A New Fractional-Order Chaotic System and Its ... - Semantic Scholar
Circuits Syst Signal Process (2012) 31:1599–1613 DOI 10.1007/s00034-012-9408-z
A New Fractional-Order Chaotic System and Its Synchronization with Circuit Simulation Diyi Chen · Chengfu Liu · Cong Wu · Yongjian Liu · Xiaoyi Ma · Yujing You
Received: 16 August 2011 / Revised: 19 March 2012 / Published online: 15 May 2012 © Springer Science+Business Media, LLC 2012
Abstract A new fractional-order chaotic system is proposed in this paper, and a list of state trajectories is presented with fractional derivative of different areas. Furthermore, a circuit diagram is studied to realize the fractional-order chaotic system. The new fractional-order chaotic system can be controlled to reach synchronization based on the nonlinear control theory, and the results between numerical emulation and circuit simulation are in agreement with each other. Keywords Fractional-order · Chaos · Synchronization · Circuit simulation 1 Introduction Fractional calculus is a mathematical topic with more than 300 year old history, but its application to physics and engineering has attracted lots of attention only in the recent D. Chen · C. Liu · C. Wu · X. Ma () · Y. You Department of Electrical Engineering, Northwest A&F University, Shaanxi Yangling 712100, P.R. China e-mail: [email protected] D. Chen e-mail: [email protected] C. Liu e-mail: [email protected] C. Wu e-mail: [email protected] Y. You e-mail: [email protected] Y. Liu School of Mathematics and Information Science, Yulin Normal University, Guangxi Yulin 537000, P.R. China e-mail: [email protected]
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years. It has been found that many systems can be described by fractional differential equations, for example, in hydrology, astronomy, archaeology, neural networks [14, 17, 22, 26]. It is known that some fractional differential systems behave chaotically, e.g., the fractional-order Chen’s system [2], fractional-order Chua’s system [19, 28], fractional-order Rössler system [8], fractional-order Lü system [11], fractional-order Liu system [23], fractional-order unified system [3]. Recently, studying fractional differential systems has become an active research field. Synchronization of chaotic fractional differential systems starts to attract increasing attention due to its potential applications in secure communication and control processing. Many methods of synchronization have been reported, for instance, as early as 2005 and 2006, the fractional Lü system and the fractional Chua system were controlled to reach synchronization in [6, 18]. In [25], a new method was proposed and applied to “generalized projective synchronization” for a class of fractional-order chaotic system via a transmitted signal. The author presented an adaptive feedback control scheme for the synchronization of two coupled chaotic fractional order system with different fractional orders in [15]. In [24], the synchronized motions in a star network of coupled fractional-order systems in which the major element is coupled to each of the non-interacting individual elements were studied. In [16, 27], linear control method was presented. In [7], the authors presented a direct approach to design a stabilizing controller based on a special matrix structure to synchronize chaotic system and extended the approach to synchronize fractional chaotic systems. The Qi oscillator with fractional order was controlled to synchronization by applying three different kinds of methods in [20]. A circuit should be and must be an important manifestation of chaotic systems. A few papers have been published, which have made great contribution to this field of research [4, 9]. For example, in [10] a novel circuit diagram is designed for hardware implementation of a fractional-order hyperchaotic Liu system. In [13], stability analysis of the fractional-order modified autonomous van der Pol–Duffing circuit is studied using the fractional Routh–Hurwitz criteria. In [23], the authors deal with the memristor-based Chua’s circuit. In [21], a particularly elegant circuit is presented whose operation is accurately described by a simple variant of that equation in which the required nonlinearity is provided by a single diode and for which the analysis is particularly straightforward. In [5], circuit simulation of a new double-wing chaos and its synchronization are presented. But there are few papers which have synchronized circuit of fractional-order chaos system, although it is very important in theory and practice. In this paper, a new fractional-order chaos and its experimental circuit simulation are presented. Our aim is to synchronize two chaotic fractional-order systems. To achieve this goal, we have used nonlinear control methods, and the results between numerical emulation and circuit experimental simulation are in agreement with each other. This paper is organized as follows. In Sect. 2, the system model is presented. This section also includes the basic definition of the fractional-order chaos and the analysis of the new fractional-order chaos. Section 3 comprises two main parts: the basic circuit unit and the circuit simulation are presented. The results of numerical
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simulation and circuit simulation are given in Sect. 4 to illustrate the effectiveness of the proposed controller for system synchronization. Conclusions and discussions in Sect. 5 complete the paper.
