A general fractional-order dynamical network: Synchronization behavior and state tuning Junwei Wang and Xiaohua Xiong Citation: Chaos 22, 023102 (2012); doi: 10.1063/1.3701726 View online: http://dx.doi.org/10.1063/1.3701726 View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v22/i2 Published by the American Institute of Physics.
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CHAOS 22, 023102 (2012)
A general fractional-order dynamical network: Synchronization behavior and state tuning Junwei Wang1,a) and Xiaohua Xiong2 1
School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510006, China School of Computer and Information Engineering, Jiangxi Normal University, Nanchang 330022, China
2
(Received 18 December 2011; accepted 19 March 2012; published online 9 April 2012) A general fractional-order dynamical network model for synchronization behavior is proposed. Different from previous integer-order dynamical networks, the model is made up of coupled units described by fractional differential equations, where the connections between individual units are nondiffusive and nonlinear. We show that the synchronous behavior of such a network cannot only occur, but also be dramatically different from the behavior of its constituent units. In particular, we find that simple behavior can emerge as synchronized dynamics although the isolated units evolve chaotically. Conversely, individually simple units can display chaotic attractors when the network synchronizes. We also present an easily checked criterion for synchronization depending only on the eigenvalues distribution of a decomposition matrix and the fractional orders. The analytic results are complemented with numerical simulations for two networks whose nodes are governed by fractional-order Lorenz dynamics and fractional-order Ro¨ssler dynamics, respectively. C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3701726] V Complex networks permeate every aspect of our daily lives, affecting both the natural and man-made world. Commonly cited examples include social networks, chemical networks, biological networks, ecological networks, organizational networks, technological networks, information networks, communication networks, and many others. Over the last few years, we have witnessed a substantial movement in the structure and dynamics of networked systems, among which the synchronization behavior inside a network has received increasing attention and becomes a hot topic in various different fields. As a typical collective behavior of complex networks, synchronization is often encountered in living systems, such as circadian rhythm, interaction of the sinus node with ectopic pacemakers, phase locking respiration with mechanical ventilator, and neurons in the brain. Indeed, synchronization has proved to serve certainly as the basic mechanism for the information processing and integration between different brain regions. Most of the research on synchronization behavior so far, however, has been restricted to integer-order dynamical networks (IDNs) of coupled “ordinary nodes” (whose dynamics are described by integer-order differential equations). Few studies have examined the synchronization behavior of fractionalorder dynamical networks (FDNs) of coupled “fractional nodes” (whose units are described by fractional-order differential equations), which provide us an excellent instrument for the description of complex systems with memory and hereditary properties. To address this, here we construct a general fractional-order dynamical network model with an important peculiarity that the connections between individual units are nondiffusive and nonlinear. We show that the synchronous behavior of such a network cannot only occur, but also be a)
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dramatically different from the behavior of its constituent units. In other words, one can obtain new synchronous motion (either simple or chaotic) by tuning a coupling parameter. We also present an easily checked criterion for synchronization depending only on the eigenvalues distribution of a decomposition matrix and the fractional orders. We believe that our study constitutes a first step towards a deeper understanding of the origins of complex collective behavior from coupled fractional-order systems.
