean norm, and β denotes the projective coefficient. Especially, master system (4) and slave system (5) are said to be globally asymptotically complete synchronized if β = 1 and are said to be globally asymptotically anti-synchronized if β = −1.
e′i (τ )
t
1
− fj (β xj (t ))] − di (t )ei (t ) . Construct the following auxiliary function V (t ) = U (t ) +
n 1 i=1
2ki
2
di (t ) − di ,
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J. Yu et al. / Neural Networks 49 (2014) 87–95
Similarly, for tk − T ≤ t ≤ tk and k = 1, 2, . . . , it follows from (9) and (10) that
where U (t ) =
n
|ei |,
U (tk ) − U (t ) ≤
i=1
each di is an adaptive constant which is determined in the later analysis. It follows from (6) and Lemma 2 that each di (t ) is nondecreasing, which together with Lemma 3, the fractional-order derivatives of V (t ) along the solutions of system (7) can be derived as Dα V (t ) = Dα U (t ) +
n 1
2ki
i=1
≤
n
sgn(ei (t )) −ci ei (t ) +
n
i=1
aij fj (yj (t )) − fj (β xj (t ))
ki
n ≤ −ci |ei (t )| + |aij |Lj |ej (t )| − di |ei (t )|
n
2
ϵ 2
,
tk − T ≤ t ≤ tk ,
n
n
i=1
j =1
ci −
|aji |Li − ci ,
Therefore,
ϵ 2
,
tk − T ≤ t ≤ tk + T ,
(11)
(12)
With inequality (8) and (11), when tk − T ≤ t ≤ tk + T , we get
|aji |Li + di |ei (t )|.
ϵ
Dα V (t ) ≤ − λ. 2 As a result, for any k = 1, 2, . . . , we have
i ∈ I.
V tk + T − V tk − T
Denote
≤− =−
di + ci −
1≤i≤n
k = 1, 2, . . . .
tk−1 + T < tk − T < tk + T < tk+1 − T .
j =1
λ = min
k = 1, 2, . . . .
j=1
Let the adaptive constants satisfy di >
(tk − t )α
Without loss of generality, we assume these interval disjoint and t1 − T > 0, then for any k = 1, 2, . . . , we have
n
=−
Γ (α + 1) ϵ ≤ ,
U (t ) ≥
n 1 − di (t )ei (t ) + (di (t ) − di )Dα di (t )
i=1
(tk − τ )α−1 dτ
t
M
=
U (t ) ≥
j =1
i=1
Γ (α)
which implies that
Dα (di (t ) − di )2
tk
M
n
|aji |Li
> 0.
Then t ≥ 0.
(8)
t ≥ 0,
which implies that ei (t ) and di (t ) are bounded on t ≥ 0. Then, there exists a constant M > 0 such that
|Dα U (t )| ≤ M ,
2α−1 ϵλT α
Γ (α + 1)
tk + T − τ
α−1
dτ
tk − T
.
(13)
V (tk−1 + T ) ≥ V (tk − T ),
k = 1, 2, . . . .
(14)
By (13) and (14),
According to Lemma 2, we have V (t ) ≤ V (0),
tk + T
On the other hand, according to Lemma 2 and inequality (8), V (t ) is monotonously non-increasing, which together with (12), we obtain
j =1
Dα V (t ) ≤ −λU (t ),
ϵ λ 2Γ (α)
t ≥ 0.
(9)
V (tk + T ) − V (0) ≤ −
2α−1 ϵλT α
Γ (α + 1)
k,
which reveals that V (tk + T ) → −∞ as k → +∞, which contradict with V (t ) ≥ 0. Hence
In the following, we will prove
lim U (t ) = 0,
lim U (t ) = 0.
t →∞
t →∞
Otherwise, there exist a constant ϵ > 0 and time series {tk } satisfying 0 < t1 < t2 < · · · < tk < tk+1 < · · · and limk→∞ tk = ∞ such that
that is, master system (4) and slave system (5) are globally asymptotically projective synchronized based on the control scheme (6).
U (tk ) ≥ ϵ,
Remark 5. Obviously, the adaptive control gains di (t ) tend to some positive constants when system (4) and system (5) achieve projective synchronization.
k = 1, 2, . . . .
