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Synchronization of Complex Networks of FractionalOrder Nonlinear Systems.
Jorge A. LeonDepartment of Automatic Control
CINVESTAV-IPNMexico, Mexico City 07360
Email: [email protected]
Armando Fabian Lugo-Penalozaand Guillermo Fernandez-Anaya
Department of Physics and MathematicsUniversidad IberoamericanaMexico, Mexico City 01219
Email: [email protected]@uia.mx
Rafael Martınez-MartınezDCA, Academy Systems
CINVESTAV-IPN, UPIITA-IPNMexico, Mexico City, 07340Email: [email protected]
Abstract—In this work we obtain analytical results for twodifferent kinds of synchronization. This is obtained by means ofthe analysis of the fractional non-linear error equation, thereforethe couplings inside the complex network do not eliminate thenon-linearities of the systems, contrary to what is usual inliterature. As an application, we present the synchronizationwith different kinds of fractional differential systems. We usesome numerical results to verify the effectiveness of the proposedmethod.
I. INTRODUCTION
Synchronization is a phenomenon that arises in manyareas of science and technology [1], [2]. The constructionof new models for the complex dynamic networks (dynamicsystems in interaction) [3], [4], [5] has led to a fascinatingset of common problems concerning the way a networkfacilitates and limits the collective behaviours. In particular,many efforts have been made in oscillators synchronization[6], [7], [8] and biological nervous networks synchronization[9], [10].
We study fractional complex networks using fractionalcalculus, which has a history almost as large as the one of theordinary calculus [11], [12]. There are many known systemswhere the modelling with fractional operators has turnedout to be useful. Namely, in encryption, the fractional orderchaotic systems have more adjustable variables than integerorder chaotic systems, therefore it is widely believed thatfractional-order chaotic systems can be applied in encryptionefficiently and thus enlarge the key space [13], [14].
There are many different works on the synchronization offractional complex systems [13], [15], [16], [17], and on thesynchronization of fractional-order chaotic systems [18], [19],[20], [21], [22], in such works several schemes are proposedthat ensure that the dynamical system of the error satisfiessimilar conditions to [23], this means that the error systemmust hold a linear relation. In this work a synchronizationscheme is proposed that asserts that the error system satisfiesconditions similar to [24], which means that is not necessaryto cancel the nonlinear part of the error dynamics. Hence,depending on the fractional order of the systems we can obtain
exact or inexact synchronization of the complex network.
This work is ordered as follows: in Section 2 basicdefinitions of fractional order systems are presented; inSection 3 the definition of a complex network is introducedand the problems of inexact and exact synchronization arestudied; in Section 4 an application of fractional orderdifferential systems is analyzed, and finally in section 5 wegive some conclusions.
II. PRELIMINARIES
There are several definitions of a fractional derivative oforder α ∈ R+ [11], [25], [26]. We will use the Caputofractional operator in the definition of fractional order systems,because the meaning of the initial conditions for systemsdescribed using this operator is the same as for integer ordersystems.
Definition 1 (Caputo Fractional derivative): The Caputofractional derivative of order α ∈ R+ of a function x is definedas (see [11])
x(α) = t0Dαt x =
1
Γ(m− α)
∫ t
t0
dmx(τ)
dτm(t−τ)m−α−1 dτ
(1)where: m− 1 ≤ α < m, d
mx(τ)dτm is the m-th derivative of x in
the usual sense, m ∈ N and Γ is the gamma function1.
We consider the initial condition problem as usual inliterature for an autonomous fractional order nonlinear system,with 0 < α < 1:
x(α) = f(x), x(0) = x0 (2)
where x ∈ Ω ⊂ Rn, f : Ω → Rn is a Lipschitz continuousfunction2, with x0 ∈ Ω ⊂ Rn and Ω is a region in Rn.
1To simplify the notation we omitted the time dependence in x(α), in whatfollows we take t0 = 0
2This assures the uniqueness of solution [27]
For fractional order lineal time invariant systems, i.e.,f(x) = Ax, A ∈ Rn×n, we can guarantee the stability inthe Lyapunov sense. The following result [23] deals with thesystem
x(α) = Ax, x(0) = x0 (3)
Theorem 1 ([23]): Let 0 < α < 1. The system (3) is:
i) Asymptotically stable if and only if | arg(λk(A))| >απ/2, where arg(λk(A)) denotes the argument of theeigenvalues of A. In this case, the component of thestate decays towards 0 like t−α;
ii) Stable if and only if either it is asymptoticallystable, or those critical eigenvalues which satisfy| arg(spec(A))| = απ/2 have geometric multiplicityone.
