Synchronization and FPGA realization of complex networks with fractional–order Liu chaotic oscillators

Applied Mathematics and Computation 332 (2018) 250–262

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Synchronization and FPGA realization of complex networks with fractional–order Liu chaotic oscillators Soriano–Sánchez A.G. a,∗, Posadas–Castillo C. a, Platas–Garza M.A. a, Arellano–Delgado A. b a b

Universidad Autónoma de Nuevo León, Av. Pedro de Alba S/N, Cd. Universitaria, San Nicolás de los Garza, N.L., C.P. 66455, México Facultad de Ingeniería, Arquitectura y Diseño, Universidad Autónoma de Baja California (UABC), Ensenada, B.C., México

a r t i c l e

i n f o

a b s t r a c t

Keywords: Synchronization Fractional–order chaotic oscillators Complex networks FPGA

In this paper synchronization of fractional–order Liu chaotic oscillators is addressed. Two networks of fractional–order Liu chaotic oscillators are synchronized by appealing to results from complex systems theory for integer–order systems. We use and implement complex dynamical networks composed by nine Liu chaotic oscillators. Two scenarios are considered: (i) complex network with regular topology, and (ii) complex network with irregular topology. Synchronization of both complex networks is achieved by coupling the fractional–order chaotic oscillators through their second state. We use numerical simulations to verify the results, we also show an FPGA realization of the complex networks previously synchronized. © 2018 Elsevier Inc. All rights reserved.

1. Introduction The first mention of the possibility of a derivative of non–integer (arbitrary) order was raised between G. Leibniz and G. L’Hôpital in 1695 [1,2]. Leibniz and L’Hôpital were discussing the meaning of a half–order derivative and its consequences. Nowadays, that branch of Mathematics is more than 300 years old and is called Fractional Calculus. The principal and most attractive characteristic of this discipline is its ability to describe and model more accurately real behavior [3], since real objects or systems are fractional [1,3]. Fractional calculus has been successfully used to describe physical systems, such as, the semi–infinite lossy transmission line [4] or the behavior of the heat diffusion into an infinite solid [5], which are the examples most cited as probably the first ones documenting fractional behavior. Recent applications in different fields have shown the importance of fractional calculus; some of the most important include earthquake and vibration engineering [6], automatic control [7], image and signal processing [8], bioengineering [9] and viscoelasticity [10], for instance. In the next section, authors will define the fractional derivative that will be used and the method to compute it. Regarding chaos synchronization, it is inevitable to think of L.M. Pecora and T.L. Carroll, who synchronized, for the first time, two identical chaotic oscillators with different initial conditions [11]. After this result, the research community paid particular attention, and the basics of chaos synchronization were established between 1990 and 2001. Among the most significant works are the definitions or investigation of essential concepts, such as asymptotic and partial synchronization, ∗

Corresponding author. E-mail addresses: [email protected] [email protected] (A.-D. A.).

(S.

https://doi.org/10.1016/j.amc.2018.03.057 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

A.G.),

[email protected]

(P.-C.

C.),

[email protected]

(P.-G.

M.A.),

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251

the relation between asymptotic synchronization and asymptotic stability [12], or the role of unstable periodic orbits in synchronous chaotic behavior and how desynchronized bursting behavior is initiated [13]. The cooperative activities related to regimes of synchronized chaos was also studied along with different types of identical chaotic oscillations [14]. The mechanisms of desynchronization and the transverse stability properties for the equilibrium point in coupled identical oscillators were also established [15]. Finally, a unified approach to analysis conventional and chaotic communications systems was provided, along with the description of chaotic synchronization schemes and the role of synchronization for chaotic communications [16]. Among the existing methods to achieve synchronization between coupled chaotic oscillators, one can find output synchronization for input–output passive systems [17], the sliding modes strategy [18], synchronization in master–slave configuration by using exponential nonlinear observers [19], synchronization through adaptive, adaptive–impulsive or active control [20–22] and Wang–Chen method [23–27], for instance. On the other hand, chaotic synchronization of fractional–order dynamical systems has recently received a big deal of attention [28–31]. As a result, the fractional–order versions of many types of chaotic oscillators, such as, fractional–order Chen [32], Lorenz [33], Rössler [34], Lü [35], Liu [36] and Chua’s [37], to mention a few, have been used to study the fractional behavior of chaos and its synchronization [3]. In this manuscript the fractional–order Liu chaotic oscillator will be used to compose the complex networks. The importance of this non–linear system lies on the deep analysis performed by V. Genjii and S. Bhalekar to determine the parameters range for chaos to exist and different values to switch between commensurate and incommensurate orders [36]. In this paper, chaotic synchronization of fractional–order oscillators is addressed. Two arranges, with regular and irregular topology, will be synchronized. The main contribution falls on the use of a method, initially designed for integer–order dynamical systems, to carry the complex networks, composed of fractional–order oscillators, to behave synchronously. This technique will be referred as the Wang–Chen method [23]. Corroborating the efficiency of this technique to synchronize fractional–order chaotic oscillators, arranged in topologies of complex network, turns to be an increase in scope of the theory, which stands as the main contribution of this article. One of the principal advantages of the Wang–Chen method lies in the fact that it is possible to analyze by separate the synchronization capacity of the complex network and the oscillator. It first requires verifying the stability of a single node of the complex network, for this particular case one single fractional–order Liu chaotic oscillator [36]. Besides, it requires the compute of the eigenvalues of the coupling matrix [23] i.e., the way that the chaotic oscillators are arranged. At the end, the method resorts to both requirements obtained before and allows us to determine the gain necessary for the complex network to reach asymptotic synchronization. Its natural interpretation, low consumption of computational resources and versatility to be applied to different types of systems make are the main advantages of this technique. In addition to numerical simulations, we generate a realization of the complex networks on FPGAs. Even they have finite precision due to its discrete nature, FPGAs have become a favorite tool for the digital realization of continuous chaos, as a result, several applications for FPGA implementations of chaotic systems have emerged [38–41]. From a practical point of view, they have advantages in comparison with analog components such as op–amps and analog multipliers. For example, the initial conditions can be easily specified, they are not sensitive to components tolerance, and they are not susceptible to saturation problems as in analog design. In addition, the analog implementation of fractional–order systems requires specific values for resistors and capacitors which could be hard to find. Many of the FPGA designs found in the literature deal with the implementation of integer–order chaotic dynamics [39], Although they are fewer, fractional–order FPGA implementations are also common [40,41], but they usually do not contemplate the deployment and synchronization of complex networks. In this paper, we show the FPGA realization of regular and irregular networks with fractional–order chaotic dynamics. The design also considers the implementation of the control laws needed for synchronization, which were obtained theoretically via the Wang–Chen method. The paper is organized as follows: In section 2, the preliminaries of fractional calculus, complex networks and the description of the fractional–order Liu chaotic oscillator will be given. The synchronization results, obtained from the solution of the fractional differential equations of the complex networks, will be provided in Section 3. The hardware realization of the complex networks is given in Section 4. Finally, the corresponding conclusions are provided in Section 5.