I. INTRODUCTION
Since Christiaan Huygens who observed that two pendulum clocks mounted on a common wall will swing in perfect synchrony in the 17th century, synchronization of regular oscillators with limit cycle attractors has been known and is in fact ubiquitous in the real world.1,2 Only recently, however, with the seminal work by Pecora and Carroll,3,4 has the synchronization of chaotic oscillators become known. Thereafter, chaos synchronization has aroused a great interest in light of its potential for technical applications and been extended from coupled two oscillators to complex dynamical networks. In general, a complex dynamical network refers to an ensemble of dynamical nodes interconnected through a nontrivial connectivity structure, in which a node is a fundamental unit with specific contents. Frequently cited examples of complex networks include gene regulatory networks, biological neural networks, ecological food webs, the World Wide Web, the Internet, scientific citation networks, the network of social contacts, etc.5–7 Among different dynamical processes on complex networks, synchronization is the most interesting and has received considerable attention in recent years.8,9 Initially, the existing work is mainly concentrated
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on synchronization of networks with completely regular topological structures (e.g., the continuous time cellular nonlinear networks10 and discrete-time coupled map lattices11). The discovery of the small-world effect and scale-free feature,12,13 two universal laws governing different kinds of real-world natural and artificial complex networks, has led to dramatic advances in exploring synchronization phenomenon in small-world and scale-free dynamical networks.14–25 These fruitful developments have largely enriched and deepened our understanding of real-world complex networks. It should be noticed that, most of the work in the literature so far has been focused on synchronization behavior of IDNs, in which individual nodes are often described by ordinary differential equations (ODEs). In this situation, the stability analysis of synchronization manifold can be carried out using classical Lyapunov stability theory. However, realistically it is inappropriate, employing IDNs, to model materials and processes with memory and hereditary properties, such as anomalous diffusion, time-dependent materials and processes with long-range dependence, allometric scaling laws, as well as power law in complex systems. Instead, FDNs provide an excellent instrument for the description of the above-mentioned complex systems due to the existence of a “memory” term in the fractional-order model. A FDN is defined as interacting dynamical units described by fractional differential equations (FDEs). As a generalization of ODEs to an arbitrary order, FDEs can capture nonlocal relations in space and time with power-law memory kernels; thus, the fractional-order models are believed to be more accurate than integer-order models. As a matter of fact, due to its extensive applications in engineering and science, the research for FDEs has grown significantly in the past years and becomes a popular developing topic (see, e.g., Refs. 26–39 and the references therein). Therefore, motivated by the converging developments in the synchronization behavior of IDNs, there is an increasingly voiced need to understand various synchronizations of the interwoven systems with fractional-order derivatives. Recent years have witnessed a growing interest in the synchronization and stabilization of coupled fractional-order systems, including FDNs.40–48 By a mode decomposition, Zhou and Li44 investigated the synchronization state of an x-symmetrically network consisting of coupled identical fractional-order differential systems including chaotic and nonchaotic units. The decomposition approach also induces a sufficient condition on synchronization of the overall system, which guarantees, if satisfied, that a group synchronization is achieved. Tang et al.45 considered the pinning control problem of fractional-order weighted complex dynamical networks and found that the fractional-order complex networks can stabilize itself by reducing the fractional order q without pinning any node. Using the open-plus-closed-loop control strategy, we propose a novel global coupling scheme for synchronization in a network of fractional-order systems.46 Based on stability criteria of fractional-order system, the synchronization of N-coupled fractional-order chaotic systems with unidirectional coupling and bidirectional coupling is achieved.48
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Although these substantial progresses toward understanding synchronization of FDNs have been made, to the best of our knowledge, the relevant researches are still in an initial stage and many open problems remain to be unsolved, e.g., how the structure affects the collective dynamics, the effect of fractional-order on the network synchronization, how can we design coupling interaction to realize the desired synchronized behavior, etc. Indeed, the desire to study the synchronization problem of the complex networks consisting of FDEs has encountered significant challenges, since most of the existing research methods for ODEs cannot be simply extended to the case of FDEs. On the other hand, most previous work on synchronization of both IDNs and FDNs focused on diffusive coupling between oscillators. In such diffusively coupled networks, the synchronized state exhibits the same dynamical behavior as their constituent units, and no new synchronized motion is emerging here.49–54 This raises a central problem of how can we tune the synchronous state to new orbits when necessary with the designed coupling schemes. In this paper, we offer a solution to this question in the context of synchronization of FDNs. The present paper will add another species to the zoo of synchronization of FDNs. A FDN model is first constructed to investigate its synchronization behavior and state tuning effect. The crucial characteristic of our model is that it does not require special structure of the individual nodes. In other words, synchronization emerges from a general FDN with any nonlinear isolated units. Moreover, the synchronous state of the network can be tuned freely through varying the coupling strength and may be dramatically different from the behavior of its constituent units. In particular, we find that simple behavior can emerge as synchronized dynamics although the isolated units evolve chaotically. Conversely, individually simple units can display chaotic attractors when the network synchronizes. We also derive a synchronization criterion that depends on the spectrum of a decomposition matrix and the fractional orders through the stability theory of FDEs. Finally, several numerical simulations are presented to manifest the effectiveness of our theoretical results.