Now for fractional order nonlinear systems we use thefollowing result [24]:
Theorem 2 ([24]): Let us consider the n-dimensional frac-tional dynamic system
x(α) = Ax+ f(x) (4)
where A is a matrix, f is a nonlinear function of x and 0 <α < 1. If:
1) the solution x(t) = 0 of x(α) = Ax is asymptoticallystable3, and αρ(A) > 14
2) f(0) = 0 and lim‖x‖→0
‖f(x)‖‖x‖ = 0
then x(t) = 0, for 0 ≤ t0 ≤ t, is a stable solution of (4).
Now we consider the n−dimensional nonlinear fractionalorder dynamic system with the Caputo derivative
x(α) = Ax+ g(t, x), t > t0, (5)
under the initial conditions
x(α−k)(t)∣∣∣t=t0
= xk−1 (k = 1, 2), (6)
where x ∈ R, matrix A ∈ Rn×n, and 1 < α < 2, g(t, x) :[t0, ∞)×Rn → Rn is a continous function in which g(t, 0) =0; moreover, g(t, x) holds the Lipschitz condition with respectto x.
Theorem 3 ([28]): If the matrix A is such that| arg(spec(A))| 6= 0, | arg(spec(A))| ≥ απ/2, α+1/‖A‖ < 2,and suppose that the function g(t, x) satisfies uniformly
limx→∞
‖g(t, x)‖‖x‖
= 0, t ∈ [t0, ∞) (7)
then the zero solution of (5) is asymptotically stable.
From these previous results we obtain that the inexactsynchronization can be obtained for systems with 0 < α < 1and the exact synchronization for systems with 1 < α < 2.
3in the sense of the Theorem 1.4ρ(A) ≡ spectral radius of matrix A
III. COMPLETE SYNCHRONIZATION OF FRACTIONALCOMPLEX NETWORKS
Here, we propose the general form for a complex fractionalnetwork. That is, this section is devoted to the study of Nsystems with fractional order, which are interacting under acertain interconnection ()see the Figure 1 (a)), of the form
x(α)i = Aixi + fi(xi) +
N∑j=1
Hij(xj) (8)
where, for every i = 1, 2, . . . , N we define
• xi : R→ Rn, is the state of the i− th system.
• Ai : Rn → Rn represents the linear dynamics of thei− th system.
• fi : Rn → Rn represents the nonlinear dynamics ofthe i− th system.
• Hij : Rn → Rn describes how the j − th system isinterconnected with the i − th one, that is, specifiesthe strength and the topology of the interconnectionof the systems.
• Aij : Rn → Rn is the linear part of Hij .
• Fij : Rn → Rn is fi added to the nonlinear part ofHij .
And so we can write equation (8)
x(α)i = Aixi +
N∑j=1
Aijxj +
N∑j=1
Fij(xj) (9)
In general, not all the systems are interconnected in the sameway, and we assume that fi and Hij are Lipschitz continuousfunctions.
Let us consider another fractional dynamical system withthe same fractional order, which will be named the mastersystem:
x(α)M = AMxM + fM(xM) (10)
where xM : R → Rn is the state vector, AM ∈ Rn×n andfM : Rn → Rn is a continuous Lipschitz function.
Let us take a complex fractional network of the form (9)plus a control law ui of the form
x(α)i = Aixi +
N∑j=1
Aijxj +
N∑j=1
Fij(xj) + ui (11)
which will be named the slave systems and ui is a newinterconnection with xM that will be determined.
In the master-slave synchronization scheme, xM describesthe target signal, while xi represents the response signal, seeFigure 1 (b). Therefore the synchronization problem can beestablished as follows:
Given the master system (10) and our slave system (11),it should be determined a interconnection ui, such that the
target signal is synchronized with the response signal. In otherwords, if we define the synchronization error by
ei = xM − xi (12)
depending on the desired behaviour of |ei(t)| we will needspecific conditions over ui.
With the master-slave scheme (12) we will prove thatinexact synchronization [29] is achieved, i. e., that the solutionei(t) = 0 is stable.
Theorem 4: Consider a complex fractional network withthe representation (11)
x(α)i = Aixi +
N∑j=1
Aijxj +
N∑j=1
Fij(xj) + ui (13)
where Aij ∈ Rn×n and that Fii(0) = 0 and
lim‖xi‖→0
‖Fii(xi)‖‖xi‖
= 0 hold, for i, j ∈ 1, . . . , N. Moreover,
assume that there are B1, . . . , Bn ∈ Rn×n such that Ai+Bisatisfies | arg(λk(Ai+Bi))| > απ/2 and αρ(Ai+Bi) > 1, fork = 1, . . . , n; finally we add a master system as in (10) to thenetwork, see Figure 1. Then we have that the interconnectionsdefined by the equation
ui = AMxM + fM(xM)−AixM −Bi (xM − xi)
−Fii(xM − xi)−N∑j=1
Aijxj −N∑j=1
Fij(xj) (14)
ensure the inexact synchronization for the complex network(11) with the master system (10).