2. Preliminaries 2.1. Fractional derivatives The fractional calculus is the theory that generalizes integration and differentiation to arbitrary non–integer order [1,3]. A fractional differential (integral) equation is an equation containing fractional derivatives (integrals); on the other hand, a fractional–order system means a system which is described by fractional differential (integral) equations [1]. The fundamental operator used in the field is a Dtα , where a and t are the limits related to the operation of fractional differentiation and α ∈  is the order of the operation. The continuous integro–differentiable operator is defined as [1,3]

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α =

a Dt

⎧ α ⎪ ⎨ dα dt 1 ⎪  ⎩ t a

( d τ )α

α > 0, α = 0, α < 0.

(1)

The integro–differentiable operator is the unification of two of the most important concepts in mathematics, i.e., the derivative of integer–order n and the n–fold integral, and allows us to interpolate between derivatives and integrals. By applying n times the definition of the derivative over the continuous function f(t), one obtains the following equation [1,3]



n dn f 1  n = lim n (−1 )k f (t − kh ), n dt k h→0 h

(2)

k=0

where

 n k

=

n × (n − 1 ) × (n − 2 ) × · · · × (n − (k − 1 )) , k!

(3)

is the standard notation for the binomial coefficients. According to [1], it is necessary that n → ∞ as h → 0 in order to prevent (2) from reaching the limit 0 as h → 0 for a fixed n, which might happen in the case that the order of (2) is negative, which would imply the opposite operation [1]. For this reason, it is suggested to use n = (t − a )/h as the integer part of the limit value [1,3]. Therefore,

lim

f (n ) (t ) =

h h→0 n=(t−a )/h

α f (t ),

a Dt

(4)

where a Dtα f (t ) denotes the operation performed on the function f(t) with a and t as the limits of the operation [1,3]. Now, (2) becomes

1 α a Dt f (t ) = lim α h→0 h

(t−a )/h

(−1 )

k=0

k

 α k

f (t − kh ),

(5)

that corresponds to the Grünwald–Letnikov definition of the fractional differintegral, which along with the Riemann– Liouville and the Caputo definitions are the most frequently used [1–3]. It is important to mention that for homogeneous initial conditions, mainly f (a ) = 0, the three above–mentioned definitions of fractional operations are equivalent [1,3]. To compute numerically the fractional–order derivatives, the method most used is the so–called Power Series Expansion of a generating function, which produces an approximation in the form of polynomials, by using a discretized fractional operator in the form of FIR filter [42,43]. The relation most commonly used is the following [1] q

(k−Lm /h ) Dtk

f (t ) ≈ h−q

k  c(jq ) f (tk − j ),

(6)

j=0

which was derived from the definition of the Grünwald–Letnikov fractional derivative, where Lm is called memory length, tk (q ) is the time step size of calculation and c j for j = 0, 1, . . . , k are the binomial coefficients, which are described as follows [1]:



c0(q ) = 1,

c(jq ) = 1 −



1 + q (q ) c j−1 . j

(7)

For further detail on this topic, please go to the references provided. In the following, the corresponding information on the fractional–order Liu chaotic oscillator will be provided. 2.2. Fractional–order Liu chaotic oscillator In 2009, C. Liu, L. Liu and T. Liu proposed a novel three–dimensional autonomous system now known as the Liu oscillator. C. Liu et al. showed that the system exhibited chaos for specific parameter values [44]. One year later, V. Gejji and S. Bhalekar reported its fractional version and determined the minimum effective dimension, necessary for the system to exhibit chaos, among other characteristics [36]. The fractional–order Liu chaotic oscillator is described by the following set of equations:

⎧ q1 ⎨0 Dt x(t ) q2 0 D y (t ) ⎩ tq3 0 Dt z (t )

=

−ax(t ) − ey2 (t ),

=

by(t ) − kx(t )z(t ),

=

−dz(t ) + mx(t )y(t ),

(8)

where the fractional–order is set at q1 = q2 = q3 = 0.95, and the parameter values for the system to exhibit chaos are a = e = 1, b = 2.5, d = 5 and k = m = 4 [3,36,44]. By using the fractional–order Liu oscillator (8) with the parameters given, one can obtain the chaotic strange attractor and the trajectory shown in Fig. 1. To end this section, authors will provide the information related to complex networks and their synchronization.