2 System Model 2.1 System Description The new fractional-order chaotic system is described as follows: ⎧ q d x ⎪ ⎪ = a(y − x) + byz + w ⎪ ⎪ dt q ⎪ ⎪ ⎪ ⎪ dq y ⎪ ⎪ ⎨ q = −cx − dxz2 + gy dt (1) qz ⎪ d ⎪ 2 ⎪ = y − kz ⎪ ⎪ dt q ⎪ ⎪ ⎪ dq w ⎪ ⎪ ⎩ = by − w dt q where x, y, z, w are state variables and q is fractional-order which satisfies 0 < q < 1; also, a, b, c, d, g, and k are known non-negative constants. One sets q = 0.80. Thus, the simulation is done with the initial value [x, y, z, w]T = [0.1, 0.1, 2.1, 0.1]T . When (a, b, c, d, g, k) = (35, 2.5, 7, 4, 28, 1/3), system (1) demonstrates a chaotic attractor. State trajectories of the four-dimensional fractionalorder system are shown in Fig. 1. State trajectories of system (1) for a different q are shown in Fig. 2. 2.2 Stability Theory and Local Stability Some stability theorems on fractional-order systems and their related results are introduced. Theorem 1 Consider the following linear system [12]: dq x = Ax, dt q
x(0) = x0
(2)
where, 0 < q < 1, x ∈ R n , and A ∈ R n×n . The system is asymptotically stable if and only if |arg(λ)| > qπ/2 is satisfied for all eigenvalues of matrix A. Also, this system is stable if and only if |arg(λ)| ≥ qπ/2 is satisfied for all eigenvalues of matrix A, and those critical eigenvalues that satisfy |arg(λ)| = qπ/2 have geometric multiplicity one. The geometric multiplicity of an eigenvalue λ of the matrix A is the dimension of the subspace of vectors v for which Av = λv is satisfied. Theorem 2 Consider the following commensurate fractional-order system [1]: dq x = f (x) dt q
(3)
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Fig. 1 State trajectories of system (6) with q = 0.8
where, 0 < q < 1, x ∈ R n . The equilibrium points of system (3) are calculated by solving the following equation: f (x) = 0. These points are locally asymptotically stable if all eigenvalues λi of the Jacobian matrix J = ∂f/∂x evaluated at the equilibrium points satisfy |arg(λ)| > qπ/2. We can use Theorem 1 to analyze the stability of system (1) at its equilibrium points. Let λi the be eigenvalues of the Jacobian matrix at a saddle point in the set-
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Fig. 2 State trajectories of the system with different q
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ting of Theorem 2. Therefore, the instability region of these saddle points can be determined by the following condition: q>
|Im(λ)| 2 arctan . π Re(λ)
(4)
We can use Theorem 2 for the fractional-order system (1) which has seven equilibrium points and their corresponding eigenvalues are as given below: Equilibrium points
Their corresponding eigenvalues
E0 (0, 0, 0, 0)
λ1 = −30.847, λ2 = 23.823, λ3 = −0.97638, λ4 = −0.33333
E1 (−2.0430, −0.9925, 0.81719, 2.0034)
λ1 = −29.546, λ2 = 21.738, λ3 = 0.4585, λ4 = −0.98297
E2 (2.0430, 0.9925, 0.81719, 2.0034)
λ1 = −28.786, λ2 = 20.924, λ3 = 0.51142, λ4 = −0.98249