Proof. We consider the error equation (12), and take itsderivative of order α
e(α)i = x
(α)M − x(α)
i= AMxM + fM(xM)
−(Aixi +
∑Nj=1Aijxj +
∑Nj=1 Fij(xj) + ui
)Replacing (14) in the last equation
e(α)i = (Ai +Bi)ei + Fii(ei) (15)
Now, due to our earlier supposition we have Fii(0) = 0 and
lim‖ei‖→0
‖Fii(ei)‖‖ei‖
= 0, we only need to choose Bi ∈ Rn×n in
such a way that Ai+Bi satisfies | arg(λk(Ai+Bi))| > απ/2and αρ(Ai + Bi) > 1, for k = 1, . . . , n, but we havealready asked for such Bi as hypothesis, thus by Theorem2 we have that the origin of each of the fractional orderdifferential nonlinear synchronization error systems of com-mensurate fractional order is stable, which gives us the inexactsynchronization of the entire network with the master system
Now we consider 1 < α < 2 and a possible timedependence for the master system (10) and the slave system(11) as follows:
x(α)M = AMxM + fM(t, xM) (16)
x(α)i = Aixi +
N∑j=1
Aijxj +
N∑j=1
Fij(t, xj) (17)
with initial conditions
x(α−r)M (t)
∣∣∣t=t0
= xMr−1(r = 1, 2). (18)
andx(α−r)i (t)
∣∣∣t=t0
= xir−1 (r = 1, 2), (19)
where Fii(t, xj) : [t0, ∞) × Rn → Rn is a continuousfunction in which Fii(t, 0) = 0; moreover, Fii(t, xj) fulfilsthe Lipschitz condition with respect to xj .
Proceeding in a similar way we will take the master-slavescheme 12 and we will prove an exact synchronization [29]is achieved, i. e., that the solution ei(t) = 0 is asymptoticallystable.
Theorem 5: Consider a complex fractional network of theform (17) and suppose that it has the representation
x(α)i = Aixi +
N∑j=1
Aijxj +
N∑j=1
Fij(t, xj) + ui (20)
where Aij ∈ Rn×n and Fii(t, xi) satisfies
limx→∞
‖Fij(t, xj)‖‖xj‖
= 0, for i, j ∈ 1, . . . , N. Moreover,
assume that there are B1, . . . , Bn ∈ Rn×n such that Ai+Bihas | arg(spec(A))| 6= 0, arg(spec(A + B))| ≥ απ/2 andα+ 1/‖Ai + Bi‖ < 2 ; finally we add a master system as in(16) to the network. Then we have that the interconnectionsdefined by the equation
ui = fM(xM)−AixM −Bi (xM − xi)−Fii(t, xM − xi)
−N∑j=1
Aijxj −N∑j=1
Fij(t, xj)
(21)ensure the inexact synchronization for the complex network
(17) with the master system (16).Proof. We consider the error equation (12), and take its
derivative of order αe(α)i = x
(α)M − x(α)
i= AMxM + fM(t, xM)
−(Aixi +
∑Nj=1Aijxj +
∑Nj=1 Fij(t, xj) + ui
)replacing (21) in the last equation
e(α)i = (Ai +Bi)ei + Fii(t, ei) (22)
In a similar way that in the proof for Theorem 4, we alreadyhave asked for Fii(t, ei) to satisfy the second hypothesis fromTheorem 3, we only need to choose Bi ∈ Rn×n in such away that Ai +Bi satisfies the first part of the hypothesis fromTheorem 3 but we also have asked for Ai + Bi to hold suchconditions, and as a consequence we get the desired result.
IV. NUMERICAL EXAMPLE
In this section we deal with a slave network whosecomponents are fractional order differential systems: twoLorenz systems and one Chen system. The master systemwill be a Lu one. The method introduced in [30] impliesthat the systems has a unique solution. For the simulationswe have taken the approximation of the fractional order
x1
x2
x3
x4
x5
x6
x7
x1
x2
x3
x4
x5
x6
x7
xM
u3 u4
u7
u2u5
u1
u6
(a) (b)
Fig. 1: The fractional complex network: Fractional Complex Network (a), Master system xM and the Slave systems xi (b)
operators presented in [31], based on [32]. First we presentthe individual systems of the network.