II. FRACTIONAL CALCULUS AND NETWORK MODEL A. Fractional calculus
As a generalization of integration and differentiation to a fractional (non-integer) order, fractional calculus (FC) goes back to times when Leibniz and Newton invented ordinary calculus. For past 300 years, the theory of factional-order derivatives was developed primarily as a pure theoretical field useful only for mathematicians. Nowadays, however, it was found that many systems in interdisciplinary fields can be described by models with FC.36–39 It is believed that FC provides an excellent tool for describing the memory and hereditary properties of various materials and processes. Up to now, there are many ways to define a fractionalorder derivatives,37 among which Caputo fractional-order derivative represents one of the most commonly used as it allows utilization of initial values of classical integer-order derivatives with known physical interpretations. The
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fractional-order derivative of the function f(t) in the Caputo sense is defined as: Dq f ðtÞ ¼ J mq f ðmÞ ðtÞ:
(1)
Here, q is the fractional order, m is an integer that satisfies m 1 q < m; f ðmÞ is the ordinary mth derivative of f, and J l is the Riemann-Liouville integral operator of order l > 0, defined by ð 1 t ðt sÞl1 gðsÞds; (2) J l gðtÞ ¼ CðlÞ 0 where CðÞ denotes the gamma function. A particularly important case in many engineering applications is 0 < q < 1. In this situation, Eq. (1) together with Eq. (2) reduces to ðt 1 ðt sÞq f 0 ðsÞds: (3) Dq f ðtÞ ¼ Cð1 qÞ 0 Dq
The operator is often called “qth-order Caputo differential operator” and will be used throughout the paper. B. Network model
In this paper, we consider a dynamical network consisting of N identical nodes with nonlinear couplings, in which each node is a n-dimensional system obeying the following fractional kinetic equations: Dq x ¼ Ax þ f ðxÞ;
(4)
where xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; …; xn ðtÞÞT 2 Rn represents the state vector, A 2 Rnn is a constant matrix, and f : Rn ! Rn represents the nonlinear part of the oscillator and is assumed to be smooth enough. q ¼ ðq1 ; q2 ; …; qn ÞT indicates the and fractional orders with all qi 2 ð0; 1Þ T D Dq x ¼ Dq1 x1 ; Dq2 x2 ; …; Dqn xn . We call system (4) a commensurate order system for q1 ¼ q2 ¼ ¼ qn , otherwise system (4) is called an incommensurate order system. The entire network is a system of nN FDEs. In particular, the state equations are Dq xi ¼ Axi þ f ðsðtÞÞ;
i ¼ 1; 2; …; N;
we shall later on restrict ourselves to the linear case, i.e., to linear integrated function h. For such networks, the state equations can be explicitly written as ! N eX q D xi ¼ Axi þ f (7) xj ; i ¼ 1; 2; …; N; N j¼1 where e 2 R is the coupling strength. The objective is to find a condition satisfied by the constant matrix A and the coupling strength e under which the solutions of network (7) globally asymptotically synchronize each other, in the sense that lim jjjxi ðtÞ xj ðtÞjj ¼ 0; for all
t!1
i; j ¼ 1; 2; ; N:
(8)
III. MAIN THEORETICAL RESULTS
The main results of this paper on global synchronization and state tuning of network (7) are derived in this section. For this purpose, we first state a stability lemma from the fractional calculus which will be used thereafter. Lemma 1: (Ref. 41). Consider the following incommensurate linear fractional-order system:
(5)
where x i 2 Rn describes the state of the ith node and sðtÞ 2 Rn is the coupling signal. These unit systems are nonlinearly coupled by the function sðtÞ ¼ hðx1 ; x2 ; …; xN Þ;
FIG. 1. Schematic form of the considered network model (5).