E3 (−2.2653i, −0.81142i, −0.90611i, −2.4631)
λ1 = −32.849 − 0.14436i, λ2 = 26.323 + 0.13977i, λ3 = −0.85676 + 0.0077466i, λ4 = −0.95049 − 0.0031571i
E4 (2.2653i, 0.81142i, 0.90611i, −2.4631)
λ1 = −32.849 + 0.14436i, λ2 = 26.323 − 0.13977i, λ3 = −0.85676 − 0.0077466i, λ4 = −0.95049 + 0.0031571i
E5 (−5.5038i, −0.72293e − 1i, −2.2015i, −14.540)
λ1 = −41.265 + 0.90343i, λ2 = 34.349 − 0.89941i, λ3 = −1.0238 − 0.0012513i, λ4 = −0.39305 − 0.0027599i
E6 (5.5038i, 0.72293e − 1i, 2.2015i, −14.540)
λ1 = −41.265 − 0.90343i, λ2 = 34.349 + 0.89941i, λ3 = −1.0238 + 0.0012513i, λ4 = −0.39305 + 0.0027599i
Therefore, the necessary condition for the appearance of chaos in that fractionalorder system is q>
|Im(λ)| 2 0.13977 2 arctan > arctan . π Re(λ) π 26.323
The aforementioned condition is a necessary but not a sufficient condition, it does not warrant chaos itself. Obviously, the theory results based on Theorem 2 are consistent with Fig. 2.
3 Circuit Diagram 3.1 System Numerical Simulation We derived the circuit diagram of integer-order chaotic system in detail, and the mathematical model of integer-order chaotic circuit system is as following:
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Fig. 3 Circuit diagram for a fractional-order cell
R5 R3 dx R3 R3 R3 =− x+ y+ yz + w, dt R4 R1 R2 C 1 R6 R2 C 1 R16 R2 C1 R25 R2 C1 R5 R9 dy R9 R5 R9 =− x− xz2 + y, dt R4 R7 R8 C 2 R4 R13 R8 C2 R10 R8 C2
(5)
R17 dz R17 R20 = y2 − z, dt R14 R15 C3 R19 R14 R15 C3 dw R39 R42 R39 y− w. = dt R37 R38 C4 R41 R38 R40 C4 The circuit parameters are: R1 = R6 = 2 K, R10 = 2.5 K,
R2 = R8 = R15 = 100 K, R13 = 17.5 K,
R4 = R5 = R7 = R14 = R19 = R20 = 10 K, R17 = 1 K, R38 = 100 K,
R3 = R9 = 7 K,
R29 = 70 K, R39 = 1 K,
R16 = 28 K,
R37 = 4 K, R40 = 10 K,
R41 = R42 = 10 K,
C1 = C2 = C3 = C4 = 1 µF. We can get the fractional order chaotic circuit just by changing the capacitors C1 , C2 , C3 , and C4 into the circuit units [10], which are shown in Fig. 3. 3.2 Circuit Diagram and Simulation Observations A circuit diagram to realize the chaotic attractor of the new fractional-order system is shown in Fig. 4. Circuit simulation observations of the chaotic attractor in different planes through Multisim software are presented in Fig. 5. We compare Fig. 1(c) with Fig. 5(a), Fig. 1(d) with Fig. 5(b), Fig. 1(e) with Fig. 5(c), Fig. 1(f) with Fig. 5(d), respectively. And it is clear that the results of circuit simulation are basically in agreement with the results of numerical simulation. It indicates that the fractional-order system can be easily achieved by the circuit. With EWB and Multisim software, the simulation results should be basically in agree-
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Fig. 4 Circuit diagram to realize the chaotic attractor of the new fractional-order system
ment with the results of the actual circuit experiment. Therefore, the simulation results are valid.