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Fig. 1. (a) Chaotic strange attractor of the fractional–order Liu oscillator (8) obtained with following randomly chosen initial conditions [x(0 ), y(0 ), z(0 )] = [−0.5936, 0.4364, −0.5044]. (b) Chaotic trajectory corresponding to y(t) state variable of the fractional–order Liu oscillator (8).

Fig. 2. Topologies considered to arrange the fractional–order chaotic oscillators: (a) Regular complex network. (b) Irregular complex network.

2.3. Complex networks and their synchronization In this section complex networks will be briefly described. Authors will provide the complex network concept and its mathematical representation, which are needed for a complete understanding of this paper. Among the possible definitions that can be found, we will use the one suggested by Wang [45]. Definition 1. A complex network is defined as an interconnected set of nodes (two or more), where each node is a fundamental unit, with its dynamic depending on the nature of the network. Each fractional–order chaotic oscillator of the complex network will be defined as follows: α x = f (x ) + u , i i i

a Dt

xi ( 0 ),

i = 1, 2, . . . , N,

(9) n

where N will denote the size of the network, xi = [xi1 xi2 . . . xin ] ∈ represents the state variables of the ith fractional– order oscillator. xi (0) ∈ n are the initial conditions for the ith fractional–order oscillator. The control law ui ∈ n establishes the synchronization between two or more fractional–order oscillators and is defined as follows [23]:

ui = c

N 

μi j x j ,

i = 1, 2, . . . , N,

(10)

j=1

where the constant c > 0 represents the coupling strength.  ∈ n × n is a constant matrix to determine the coupled state variable of each oscillator. Assume that  = diag(r1 , r2 , . . . , rn ) is a diagonal matrix. If two oscillators are linked through their kth state variables, then, the diagonal element rk = 1 for a particular k and r j = 0 for j = k. The matrix M ∈ N × N with elements μij is the coupling matrix, which describes the topology of the network, i.e., it shows the connections between oscillators, if the oscillator ith is connected to the oscillator jth, then μi j = 1, otherwise μi j = 0 for i = j. The diagonal elements of matrix M are defined as

μii = −

N  j=1, j=i

μi j = −

N 

μ ji

i = 1, 2, . . . , N.

(11)

j=1, j=i

The dynamical complex network (9) and (10) achieves synchronization if

x1 (t ) = x2 (t ) = · · · = xN (t ),

t → ∞,

(12)

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which means that the following equation:





lim xi (t ) − x j (t ) = 0,

t→∞

i = 1, . . . , N − 1; j = i + 1, . . . , N,

(13)

holds for every pair of fractional–order chaotic oscillators, which implies that the complex network has reached synchrony. The conditions that must be met for the complex network to reach synchrony are the following: suppose that there are no isolated chaotic oscillators in the complex network, then the coupling matrix M, obtained as explained above, is a symmetric irreducible matrix, so one eigenvalue of M is zero, and all the other eigenvalues are strictly negative, this means, λ2,...,N (M ) < 0. Theorem 1. Consider the dynamical network (9). Let

0 = λ1 > λ2 ≥ λ3 · · · ≥ λN ,

(14)

be the eigenvalues of its coupling matrix M. Suppose that there exist an n × n diagonal matrix D > 0 and two constants d¯ < 0 and τ > 0, such that

[D f (s(t ) ) + d] D + D[D f (s(t ) ) + d] ≤ −τ In , T

(15)

for all d ≤ d¯, where In ∈ n × n is an unit matrix. If moreover,

cλ2 ≤ d¯,

(16)

then the synchronization state (12) is exponentially stable [45]. The coupling strength c is computed according to the following lemma: Lemma 1. Consider network (9). Let λ2 be the largest nonzero eigenvalue of the coupling matrix M of the complex network. The synchronization state of network (9) defined by x1 (t ) = x2 (t ) = · · · = xN (t ) is asymptotically stable, if

λ2 ≤ −

T c

(17)

where c > 0 is the coupling strength of the network and T > 0 is a positive constant such that zero is an exponentially stable point of the n–dimensional system

⎧ z˙ 1 ⎪ ⎪ ⎨ z˙ 2 .. ⎪ ⎪ ⎩. z˙ n

= = .. . =

f 1 ( z ) − T z1 , f 2 ( z ), .. . f n ( z ).

(18)

Note that system (18) is actually a single chaotic oscillator with self–feedback −T z1. Condition (17) means that the entire network will synchronize provided that λ2 is negative enough, e.g., it is sufficient to be less than −T /c, where T is a constant so that the self–feedback term −T z1 could stabilize an isolated chaotic oscillator [23]. In this paper, complex networks composed of N identical chaotic oscillators are considered. Each dynamical system will be a fractional–order Liu chaotic oscillator, and the arrangements considered are the ones shown in Fig. 2, which correspond to regular and irregular topologies, respectively. We inform the reader that these examples are provided to illustrate the difference between the types of topologies of the complex networks that are about to be synchronized. From the knowledge of the authors, the topology of a network is closely related with its origin, i.e., regular ones are usually presented on artificial or man–made networks, while the irregular ones are commonly presented in nature [27]. For this particular case, the irregular topology of Fig. 2 (b) was arbitrary created. In the next section, numerical simulation of synchronization of the fractional–order Liu chaotic oscillator will be provided to show the effectiveness of the technique just described. 3. Synchronization of N identical fractional-order Liu chaotic oscillators In this section, numerical simulation of synchronization of N fractional–order Liu chaotic oscillators will be provided. The topologies considered to arrange the chaotic dynamical systems are regular and irregular, as shown in Fig. 2. In order to meet the requirements described in Section 2.3, Lemma 1, the self–feedback signal −T z was applied in each state variable of an isolated fractional–order Liu chaotic oscillator (8) at a time. It was determined that only −T z2 , i.e., the self–feedback applied in the second state, can stabilize a single chaotic oscillator for T ≥ 2.8 so that zero is an exponential equilibrium point. Fig. 3 shows the state variables of an isolated fractional–order Liu oscillator for T = 3. The synchronization results were obtained by computing the coupling strength c for T = 3 and the largest nonzero eigen value λ2 of the corresponding coupling matrix M. The control laws ui2 = c Nj=1 μi j y j for i = 1, 2, . . . , N were applied in the second state, i.e.,  = diag(0, 1, 0 ), since y(t) is the only state through which an isolated fractional–order Liu oscillator can be stabilized.