(6)
where h : RnN ! Rn integrates the state of all units forming a complex network, as illustrated in Fig. 1. It should be noted that the integrated function h can be designed in either linear P form (e.g., h(x1, x2 ; …; xN Þ ¼ N1 Nj¼1 xj ) or nonlinear form P (e.g., h(x1, x2 ; …; xN Þ ¼ N1 Nj¼1 gðxj Þ with smooth function g : Rn ! Rn ). From the view of synchronization, both kinds of function h would achieve network synchronization, but with different synchronized state. Therefore, for simplicity,
8 q1 D x1 ¼ c11 x1 þ c12 x2 þ þ c1n xn > > > < Dq2 x2 ¼ c21 x1 þ c22 x2 þ þ c2n xn .. > . > > : qn D xn ¼ cn1 x1 þ cn2 x2 þ þ cnn xn ;
(9)
where qi 2 ð0; 1Þ are rational numbers. Assume M be the lowest common multiple of the denominators ui of qi , where qi ¼ vi =ui ; ðui ; vi Þ ¼ 1; ui ; vi 2 Z þ ; i ¼ 1; 2; …; n. Define 0 Mq 1 c12 c1n k 1 c11 B c21 kMq2 c22 c2n C B C DðkÞ ¼ B C: (10) .. . . . .. .. .. @ A . Mqn cn1 cn2 k cnn
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Then the zero solution of system (9) is globally asymptotically stable in the Lyapunov sense if all roots k of the equation detðDðkÞÞ ¼ 0 satisfy jargðkÞj > p=2M. A. Synchronization analysis
Define the synchronization error as ei ¼ xi x1 ;
i ¼ 2; 3; …; N:
(11)
words, the synchronous motion of the network can be tuned quite different from the behavior of isolated units by selecting a suitable tuning parameter e. To understand the effect of the coupling strength e on the synchronous state, we here analyze the stability of the equilibrium points of the synchronous system (16). The equilibrium points of system (16) are determined by solving the following equations: AgðtÞ þ f ðegðtÞÞ ¼ 0:
Then based on the Eq. (7), we get the error system: Dq ei ¼ Aei ;
i ¼ 2; 3; …; N:
(12)
According to Lemma 1, we obtain the following theorem: Theorem 1: Consider the FDN (7) with qi ¼ vi =ui ; ðui ; vi Þ ¼ 1; ui ; vi 2 Zþ , and assume that M is the lowest common multiple of the denominators ui of qi (i ¼ 1; 2; …; n). The zero solution of the error system (12) is asymptotically stable if all roots k of the characteristic equation (13) det diagðkMq1 ; kMq2 ; …; kMqn Þ A ¼ 0; satisfy the following condition: jargðkÞj >
p ; 2M
(14)
which means that network (7) has achieved network synchronization. Remark 1: Different from previous works, whether synchronization of network (7) is achieved do not depend on the coupling strength e. However, the coupling strength e will tune the synchronization dynamics (see Subsection III B). Remark 2: The inequality (14) means that the stability region of synchronous manifold of network (7) is only determined by the eigenvalue distribution of constant matrix A and is bounded by a cone (with vertex at the origin, extending into the right half of the complex k-plane such that it encloses an angle of 6p=2M with the positive real axis). Therefore, the synchronization manifold of network (7) is stable if all roots of the characteristic polynomial (13) are placed outside this cone by selecting appropriate constant matrix A. B. Tuning the synchronized motion
When network (7) realized synchronization, then x1 ðtÞ ¼ x2 ðtÞ ¼ ¼ xN ðtÞ ¼ gðtÞðt ! 1Þ;
(15)
where gðtÞ satisfies Dq gðtÞ ¼ AgðtÞ þ f ðegðtÞÞ:
(16)
In the rest of this paper, we call fractional-order system (16) the synchronous system. From Eq. (16), if e ¼ 1, the synchronous state is determined by Dq gðtÞ ¼ AgðtÞ þ f ðgðtÞÞ, which is the dynamics of an uncoupled node; or else e 6¼ 1, the original synchronous state will be changed. In other
(17)
Let g denotes one of the equilibrium points of system (17), i.e., Ag þ f ðeg Þ ¼ 0. To evaluate the local asymptotic stability of this point, we define: nðtÞ ¼ gðtÞ g :
(18)
Supposing that the function f has second continuous partial derivatives in a ball Rn centered at point g and using the Taylor series expansion, fðegðtÞÞ in the right hand side of Eq. (16) can be expanded as: f ðegðtÞÞ ¼ f ðeg Þ þ e
@f j nðtÞ þ : @x eg
(19)