(10)
1 ( Γ (α+1)ϵ ) α
> 0. When tk ≤ t ≤ tk + T for any Denote T = 2M k = 1, 2, . . . , it follows from (9) and (10) that U (t ) − U (tk ) ≥ −
M
Γ (α)
t
(t − τ )
dτ
tk
M
=−
α−1
Γ (α + 1) ϵ ≥− ,
(t − tk )α
2
ϵ
2
,
tk ≤ t ≤ tk + T ,
ui (t ) = −di (t )(yi (t ) − xi (t )), Dα di (t ) = ki |yi (t ) − xi (t )|.
(15)
In this case, the following result is directly derived from Theorem 1.
which shows that U (t ) ≥
When β = 1, the control input (6) is reduced to the following form
k = 1, 2, . . . .
Corollary 1. Assume that (H1 ) holds. System (4) and system (5) are globally asymptotically complete synchronized under the adaptive control (15).
J. Yu et al. / Neural Networks 49 (2014) 87–95
Evidently, when β rewritten as
=
91
−1, the control input (6) can be
n aij −fj (xj (t )) − fj (−xj (t )) ui (t ) = j =1 − 2Ii − di (t ) yi (t ) + xi (t ) , α D di (t ) = ki |yi (t ) + xi (t )|.
(16)
As a special case of Theorem 1, we have the following result. Corollary 2. Assume that (H1 ) holds. System (4) and system (5) are globally asymptotically anti-synchronized under the control strategy (16). Remark 6. If each activation function fj is an odd function in model (4), the controller (16) can be rewritten in the following form
Fig. 1. Chaotic behavior of system (4) with initial value (0.2, −0.5, 0.8).
ui (t ) = −2Ii − di (t ) yi (t ) + xi (t ) , Dα di (t ) = ki |yi (t ) + xi (t )|.
It follows from Corollary 2 that system (4) and system (5) are globally asymptotically anti-synchronized in this case. If β = 0, the control scheme (6) is degenerated to n u (t ) = − a f (0) − I − d (t )y (t ), i
ij j
j=1 Dα d (t ) = k |y (t )|. i
i
i
i
i
(17)
i
The following statements are directly obtained from Theorem 1 in this case. Corollary 3. Suppose that (H1 ) holds. System (5) is globally asymptotically stabilized to the origin under the control scheme (17). Fig. 2. Synchronization errors with β = 3.
Remark 7. The projective synchronization for a class of fractional neural networks is proposed via combining open loop control with adaptive control in this paper. The derived criteria are novel and effective. In fact, the designed control scheme and analysis technique can be extended to the investigation of synchronization for fractional chaotic systems. Remark 8. In this paper, a fractional adaptive control is introduced to realize projective synchronization. It is easy to see that the adaptive control is related to the differential order of systems, i.e., the different orders of systems affect the synchronized adaptive feedback gains. Hence, the synchronization of fractional neural network is influenced by the order of fractional system. Remark 9. Recently, projective synchronization of the chaotic and hyperchaotic systems with fractional-order has been extensively investigated based on the stability theorems of a fractional-order linear system (Si et al., 2012; Wang & He, 2008; Wang & Song, 2009; Wang & Wang, 2007; Wang et al., 2012; Xin et al., 2011), in which the eigenvalues of the coefficient matrix or Jacobian matrix were required to be calculated, this leads to a large amount of calculation and the obtained synchronization may be local such as Wang and Wang (2007). Different from these results, in this paper, the globally projective synchronization of fractional-order neural networks is proposed for the first time by introducing a new fractional-order differential inequality and by means of the Lyapunov-like method, some effective analysis techniques such as reduction to absurdity, and the fractional-order adaptive control. Evidently, the proposed methods in this paper are meaningful and new and can be extended to the study of the projective synchronization of the chaotic and hyperchaotic systems with fractional-order.
Fig. 3. Evolutions of master–slave systems with β = 3.
5. Numerical simulations In this section, several numerical examples are given to verify the effectiveness of the theoretical results. In system (4), select x = (x1 , x2 , x3 )T , α = 0.98, fj (xj ) = tanh(xj ) for j = 1, 2, 3, c1 = c2 = c3 = 1, I1 = I2 = I3 = 0 and 2 1.8 −4.75
A = (aij )3×3 =
−1.2 1.71 0
0 1.15 . 1.1
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Fig. 4. Trajectories of master–slave systems with β = 3.