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Fig. 3. A single fractional–order Liu chaotic oscillator with self–feedback −T z2 , where [x(0 ), y(0 ), z(0 )] = [−2, −4, −2] and T = 3 is enough to stabilize all states of the isolated oscillator.

3.1. Synchronization of a regular complex network Each fractional–order Liu chaotic oscillator of the regular complex network under consideration (Fig. 2 (a)) will be described as

⎧ q1 ⎨0 Dt xi (t ) Dq2 y (t ) ⎩0 tq3 i 0 Dt zi (t )

=

−axi (t ) − ey2i (t ),

=

byi (t ) − kxi (t )zi (t ) + c

=

−dzi (t ) + mxi (t )yi (t ),

N j=1

μi j y j ,

(19)

with q1 = q2 = q3 = 0.95, a = e = 1, b = 2.5, d = 5, k = m = 4 [3,36,44] for i = 1, 2, . . . , 9. The coupling matrix M ∈ 9 × 9 , corresponding to the topology shown in Fig. 2 (a) is given by

⎡

−8

1 .. . .. . ···

⎢ ⎢1 ⎣ ..

M=⎢

. 1

··· ··· .. . 1

1 ... 1 −8

⎤ ⎥ ⎥ ⎥, ⎦

(20)

with the following eigenvalues σ (M ) = {λ1 , λ2 , . . . , λ9 } = {0, −9, . . . , −9}. The control laws ui2 will be given by the elements of the coupling matrix μij as follows:

⎧ u 1,2 ⎪ ⎪ ⎨u2,2 .

. ⎪ ⎪ ⎩.

u 9,2

= = .. . =

c (−8y1 + y2 + y3 + y4 + y5 + y6 + y7 + y8 + y9 ), c ( y1 − 8y2 + y3 + y4 + y5 + y6 + y7 + y8 + y9 ), .. .. .. . . . c ( y1 + y2 + y3 + y4 + y5 + y6 + y7 + y8 − 8y9 ).

(21)

The coupling strength c will be computed from (17), since we know the constant T to stabilize the isolated fractional– order Liu oscillator and the largest nonzero eigenvalue λ2 . Thus, one obtains

c≥

T

λ2

≥

3 ≥ 0.33. 9

(22)

By using c = 0.5 and initial conditions randomly chosen within the range x(0 ) ∈ [−3, 3], the following synchronization results were obtained: Fig. 4(a) shows the time evolution of the xj (t) state variables for j = 1, 2, . . . , 9, where synchronization can be observed. Embedded in the same figure, a close up of the first seconds of the simulation is provided, where one can observe every fractional–order Liu chaotic oscillator starting from different initial conditions. On the other hand, synchronization can be corroborated by mean of the phase portrait, which will produce a 45° line when the variables involved reach synchrony. Fig. 4 (b) presents the phase portraits between zi (t) vs zj (t) for i = 1, 4, 7 and j = 3, 6, 9, where synchronization can be confirmed in the third state variable. The chaotic attractor of the final dynamic of the complex network is also provided in this figure. In the remaining of the section, the synchronization results corresponding to the irregular complex network will be provided.

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Fig. 4. (a) Evolution of the synchronization of the first state xj for j = 1, 2, . . . , 9 using c = 0.5. (b) Phase portraits of the complex network in Fig. 2 (a): Confirmed synchronization of third state zi vs zj , i = 1, 4, 7; j = 3, 6, 9 and the corresponding chaotic strange attractor of the final dynamic.

3.2. Synchronization of an irregular complex network It is time now to present the synchronization results of the irregular complex network from Fig. 2 (b), which is also composed by N = 9 fractional–order Liu chaotic oscillators. Each chaotic oscillator of the network will be described by the set of Eq. (19) with the same values for the parameters. The coupling matrix M corresponding to the irregular topology will be given as follows:

⎡

−3 ⎢1 ⎢0 ⎢ ⎢1 ⎢ M=⎢ 1 ⎢0 ⎢ ⎢0 ⎣ 0 0

1 −2 0 0 1 0 0 0 0

0 0 −1 0 1 0 0 0 0

1 0 0 −3 0 0 1 1 0

1 1 1 0 −6 1 1 0 1

0 0 0 0 1 −1 0 0 0

0 0 0 1 1 0 −3 1 0

0 0 0 1 0 0 1 −2 0

⎤

0 0⎥ 0⎥ ⎥ 0⎥ ⎥ 1 ⎥, 0⎥ ⎥ 0⎥ ⎦ 0 −1

(23)

λ7 = −3.5044, λ8 = −4.4225, λ9 = −7.1467.