4 Design of Synchronization Controller 4.1 Nonlinear Feedback Method Through a nonlinear feedback system, we can transform the nonlinear synchronization errors of a nonlinear fractional-order chaotic system into linear errors, which will change the complex synchronization problem of a nonlinear chaotic system into a synchronization problem of a linear system. Therefore, we can achieve synchronization of the nonlinear fractional-order system easily. This method will be used in this section.
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Fig. 5 Circuit simulation observations of the chaotic attractor in different planes
The drive system is described by: ⎛ dq x ⎞ m ⎛ ⎞ ⎞ ⎛ ⎞⎛ q 2.5ym zm −35 35 0 1 xm ⎜ ddt ⎟ qy ⎜ qm ⎟ ⎜ 2 ⎟ ⎟ ⎜ ⎜ 0 0 ⎟ ⎜ dt ⎟ ⎜ −7 28 ⎟ ⎜ ym ⎟ + ⎜ −4xm zm ⎟ . ⎜ dq z ⎟ = ⎝ 2 0 0 −1/3 0 ⎠ ⎝ zm ⎠ ⎝ ym ⎠ ⎜ qm ⎟ ⎝ dt ⎠ wm 0 2.5 0 −1 0 q d w
(6)
m
dt q
The response system is defined by: ⎛ dq x ⎞ s ⎛ ⎞ ⎞⎛ ⎞ ⎛ q 2.5ys zs −35 35 0 1 xs ⎜ ddtq y ⎟ ⎜ qs ⎟ ⎜ 2⎟ ⎜ ⎟ ⎜ 0 0 ⎟ ⎜ dt ⎟ ⎜ −7 28 ⎟ ⎜ ys ⎟ + ⎜ −4xs zs ⎟ + u, ⎜ dq z ⎟ = ⎝ 2 ⎠ ⎝ ⎠ ⎝ 0 0 −1/3 0 zs ys ⎠ ⎜ qs ⎟ ⎝ dt ⎠ ws 0 2.5 0 −1 0 dq w dt q
s
(7)
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Fig. 6 Synchronization errors of system (11) and systems (6) and (7) with the initial values xm (0) = 2, ym (0) = 0.8, zm (0) = 3.8, wm (0) = 0.7; xs (0) = 0.1, ys (0) = 0.15, zs (0) = 5, and ws (0) = −1.1
where the subscripts m and s stand for the elements belonging to the drive and response oscillators, respectively, u is a nonlinear controller to be designed, and q = 0.8. The error system is defined as follows: ⎧ e1 = xs − xm , ⎪ ⎪ ⎪ ⎨e = y − y , 2 s m ⎪ e = z − z 3 s m, ⎪ ⎪ ⎩ e4 = ws − wm .
(8)
The controller was designed as follows: ⎞ ⎛ 2.5ym zm ⎞ ⎛ 2.5ys zs ⎞ u1 ⎜ 2 ⎟ 2⎟ ⎜ u2 ⎟ ⎜ −4xm zm ⎟ ⎜ −4xs zs ⎟ ⎟ ⎜ u=⎜ ⎟−⎜ ⎟ − Be ⎝ u3 ⎠ = ⎜ 2 2 ⎝ ym ⎠ ⎝ ys ⎠ u4 0 0 ⎛
(9)
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Fig. 7 Circuit diagram of synchronization of system (11) and systems (6) and (7)
where B ∈ R 4×4 , e = (e1 , e2 , e3 , e4 )T . Also ⎛ dq e ⎞ 1 ⎛ ⎞ q −35 35 0 1 ⎜ ddtq e ⎟ ⎜ q2 ⎟ ⎜ 0 0 ⎟ ⎜ dt ⎟ ⎜ −7 28 ⎟ e − Be = (A − B)e. ⎜ dq e ⎟ = ⎝ 0 0 −1/3 0 ⎠ ⎜ q3 ⎟ ⎝ dt ⎠ 0 2.5 0 −1 dq e 4
dt q
One sets
⎛
⎞ 0 −35 0 0 ⎜ 0 29 0 0 ⎟ ⎟. B =⎜ ⎝0 0 1 0⎠ 1 0 0 1
(10)
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Fig. 7 (Continued)
Thus, (A − B) has the all the eigenvalues λi (i = 1, 2, . . . , n) satisfying the condition | arg(λi )| > qπ/2. According to the principle of stability for a fractional-order linear system, the zero point of the error system is asymptotically stable. Using Theorem 1, we know that the synchronization errors ei (t) (i = 1, 2, 3, 4) converge to zero, which means that the chaotic synchronization of the systems (6) and (7) is achieved. The simulation results with the initial values xm (0) = 2, ym (0) = 0.8, zm (0) = 3.8, wm (0) = 0.7, xs (0) = 0.1, ys (0) = 0.15, zs (0) = 5, and ws (0) = −1.1 are shown in Fig. 6. 4.2 Circuit Diagram for Synchronization of Fractional-Order Systems The circuit diagram of these two chaotic systems was designed in detail. The mathematical model of the systems is given as follows: d q xs R48 R51 R48 R48 R48 =− xs + ym + ym zm + ws q dt R50 R52 R36 C0 R30 R36 C0 R49 R36 C0 R139 R36 C0