Keeping the first order term in Eq. (19) and substituting it in Eq. (16), we obtain the system @f Dq nðtÞ ¼ Ag þ f ðeg Þ þ A þ e jeg nðtÞ ¼ HnðtÞ; @x (20) where D
H ¼Aþe
@f j : @x eg
(21)
Equation (20) with (21) already shows that the overall coupling strength e will affect the eigenvalue distribution of matrix H. At this stage, one can make the synchronous motion either simple or chaotic by choosing the tuning parameter e. In order to obtain a simple synchronous behavior, according to Lemma 1, we can tune the parameter e such that all eigenvalues ki ði ¼ 1; 2; …; nÞ of matrix H satisfy jarg½ki ðHÞj >
p ; 2M
(22)
then the equilibrium point g of the system (16) is locally asymptotically stable, indicating that the whole synchronized network will display simple dynamical behavior with appropriate coupling strength e. Such a simple synchronous behavior may arise by tuning parameter e from two aspects: First, if the original uncoupled units already behave simple dynamics, the tuning parameter e drives the network to a new equilibrium point different from its constituent units; second, if the original uncoupled units evolve chaotically, the tuning parameter e will suppress the individual chaotic behavior such that the FDN (7) displays simple behavior. On the other hand, we can also obtain a complex synchronous behavior (e.g., chaotic oscillations) via tuning the
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parameter e. Suppose that X is the set of equilibrium points of the synchronous system (16). Hence, a necessary condition for the synchronous system (16) to exhibit the chaotic attractor is instability of the equilibrium points in X. Otherwise, one of these equilibrium points becomes asymptotically stable and attracts the nearby trajectories. Mathematically, this necessary condition is equivalent to the non-negativity of the instability measure for all equilibrium points of synchronous system (16),55,56 i.e., p min½jargðki Þj 0; i 2M
(23)
dq x ¼ Bx þ f ðxÞ; dtq
where ki are roots of equations: det diagðkMq1 ; kMq2 ; …; kMqn Þ A @f =@xjeg ¼ 0; for any g 2 X:
the common Lorenz system exhibiting chaotic behavior. By defining an effective dimension R as a sum of the orders of all involved derivatives, i.e., R ¼ q1 þ q2 þ q3 , it has been found that the fractional-order Lorenz system with R < 3 can exhibit chaotic behavior.67 For q1 ¼ q2 ¼ q3 ¼ 0:99, the chaotic attractor of the fractional-order Lorenz system is shown in Fig. 2 with initial conditions ðx1 ð0Þ; x2 ð0Þ; x3 ð0ÞÞT ¼ ð6; 10; 18ÞT : In order to study the synchronization of the FDN with the aforesaid fractional-order Lorenz system being the node, we decompose Eq. (25) into the form of Eq. (4):
(24)
The above analysis suggests that by tuning the parameter e to satisfy the necessary condition (24), the synchronous system (16) may exhibit chaotic behavior. IV. NUMERICAL EXAMPLES
In this section, illustrative examples will be provided to verify the correctness and effectiveness of the theoretical analysis obtained in Sec. III. Here, the FDEs of Caputo type is numerically solved by a predictor-corrector or, more precisely, Predict, Evaluate, Correct, Evaluate (PECE) scheme,57 which is a generalization of the classical one-step AdamsBashforth-Moulton method that is well known for the numerical solution of first-order ODEs. Later, some further analysis58–60 and improved version61–63 of this scheme have been presented, which all were proved to be very useful and efficient for numerical integration of various FDEs. In particular, this algorithm has been successfully and reliably applied in detecting chaotic attractors of fractional-order systems.64
with x ¼ ðx1 ; x2 ; x3 ÞT , q ¼ ðq1 ; q2 ; q3 ÞT , and 0 1 0 1 a a 0 0 B ¼ @ c 1 0 A; f ðxÞ ¼ @ 2cx1 x1 x3 A: 0 0 b x1 x2
(26)
(27)
With the coupling signal N eX sðtÞ ¼ xj ¼ N j¼1
! N N N eX eX eX xj1 ; xj2 ; xj3 ; N j¼1 N j¼1 N j¼1
the network (7) reads: Dq1 xi1 ! Dq2 xi2 ¼ Dq3 xi3
a c 0 0
a 1 0
0 0 b
!
xi1 xi2 xi3
!