(24)

whose eigenvalues are the following:

 λ1 = 0.0 0 0 0, σ (M ) = λ2 = −0.6238, λ3 = −1.0 0 0 0,

λ4 = −1.0 0 0 0, λ5 = −1.3027, λ6 = −3.0 0 0 0,

As in the previous example, the control laws ui2 will be given by the nonzero elements of the coupling matrix (23) as follows:

⎧ u1,2 = c (−3y1 + y2 + y4 + y5 ), ⎪ ⎪ ⎪u2,2 = c(y1 − 2y2 + y5 ), ⎪ ⎪ ⎪ ⎪u3,2 = c(−y3 + y5 ), ⎪ ⎪ ⎨u4,2 = c(y1 − 3y4 + y7 + y8 ), u 5,2 = c ( y1 + y2 + y3 − 6 y5 + y6 + y7 + y9 ) , ⎪ ⎪ u 6,2 = c ( y5 − y6 ) , ⎪ ⎪ ⎪ u 7,2 = c ( y4 + y5 − 3 y7 + y8 ) , ⎪ ⎪ ⎪ ⎪ ⎩u8,2 = c(y4 + y7 − 2y8 ), u 9,2 = c ( y5 − y9 ) .

(25)

The coupling strength c that corresponds to the irregular topology is computed as

c≥

T

λ2

≥

3 ≥ 4.8092. 0.6238

(26)

By setting c = 5 and assigning randomly chosen initial conditions within the range x(0 ) ∈ [−2, 2], the following synchronization results were obtained:

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Fig. 5. (a) Evolution of the synchronization of the third state zj for j = 1, 2, . . . , 9 using c = 5. (b) Phase portraits of the complex network in Fig. 2 (b): Confirmed synchronization of first state xi vs xj , i = 2, 5, 8; j = 1, 4, 7 and the corresponding chaotic strange attractor of the final dynamic.

The evolution of the zj (t) state variables for j = 1, 2, . . . , 9, where synchronization can be observed as time passes, is provided in Fig. 5 (a). In order to observe that every fractional–order Liu chaotic oscillator started from different initial conditions, a close up of the first seconds of the simulation is embedded in the same figure. Fig. 5(b) presents the phase portraits between state variables of some fractional–order oscillators arbitrary chosen, xi (t) vs xj (t) for i = 2, 5, 8 and j = 1, 4, 7, where synchronization can be confirmed in the first state variable. The chaotic strange attractor of the final dynamic of the complex network is also embedded in this figure. Two things are worth mentioning from these results. First, the fact that even when the magnitude of the coupling strength c was very close to its lower boundary, it was sufficient enough to achieve synchronization on the complex network. On the other hand, determining that the topology of the complex network, i.e., the way the chaotic oscillators are arranged, plays a critical role when the coupling strength c, necessary to carry the complex network to synchrony, is computed. This is easy to observe from the difference between both coupling strength values used to synchronize the regular and irregular complex networks. In the next section, the results just presented will be corroborated by simulating the electrical behavior of the fractional– order Liu chaotic oscillator, arranged in complex networks of regular and irregular topologies. 4. FPGA realization This section presents the FPGA realization of the two fractional–order complex networks previously shown in Section 3. Two realizations are shown, the first one is related to the regular network addressed in Section 3.1, while the second represents the irregular network covered in Section 3.2. 4.1. The Hardware and software used Both implementations were done into a Xilinx Zynq-70 0 0 XC7Z020 FPGA chip. We use the LabVIEW–FPGA compiler software and the National Instruments NI cRIO–9068 hardware. The NI cRIO–9068 contains the FPGA chip used and a NI– 9381 multifunction module. The NI–9381 module was used as a digital to analog converter for the FPGA outputs related to the states of the network oscillators. We also use a Tektronix TDS5104B digital oscilloscope to visualize the analog outputs of the NI–9381 module. The hardware used is shown in Fig. 6. Concerning the software, the NI LabVIEW–FPGA software was chosen to simplify the simulation stage and reduce the project completion time, even it could use more hardware resources in synthesis in comparison with hardware description languages as VHDL. 4.2. Description of the design Both networks consider nine identical fractional-order Liu oscillators as nodes and synchronize through the second state of each node. As both realizations share these general characteristics, the description of both networks is covered through the block descriptions shown in Figs. 7 and 8. The mathematical operations were performed using fixed point (FXP) arithmetics. In almost all the cases a 24–bit word– length was used, with 4–bits for the real part and 20–bits for the fractional part of each FXP number. A variable word–length was also used to represent the binomial coefficients (7) and reduce rounding errors induced by truncation. Although FXP

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Fig. 6. Hardware used for the FPGA realization. The connections between the cRIO–9068 hardware and the Tektronix TDS5104B digital oscilloscope are illustrated.

Fig. 7. Block diagram describing the top level description of the regular and irregular networks implemented. (a) The control law and the network connections according to (21) and (25) for the regular and irregular cases. (b) The dynamics of the nine identical nodes of the network. The dynamics of each node are given by (27).

representation may introduce extra round-off errors, we preferred to use FXP in place of floating point arithmetics to save FPGA resources. 4.2.1. Complex network level The top level, i.e., the complex network level, description of the design is reflected in Fig. 7. This representation applies to both networks. We separate the blocks of Fig. 7 in two stages for better interpretation. The stage (a) represents the coupling of the network. It defines the network topology and the control laws needed to achieve the synchronization for each one of the networks analyzed. For the regular network this stage represents the relations shown in (21), while for the irregular case represents (25). Then stage (b) is used to generate the digital approximation for each one of the nine nodes. This stage applies the control outputs ui, 2 [tk ] to the nodes of the network. As nodes are identical, stage (b) is composed of nine copies of the same system, where each node process a different control input and its state registers have been initialized with different values. 4.2.2. Single node level Each network is composed of nine fractional–order Liu chaotic oscillators. The single node level is responsible for the generation of the node dynamics. The Grunwald–Letnikov method was implemented to solve the fractional–order numerical integration. Then, the digital approximation for the ith node is given by