Fig. 7. Evolutions of master–slave systems with β = −2.
Fig. 5. Time responses of di (t ) with β = 3.
Fig. 6. Synchronization errors with β = −2.
Under those parameters, system (4) has a chaotic attractor, which is shown in Fig. 1. In control scheme (6), choose k1 = 0.05, k2 = 0.06, k3 = 0.08, d1 (0) = d2 (0) = d3 (0) = 0.05. Select the projective coefficient β = 3. According to Theorem 1, master system (4) and response system (5) are globally asymptotically projective synchronized, which is verified in Figs. 2–5. In Fig. 2, the projective synchronization errors with different initial values converge to zero, which show that master system (4) and slave system (5) are globally asymptotically projective
Fig. 8. Trajectories of master–slave systems with β = −2.
Fig. 9. Time responses of di (t ) with β = −2.
synchronized with projective coefficients β = 3 and they are verified in Figs. 3 and 4. In addition, the adaptive gains di (t ) (i = 1, 2, 3) converge to some positive constants in this case, see Fig. 5. Similarly, the projective synchronization with projective coefficient β = −2 is simulated in Figs. 6–9. In the following, we consider the globally asymptotically complete synchronization of system (4) and system (5). In this case, β = 1. Choose k1 = k2 = k3 = 0.03, d1 (0) = d2 (0) = d3 (0) = 0.01. It follows from Corollary 1 that master system (4) and slave system (5) are globally asymptotically complete synchronized.
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Fig. 10. Synchronization errors with β = 1.
Fig. 11. Evolutions of master–slave systems with β = 1.
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Fig. 13. Time responses of di (t ) with β = 1.
Fig. 14. Synchronization errors with β = −1.
Fig. 12. Trajectories of master–slave systems with β = 1.
The synchronization errors and synchronization evolutions are presented in Figs. 10–12, respectively. The adaptive gains are also shown in Fig. 13, from which we can see that di (t ) (i = 1, 2, 3) approach to some positive constant. Similarly, when we choose β = −1, and select k1 = k2 = k3 = 0.1, d1 (0) = d2 (0) = d3 (0) = 0.01, Figs. 14–17 imply that system (4) and system (5) are globally asymptotically anti-synchronized. When β = 0, according to Corollary 3, system (5) is globally asymptotically stabilized to the origin, which is verified in Figs. 18–20 with control parameters k1 = k2 = k3 = 0.2 and d1 (0) = d2 (0) = d3 (0) = 0.01.
Fig. 15. Evolutions of master–slave systems with β = −1.
6. Conclusion The globally asymptotically projective synchronization of fractional-order neural networks is concerned by designing a novel control input in this paper. To obtain the main results, a sufficient condition in the sense of Caputo’s fractional derivation is given to ensure the monotonicity of the continuous and differential functions. Besides, a new fractional-order differential inequality
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Fig. 16. Trajectories of master–slave systems with β = −1.
Fig. 19. Trajectory of system (5) with β = 0.
Fig. 17. Time responses of di (t ) with β = −1.
Fig. 20. Time responses of di (t ) with β = 0.
Acknowledgments This work was supported by Natural Science Foundation of Xinjiang (Grant No. 2013211B06), China Postdoctoral Science Foundation funded project (2013M540782), Natural Science Foundation of Xinjiang University (Grant No. BS120101), and National Natural Science Foundation of People’s Republic of China (Grant No. 61164004). References
Fig. 18. Time response of system (5) with β = 0.
is derived, which differs from the corresponding integer-order case and plays a central role in the investigation of the fractional adaptive control. Based on those preparations, some criteria are derived to realize projective synchronization of fractional-order neural networks via combining open loop control and adaptive control. It is worthwhile to note that the complete synchronization, anti-synchronization and the stabilization of the addressed neural networks are guaranteed as some special cases of our main results. It is believed that our results should provide some practical guidelines for engineering applications of fractional-order neural networks. The future work will focus on the synchronization of fractional-order delayed neural networks.
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