1 0 ! ! N N N B eX C eX eX B 2c x x xj3 C j1 j1 B N C N j¼1 N j¼1 C: j¼1 þB ! ! B C N N B C X X e e @ A xj1 xj2 N j¼1 N j¼1 (28)
A. Suppressing individual chaos through synchronization
First, consider a network with N ¼ 10 nodes. In this network, we take a three-dimensional fractional-order Lorenz system as the dynamics of nodes. The classical integer-order Lorenz system, found in 1963, was originally derived from a model of the Earth’s atmospheric convection flow heated from below and cooled from above, and it can be applied to describe many interesting nonlinear systems, ranging from thermal convection to laser dynamics.65 It has been shown that the Lorenz system does indeed define a robust chaotic attractor.66 In 2003, the following fractional-order version of the Lorenz system67 was introduced: 8 q < D1 x1 ¼ aðx2 x1 Þ Dq2 x2 ¼ cx1 x2 x1 x3 (25) : q3 D x3 ¼ x1 x2 bx3 ; where qi 2 ð0; 1Þði ¼ 1; 2; 3Þ is the fractional orders. Here, we fix parameter values as: a ¼ 10, b ¼ 8=3, c ¼ 28, so that in the case q1 ¼ q2 ¼ q3 ¼ 1 the system Eq. (25) reduces to
FIG. 2. Chaotic behavior of the fractional-order Lorenz system (25) with ðq1 ; q2 ; q3 Þ ¼ ð0:99; 0:99; 0:99Þ and initial conditions ðx1 ð0Þ; x2 ð0Þ; x3 ð0ÞÞ ¼ ð6; 10; 18Þ. (a) Time evolution of state variables x1 ; x2 , and x3 and (b) view on ðx1 ; x2 ; x3 Þ space.
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FIG. 3. Synchronized dynamics of the FDN (28) consisting of coupled fractional-order Lorenz systems with ðq1 ; q2 ; q3 Þ ¼ ð0:99; 0:99; 0:99Þ and e ¼ 1. (a) The time evolutions of xi1 ðtÞði ¼ 1; 2; …; 10Þ; (b) the time evolutions of xi2 ðtÞði ¼ 1; 2; …; 10Þ; and (c) the time evolutions of xi3 ðtÞði ¼ 1; 2; …; 10Þ.
With the parameters specified above, we can find that the corresponding eigenvalues of B are k1;2 ¼ 5:5616:12i and k3 ¼ 2:67. It can be easily verified that all ki ði ¼ 1; 2; 3Þ lie in the stability region, i.e., jarg½ki ðBÞj > 99p=200. Thus, the FDN composed of coupled fractional-order chaotic Lorenz system will achieve global synchronization behavior according to Theorem 1. For the special case e ¼ 1, simulation results are shown in Fig. 3. Figures 3(a)–3(c) show the time evolution of
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FIG. 5. Synchronized dynamics of the FDN (28) consisting of coupled fractional-order Lorenz systems with ðq1 ; q2 ; q3 Þ ¼ ð0:99; 0:99; 0:99Þ and e ¼ 0:6. (a) The time evolutions of xi1 ðtÞði ¼ 1; 2; …; 10Þ; (b) the time evolutions of xi2 ðtÞði ¼ 1; 2; …; 10Þ; and (c) the time evolutions of xi3 ðtÞði ¼ 1; 2; …; 10Þ.
xi1 ; xi2 , and xi3 ði ¼ 1; 2; …; NÞ, respectively. It is seen from the figure that the network synchronization is attained after a quite short transient period. Moreover, from Fig. 3, we can see that the synchronized state is kept chaotic, exhibiting the same dynamical behavior as each isolate chaotic system (25) with certain initial conditions. From the view point of our theoretical analysis above, the synchronous solution can be tuned via varying the overall coupling strength e. Figure 4 shows the synchronization behavior with e ¼ 2:8, in which the synchronized dynamics still behaves chaotically, but with different chaotic attractor from its uncoupled individual units. Figure 5 displays the global synchronization behavior with e ¼ 0:6, in which the coupled fractional-order Lorenz oscillators stop oscillating and arrive at a stable steady state. So in this case, although all constituent units of the FDN are chaotic, the whole synchronized network displays simple dynamical behavior. B. Generating synchronized chaos from coupled simple units
In this subsection, we consider a FDN consisting of N ¼ 5 identical simple units and take fractional-order Ro¨ssler system as local node dynamics. The single fractional-order Ro¨ssler system is described by 8 q1 < D x1 ¼ x2 x3 (29) Dq2 x ¼ x1 þ ax2 : q3 2 D x3 ¼ cx3 þ b þ x1 x3 ; FIG. 4. Synchronized dynamics of the FDN (28) consisting of coupled fractional-order Lorenz systems with ðq1 ; q2 ; q3 Þ ¼ ð0:99; 0:99; 0:99Þ and e ¼ 2:8. (a) The time evolutions of xi1 ðtÞði ¼ 1; 2; …; 10Þ; (b) the time evolutions of xi2 ðtÞði ¼ 1; 2; …; 10Þ; and (c) the time evolutions of xi3 ðtÞði ¼ 1; 2; …; 10Þ.