⎧ ⎪ ⎨xi [tk+1 ]

=

yi [tk+1 ]

=

k+1 ]

=

⎪ ⎩z [t i





 Lm −1



c(jq1 ) xi [tk − j] ,  Lm −1 (q2 )  c j yi [tk − j] , (byi [tk ] − kxi [tk ]z[tk ] + ui,2 (tk ) )hq2 − j=0  Lm −1 (q3 )  c j zi [tk − j] . (−dzi [tk ] + mxi [tk ]yi [tk ] )hq3 − j=0 −axi [tk ] − ey2i [tk ] hq1 −

j=0

(27)

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Fig. 8. Block diagram describing the proposed implementation for each node. Where the next state logic for each state register is given by the right part of (27). The nine nodes of the networks considered in this work are represented by this implementation.

Table 1 Xilinx Zynq-70 0 0 XC7Z020 FPGA chip resource usage. The resources used for the implementation of the regular and irregular networks from Section 3 are shown. Regular network Used Slice Registers Slice LUT DSP48s Block RAMs Clock Latency Throughput Estimated power

Total

23679 106400 26340 53200 148 220 16 140 40 MHz 41.25 μs. 15.7 Mbps 1.428 W

Irregular network Percent

Used

Total

Percent

22.3 % 49.5 % 67.3 % 11.4 %

23435 25892 184 16 40 MHz 67.5 μs. 9.6 Mbps 1.432 W

106400 53200 220 140

22.0 % 48.7 % 83.6 % 11.4 %

Fig. 9. FPGA implementation results for the first node of the regular network shown in Section 3. Phase portraits for (a) y1 [tk ] vs x1 [tk ], (b) x1 [tk ] vs z1 [tk ] and (c) y1 [tk ] vs z1 [tk ]. The remaining eight nodes exhibit similar chaotic behavior.

where k ∈ Z + represents the sampling index and h = 0.002 the step size. The rest of the parameters in (27) are defined in Section 2.2. The block diagram description of (27) is shown in Fig. 8 for one node. We remark that as nodes are identical this description applies to any node in the network. Note that the right side summations in (27) can be seen as an FIR filtering process; and that for commensurate orders the three filters involved are defined by the same coefficients (7). We use single MAC traditional FIR filters with a finite memory length of Lm = 8 coefficients to save FPGA resources. For all the nodes, all initial conditions for the state registers were arbitrarily chosen within the range [−3, 3].

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Fig. 10. Evidence of synchronization for the regular network. Synchronization is confirmed through the phase portraits (a) x1 [tk ] vs x3 [tk ], (b) x1 [tk ] vs x5 [tk ] and (c) x3 [tk ] vs x5 [tk ].

Fig. 11. Evidence of synchronization for the irregular network. Synchronization is confirmed through the phase portraits (a) x2 [tk ] vs x4 [tk ], (b) x2 [tk ] vs x6 [tk ] and (c) x4 [tk ] vs x6 [tk ].

Table 2 Comparison of the main differences/advantages between the described results and the ones already published. Paper contributions

Other results

Synchronization of complex networks composed of fractional–order chaotic oscillators. Theory for inter–order chaotic systems was efficient to synchronize fractional–order chaotic oscillators [23]. The network ability to synchronize is analyzed by separate. First, the stability characteristics of the dynamical system are determined. Second, we analyze the topology effect through the eigenvalues of the coupling matrix. Undirected configurations are allowed, i.e., chaotic dynamics flow in both ways. The resulting synchronous behavior is unknown and unpredictable, making it ideal for applications of secret communications [25,27]. Control laws to lead the complex network to behave synchronously are linear combinations of the state variables.

Synchronization of pairs of fractional–order oscillators [17,18]. For the standard control scheme, theory and controllers need to be redesigned on every model of chaotic oscillator [20,21]. The analysis to determine the network ability to synchronize is performed over the whole system, which results in complicated control laws difficult to implement [19,22]. The configuration is restricted to directed network, i.e., only master–slave interactions are considered [19]. The final state is described by the dynamics of the master chaotic oscillator. Control laws are generally complex nonlinear functions depending on the feedback error and the system parameters [20–22].

4.2.3. Results Table 1 presents the total amount of FPGA resources used in the implementation of each network. These resources were obtained from the final device utilization report from the NI LabView–FPGA compiler. Latency and throughput were computed experimentally by increasing the operating frequency. The estimated power was computed employing the Xilinx power estimator. Note that the irregular network exhibits a lower performance and uses a more significant amount of slices in comparison with the regular network. The compiler may infer fewer resources during synthesis for the regular case due to the symmetry of (21) in comparison with (25), as the only change between these networks is the topology. For the regular case, the network attractor is shown in Fig. 9, where XY graphs of the NI–9381 analog outputs taken with the TDS5104B oscilloscope are shown. The outputs shown are related to the states of the first node. Fig. 10 shows the verification of the synchronization for the x[tk ] state via phase portraits. In this figure, XY graphs relating the x[tk ] state of the first, third and fifth node are shown. The presence of a 45-degree line in all the cases confirms synchronization of these nodes. Similar behavior was presented for the rest of the nodes. Similar results were obtained for the irregular case, evidence of synchronization for the irregular network is shown in Fig. 11. Note that these results are similar than those obtained through the numerical simulations presented in Section 3.