We take into account the modified fractional order Lorenzsystem [24].
x(α)i (t) = Ai xi + fi(xi) (23)
=
−ai ai 0bi −ci 00 0 −di
xi +
0−xi1 xi3xi1 xi2
where xi = [xi1, xi2, xi3]T . The case that ai = 10, bi = 28,ci = −8, di = 8/3 and α = 0.8 is shown in the Figure 2 (a).
Now the fractional order Lu system [17] is
x(α)M = AM xM + f(xM) (24)
=
−aM aM 00 cM 00 0 −bM
xM +
0−xM1xM3
xM1xM2
where xM = [xM1, xM2, xM3]T . Figure 2 (b) shows thebehavior in this case when aM = 35, bM = 3, cM = 28, andα = 0.8.
And finally we present the fractional order Chen system[33]:
x(α)i = Ai xi + fi(xi) = (25) −ai ai 0
ci − ai ci 00 0 −bi
xi +
0−xi1 xi3xi1 xi2
where xi = [xi1, xi2, xi3]T . See Figure 2 (c) to observe thebehavior of this system for ai = 35, bi = 3, ci = 28, andα = 0.8.
.
Let us take a complex network with, two Lorenz’s systemsfor i = 1, 2, and a Chen’s system for i = 3; the bidirectionalcoupling scheme proposed in [34] is used:
x(α)i (t) = Ai xi + fi(xi) +
3∑j=1
Hij(xj) (26)
where
Hij(xj) =
σijdi1 0 0
0 σijdi2 0
0 0 σijdi3
xj1
xj2xj3
, σij =
−2, if i = j1, if i 6= j
and the network can be separated in a linear plus a nonlinearpart; in such case the nonlinear part in each system is F11 =F22 = F33 and we have that
0 ≤
√x2i1x
2i3 + x2i1x
2i2
x2i1 + x2i2 + x2i3≤√x2i3 + x2i2
Applying the limit when ‖x‖ → 0 we have that hypothesis2 from Theorem 2 holds, now let us consider a Lu system asthe master, and construct the interconnections as in (14), thus
ui = fM(xM)−AixM −Bi (xM − xi)−Fii(xM − xi)
−3∑j=1
Aijxj − Fii(xi)
(27)
Here Aij = Hij , which implies having errorsas in equation (15). Now, with the election ofB1 = B2 = B3 = diag(−50,−50,−50) condition 1from Theorem 2 holds, e. g., A1 + B1 has the eigenvalues−70,−32,−52.66 and αρ(A) = 56. We show the simulationsin the Figure 3.
We take the parameters of Hij as d11 = −1, d12 = 2,d13 = −3, d21 = 4, d22 = 2, d23 = −3, d31 = 100/29,d32 = −1, d33 = 17/48, the selection of these values wasarbitrary.
In the simulation, see Figure 3, we turn on theinterconnection ui on t = 1.5s, and then we turn it offon t = 3s. Before the activation of the interconnection weobserve the behavior in each of the states of the complexnetwork, and when it is activated then the corresponding statesof the network synchronize with the corresponding master(Lu) system. We can also observe that the error approximateszero when the interconnection is turned on.
V. CONCLUSIONS
In this work a coupled scheme is used to achieve thefractional complex network synchronization in a Master-Slavescheme, where all the elements of the network are fractionalorder dynamical systems. As a specific difference with other
(a) Lorenz system (b) Lu system (c) Chen system
Fig. 2: Graphs of the fractional order non-linear systems in the network
(a) Slave systems states
(b) Master system states
(c) Error system states
Fig. 3: Synchronization of fractional complex network
works, the results used herein allowed us to let the dynamicalerror to be nonlinear. The coupling ui is nonlinear and doesnot cancel completely the nonlinear part of each system, thusthe inexact synchronization and the exact synchronization arerespectively guaranteed, despite this nonlinearities.
In particular, the results obtained are applicable when
we have different fractional linear systems, when we havefractional linear systems of the same type, or when we areworking with a linearization of the original nonlinear systemsor when consider a system-to-system synchronization scheme,because it turns out that these schemes satisfy particularconditions of the family of complex networks treated herein.For linear systems with their orders lying in (1,2) we onlyneed to use Theorem 3.3 from [28].
It is important to notice that the hypothesis on each nodeare applicable also to other kinds of fractional dynamicalsystems [35], [36], and not only to that used in the examplepresented here.
Besides it is not necessary a specific kind of couplingbetween the elements of the complex network, because wehave the action of the interconnection over the entire network.
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