where q ¼ ðq1 ; q2 ; q3 ÞT is subject to 0 < q1 ; q2 ; q3 < 1, a, b, and c are system parameters. When q ¼ (1, 1, 1), system (29) is the original Ro¨ssler system, designed by Ro¨ssler system in 1976, which is chaotic for appropriate parameter
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FIG. 6. Damped oscillations in the fractional-order Ro¨ssler system (29) with ðq1 ; q2 ; q3 Þ ¼ ð0:8; 0:9; 0:9Þ and initial conditions ðx1 ð0Þ; x2 ð0Þ; x3 ð0ÞÞ ¼ ð0:4; 0:7; 0Þ. (a) Time evolution of state variables x1 ; x2 , and x3 and (b) view on ðx1 ; x2 ; x3 Þ space.
values.68 Simulations have been performed to obtain chaotic or hyperchaotic behavior of the fractional-order Ro¨ssler equations and the results demonstrated that chaos indeed exists with R < 3.69,70 Here, in order to illustrate the emergence of synchronized chaos from simple fractional-order systems, we choose incommensurate order q ¼ ð0:8; 0:9; 0:9ÞT and parameters a ¼ 0.4, b ¼ 0.2, and c ¼ 10 such that the fractional-order Ro¨ssler system possesses stable equilibria, see Fig. 6. According to our approach, the whole network of coupled fractional-order Ro¨ssler systems is written as
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FIG. 8. Synchronized dynamics of the FDN (30) consisting of coupled fractional-order Ro¨ssler systems with ðq1 ; q2 ; q3 Þ ¼ ð0:8; 0:9; 0:9Þ and e ¼ 1:2. (a) The time evolutions of xi1 ðtÞði ¼ 1; 2; …; 5Þ; (b) the time evolutions of xi2 ðtÞði ¼ 1; 2; …; 5Þ; and (c) the time evolutions of xi3 ðtÞði ¼ 1; 2; …; 5Þ.
1 0 Dq1 xi1 0 1 @ Dq2 xi2 A ¼ @ 1 a Dq3 xi3 0 0 0 0
B B B þB B B @b þ
10 1 xi1 1 0 A@ xi2 A c xi3
1 0 N C eX C xj2 2a C N j¼1 ! !C C; N N C eX eX xj1 xj3 A N j¼1 N j¼1
(30)
where e is the overall coupling strength and let
FIG. 7. Synchronized dynamics of the FDN (30) consisting of coupled fractional-order Ro¨ssler systems with ðq1 ; q2 ; q3 Þ ¼ ð0:8; 0:9; 0:9Þ and e ¼ 1. (a) The time evolutions of xi1 ðtÞði ¼ 1; 2; …; 5Þ; (b) the time evolutions of xi2 ðtÞði ¼ 1; 2; …; 5Þ; and (c) the time evolutions of xi3 ðtÞði ¼ 1; 2; …; 5Þ.
FIG. 9. Synchronized chaos in the 3rd node of the FDN (30) consisting of coupled fractional-order Ro¨ssler systems with ðq1 ; q2 ; q3 Þ ¼ ð0:8; 0:9; 0:9Þ and e ¼ 1:2. (a) Time evolution of state variables x31 ; x32 , and x33 and (b) view on ðx31 ; x32 ; x33 Þ space.
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FIG. 10. Synchronized dynamics of the FDN (30) consisting of coupled fractional-order Ro¨ssler systems with ðq1 ; q2 ; q3 Þ ¼ ð0:8; 0:9; 0:9Þ and e ¼ 0:1. (a) The time evolutions of xi1 ðtÞði ¼ 1; 2; …; 5Þ; (b) the time evolutions of xi2 ðtÞði ¼ 1; 2; …; 5Þ; and (c) the time evolutions of xi3 ðtÞði ¼ 1; 2; …; 5Þ.