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It is important to remark that the quantization and the finite window length were selected to derive a feasible design for the hardware used. These parameters may produce a degradation of the dynamics in the digital realization [46]. However, this degradation is inevitable and represents a repercussion of digital simulations or implementations. Another critical point that must be discussed is the comparison of the performance of this design with other designs found in the literature [39–41]. In general, the proposed design has a lower performance compared to the literature, especially concerning latency. The performance degradation may be due to (1) the high-level software platform used in this work may affect low-level optimization, (2) the fact that most of the works in the literature deal with single node implementations, and (3) the use of FPGAs with different resource capacity. 5. Conclusion In this paper, synchronization of fractional–order Liu chaotic oscillators arranged in topologies of complex networks was addressed. A theory initially developed to synchronize integer–order systems was employed. Authors have proven the effectiveness of the coupling matrix technique, Theorem 1 and Lemma 1 on synchronizing complex networks composed of fraction–order dynamical systems. It was studied the ability of the fractional–order Liu chaotic oscillator to be synchronized by mean of the theory described in Section 2.3. It was determined that this chaotic dynamical system is able to synchronize only when is coupled through its second state variable. Furthermore, it was possible to show the importance of the topology through the coupling strength c since, for both cases, the only difference between the complex networks considered was the way the chaotic oscillators were arranged. However, the lower boundaries for both cases were different, being the irregular complex network more difficult to synchronize. The most relevant thing from this results is related to the fact that synchronization was possible in complex networks composed of fractional–order oscillators. We corroborated that data obtained from the Wang–Chen method was sufficient enough to achieve synchronization. The results obtained from the FPGAs allowed us to confirm that the Wang–Chen method is efficient to synchronize fractional–order chaotic oscillators, which means an increase in scope of this theory. In Table 2 the main contribution of this paper is briefly summarized. We mention significant differences between this results and some of the most important works. Two things are worth mentioning from the FPGA realization presented in this paper. First, the fact that the software/hardware platform allowed the realization of the fractional–order chaotic oscillator prototype as well as the synchronization process. And, on the other hand, it is highlighted that the FPGA results were consistent with the numerical simulations. Acknowledgment This work was supported by CONACYT México under Research Grant no. 166654 and “Facultad de Ingeniería Mecánica y Eléctrica (FIME–UANL)”. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

I. Podlubny, Fractional Differential Equations, 198, Academic press, 1998. K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiationand Integration to Arbitrary Order, 111, Elsevier, 1974. I. Petráš, Fractional–Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer Science & Business Media, 2011. J.C. Wang, Realizations of generalized warburg impedance with RC ladder networks and transmission lines, J. Electrochem. Soc. 134 (8) (1987) 1915–1920. T.T. Hartley, C.F. Lorenzo, H.K. Qammer, Chaos in a fractional order Chua’s system, IEEE Trans. Circuits Syst.-I: Fundam. Theory Appl. 42 (8) (1995) 485–490. M.P. Singh, T.S. Chang, H. Nandan, Algorithms for seismic analysis of MDOF systems with fractional derivatives, Eng. Struct. 33 (8) (2011) 2371–2381. S.H. Hosseinnia, I. Tejado, V. Milanés, J. Villagrá, B.M. Vinagre, Experimental application of hybrid fractional–order adaptive cruise control at low speed, IEEE Trans. Control Syst. Technol. 22 (6) (2014) 2329–2336. C.C. Tseng, Design of FIR and IIR fractional order Simpson digital integrators, Signal Process. 87 (5) (2007) 1045–1057. R.L. Magin, M. Ovadia, Modeling the cardiac tissue electrode interface using fractional calculus, J. Vib. Control 14 (9–10) (2008) 1431–1442. T. Margulies, Wave propagation in viscoelastic horns using a fractional calculus rheology model, J. Acoust. Soc. Am. 114 (4) (2003) 22442. L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (8) (1990) 821. C.W. Wu, L.O. Chua, A unified framework for synchronization and control of dynamical systems, Int. J. Bifurc. Chaos 4 (4) (1994) 979–998. J.F. Heagy, T.L. Carroll, L.M. Pecora, Desynchronization by periodic orbits, Phys. Rev. E 52 (2) (1995) R1253. N.F. Rulkov, Images of synchronized chaos: experiments with circuits, Chaos 6 (3) (1996) 262–279. S. Yanchuk, Y. Maistrenko, E. Mosekilde, Loss of synchronization in coupled Rössler systems, Phys. D 154 (1) (2001) 26–42. G. Kolumbán, M.P. Kennedy, L.O. Chua, The role of synchronization in digital communications using chaos. i. fundamentals of digital communications, IEEE Trans. Circuits Syst.-I: Fundam. Theory Appl. 44 (10) (1997) 927–936. Z.M. Zhang, Y. He, M. Wu, Q.G. Wang, Exponential synchronization of chaotic neural networks with time-varying delay via intermittent output feedback approach, Appl. Math. Comput. 314 (2017) 121–132. X. Chen, J.H. Park, J. Cao, J. Qiu, Sliding mode synchronization of multiple chaotic systems with uncertainties and disturbances, Appl. Math. Comput. 308 (2017) 161–173. Z. Zhang, H. Shao, Z. Wang, H. Shen, Reduced–order observer design for the synchronization of the generalized Lorenz chaotic systems, Appl. Math. Comput. 218 (14) (2012) 7614–7621. H. Yang, X. Wang, S. Zhong, L. Shu, Synchronization of nonlinear complex dynamical systems via delayed impulsive distributed control, Appl. Math. Comput. 320 (2018) 75–85. A.Y.T. Leung, X.-F.Li, Y.-D. Chu, X.-B. Rao, A simple adaptive–feedback scheme for identical synchronizing chaotic systems with uncertain parameters, Appl. Math. Comput. 253 (2015) 172–183.