0
0 C ¼ @1 0
1 a 0
1 1 0 A: c
In view of incommensurate order q ¼ (0.8, 0.9, 0.9), as in The9 9 orem 1, we have q1 ¼ 45 ; q2 ¼ 10 ; q3 ¼ 10 , and thus M ¼ 10. Then the characteristic equation detðdiagðk8 ; k9 ; k9 Þ CÞ ¼ 0 can be written as ðk9 þ cÞðk17 þ ak8 þ 1Þ ¼ 0:
(31)
Using simple calculations, we can show that all roots of the above equation lie in the stability region jargðkÞj > p=20. Therefore, according to Theorem 1, the global synchronization behavior is realized in the network of coupled fractional-order stable Ro¨ssler systems. Figure 7 shows the simulation results for the special case e ¼ 1, which indicates that the network synchronization is indeed achieved after a quite short transient period. From this figure, we can see that the synchronized state is in its stable steady state, which shows the dynamical behavior of an isolated system (29) with certain initial conditions. By tuning the global coupling strength e, other interesting synchronous dynamics can be observed, as shown in Figs. 8–10. Figures 8 and 9 depict the synchronized time series with e ¼ 1:2, in which the synchronized dynamics becomes chaotic attractors although its individual units are at their stable steady states. Figure 10 displays the global synchronization behavior with e ¼ 0:1, in which the coupled fractional-order Ro¨ssler oscillators are still at a stable equilibria, but different from the isolated nodes. V. CONCLUSIONS
Even though synchronization dynamics on IDNs has been extensively studied, investigation on synchronization of
FDNs is still at the initial stage. To the best of our knowledge, there have been only a few papers in the literature that focus on complete synchronization of FDNs, whose coupling schemes are all diffusive type. In contrast, our present work proposes a general fractional-order network model, which can exhibit interesting collective behavior. Our model differs from previous ones and is unique in several aspects: First, instead of introducing extra coupling function on constituent units, the present model directly employs the nonlinear term of each node as the coupling functions (see Eq. (5)). Second, the interaction among coupled fractional units is nonlinear despite of the linear form of integrated function h (see Eq. (7)). Third, the model is very general in the sense that nearly all fractional-order dynamical systems can be chosen as the node dynamics to form such a FDN. It is claimed that the coupling strength e of Eq. (7) does not destroy the network synchronization behavior, but will tune the synchronous sate to new collective behavior. In particular, we have shown that the chaos dynamics in individual units is suppressed in networks of chaotic systems through synchronization, and conversely, synchronous chaotic behavior can emerge in networks of simple units. Moreover, based on the stability theory of incommensurate-order FDEs, we have established a global stability criterion for the network synchronization. The stability criterion is easily checked by calculating the eigenvalue distribution of the decomposed constant matrix A without the requirement of calculating the conditional Lyapunov exponents. In the end of the paper, some simulation examples in networks with fractional-order Lorenz system or fractional-order Ro¨ssler system as node dynamics are exploited to demonstrate the applicability of the theoretical results. In the real instances of synchronized behavior, synchronization is usually accompanied by a qualitative change in the global dynamics of the network. For example, in an insect society, complex functional collective dynamics can emerge from the behaviors of simple agents interacting locally with each other and=or their environment.71 During an epileptic episode, contrasting strongly with the seemingly chaotic patterns of normal brain function, the coherent population activity in many regions of the cerebral cortex is almost periodic.72 On the other hand, in recent years the fractional calculus has grown to be a fruitful field of research in science and engineering, and many scientific areas are currently paying attention to the fractional calculus concepts, such as signal processing, viscoelasticity and damping, electromagnetism, fractals, heat transfer, biology, electronics, system identification, traffic systems, genetic algorithms, percolation, to name just a few.36–39 At this stage, it is very relevant to consider networks of coupled fractional-order dynamical systems able to exhibit coherent behavior, but such that the synchronized motion differs from that of the individuals. Therefore, our work presents a first step towards the understanding of how complex collective behavior emerges in networks of interacting dynamical systems described by FDEs. In future studies, we intend to complement our model with nonlinear integrated functions or communication time-delays to predict more interesting collective behavior. It is also worth noting that the consensus algorithms of
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multi-agent systems have been recently extended to the networked fractional-order linear systems because of the fact that many practical vehicles often demonstrate fractionalorder dynamics.73–75 Hence, another possible extension of our work is to introduce distributed consensus protocols between networked fractional-order linear=nonliear units to study the relationship between the interaction graph and the fractional orders in ensuring consensus of FDNs. ACKNOWLEDGMENTS
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions in helping us improve this paper. This research was supported by the National Natural Science Foundation of China (Grant No. 61104138), Guangdong Natural Science Foundation (Grant No. S2011040001704), and the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (Grant No. LYM10074). 1
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