262

S. A.G. et al. / Applied Mathematics and Computation 332 (2018) 250–262

[22] J. Peng, L. Wang, Y. Zhao, Y. Zhao, Bifurcation analysis in active control system with time delay feedback, Appl. Math. Comput. 219 (19) (2013) 10073–10081. [23] X.F. Wang, G. Chen, Synchronization in small–world dynamical networks, Int. J. Bifurc. Chaos 12 (1) (2002) 187–192. [24] A.G. Soriano-Sánchez, C. Posadas-Castillo, M.A. Platas-Garza, Synchronization of generalized Chua’s chaotic oscillators in small–world topology, J. Eng. Sci. Technol. Rev. 8 (2) (2015) 185–191. [25] A.G. Soriano-Sánchez, C. Posadas-Castillo, M.A. Platas-Garza, D.A. Diaz-Romero, Performance improvement of chaotic encryption via energy and frequency location criteria, Math. Comput. Simul. 112 (2015) 14–27. [26] A.G. Soriano-Sánchez, C. Posadas-Castillo, M.A. Platas-Garza, C. Cruz-Hernández, R.M. López-Gutiérrez, Coupling strength computation for chaotic synchronization of complex networks with multi–scroll attractors, Appl. Math. Comput. 275 (2016) 305–316. [27] A.G. Soriano-Sánchez, C. Posadas-Castillo, M.A. Platas-Garza, C. Elizondo-González, Chaotic synchronization of CNNS in small–world topology applied to data encryption, in: Advances and Applications in Chaotic Systems, Springer, 2016, pp. 337–362. [28] R. Martínez-Guerra, C.A. Pérez-Pinacho, G.C. Gómez-Cortés, J.C. Cruz-Victoria, Synchronization of incommensurate fractional order system, Appl. Math. Comput. 262 (2015) 260–266. [29] J.C. Cruz-Victoria, R. Martínez-Guerra, C.A. Pérez-Pinacho, G.C. Gómez-Cortés, Synchronization of nonlinear fractional order systems by means of pi rα reduced order observer, Appl. Math. Comput. 262 (2015) 224–231. [30] G.-S. Wang, J.-W. Xiao, Y.-W. Wang, J.-W. Yi, Adaptive pinning cluster synchronization of fractional-order complex dynamical networks, Appl. Math. Comput. 231 (2014) 347–356. [31] A. Ouannas, G. Grassi, A new approach to study the coexistence of some synchronization types between chaotic maps with different dimensions, Nonlinear Dyn. 86 (2) (2016) 1319–1328. [32] J.G. Lu, G. Chen, A note on the fractional–order Chen system, Chaos Solitons Fract. 27 (3) (2006) 685–688. [33] C. Li, J. Yan, The synchronization of three fractional differential systems, Chaos Solitons Fract. 32 (2) (2007) 751–757. [34] C. Li, G. Chen, Chaos and hyperchaos in the fractional–order Rössler equations, Phys. A 341 (2004) 55–61. [35] W.H. Deng, C.P. Li, Chaos synchronization of the fractional lü system, Phys. A 353 (2005) 61–72. [36] V. Daftardar-Gejji, S. Bhalekar, Chaos in fractional ordered LIU system, Comput. Math. Appl. 59 (3) (2010) 1117–1127. [37] I. Petráš, A note on the fractional–order Chua’s system, Chaos Solitons Fract. 38 (1) (2008) 140–147. [38] S. Chen, S. Yu, J. Lu, G. Chen, J. He, Design and FPGA-based realization of a chaotic secure video communication system, IEEE Trans. Circuits Syst. video Technol. PP (99) (2017) 1. [39] E. Tlelo-Cuautle, A.D. Pano-Azucena, J.J. Rangel-Magdaleno, V.H. Carbajal-Gomez, G. Rodriguez-Gomez, Generating a 50-scroll chaotic attractor at 66 MHz by using FPGAS, Nonlinear Dyn. 85 (4) (2016) 2143–2157. [40] K. Rajagopal, A. Karthikeyan, A.K. Srinivasan, Fpga implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization, Nonlinear Dyn. 87 (4) (2017) 2281–2304. [41] M.F. Tolba, A.M. AbdelAty, N.S. Soliman, L.A. Said, A.H. Madian, A.T. Azar, A.G. Radwan, FPGA implementation of two fractional order chaotic systems, AEU – Int. J. Electron. Commun. 78 (Supplement C) (2017) 162–172. [42] Y.Q. Chen, K.L. Moore, Discretization schemes for fractional–order differentiators and integrators, IEEE Trans. Circuits Syst.-I: Fundam. Theory Appl. 49 (3) (2002) 363–367. [43] B.M. Vinagre, Y.Q. Chen, I. Petráš, Two direct Tustin discretization methods for fractional–order differentiator / integrator, J. Frankl. Inst. 340 (5) (2003) 349–362. [44] C. Liu, L. Liu, T. Liu, A novel three–dimensional autonomous chaos system, Chaos, Soliton Fract. 39 (4) (2009) 1950–1958. [45] X.F. Wang, Complex networks: topology, dynamics and synchronization, Int. J. Bifurc. Chaos 12 (5) (2002) 885–916. [46] S. Li, G. Chen, X. Mou, On the dynamical degradation of digital linear chaotic maps, Int. J. Bifurc. Chaos 15 (10) (2005) 3119–3151.