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Fig. 8 Circuit simulation observations of synchronization of system (11) and systems (6) and (7)
d q ys R55 R51 R55 2 =− xs − xm zm q dt R50 R53 R54 C0 R57 R54 C0 R55 R55 R96 R94 + ym − ys R95 R56 R54 C0 R93 R115 R54 C0 R60 R60 R63 d q zs 2 = ym − zs q dt R58 R59 C0 R62 R61 R59 C0 d q ww R81 R81 R84 = ys − ws . dt q R79 R80 C0 R83 R82 R80 C0
(11)
The parameters of system (11) are: R36 = R54 = R59 = R80 = 100 K, R49 = 28 K,
R30 = R48 = 2 K,
R57 = 17.5 K,
R50 = R51 = R53 = R58 = R62 = R63 = R82 = R83 = R84 = R93 = R94 = 10 K, R48 = R55 = 7 K,
R61 = 30 K,
R79 = 4 K,
R56 = R115 = R139 = 70 K, R60 = R81 = R95 = 1 K,
R96 = 29 K,
C0 = 1 uF,
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where C0 is the equivalent capacitance for the fractional-order cell circuit. Circuit diagrams for synchronization of system (6) and system (7) are shown in Fig. 7. And circuit simulation observations of synchronization of system (6) and system (7) are shown in Fig. 8, which are smooth lines with a 45◦ angle. And that means they achieve precise synchronization. As shown in the simulation results, the circuit simulation provided by Multisim software is in full agreement with the numerical simulation using MATLAB, which proves that system (6) has been synchronized with system (7) and indicates that synchronization can be achieved by the circuit diagram successfully.
5 Conclusions and Discussion In this paper, a new 4D fractional-order chaos was proposed and its local stability was studied. In addition, the circuit experimental simulation was presented. Furthermore, the new fractional-order chaotic system could be controlled to reach synchronization based on the nonlinear control theory, and the results between numerical emulation and circuit experimental simulation were in agreement with each other. Better synchronization methods of a fractional-order system and its circuit diagram need to be studied. This knowledge should be applied in engineering, e.g., in communications and so on. Acknowledgements The authors would like to thank the reviewers for their useful comments and suggestions on our manuscript. This work was supported by the scientific research foundation of National Natural Science Foundation (No. 51109180), (No. 11161051), Personnel Special Fund of North West A&F University (RCZX-2009-01), the Natural Science Foundation of Guangxi Province (Grant No. 2012GXNSFAA053014).
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A New Fractional-Order Chaotic System and Its Synchronization with Circuit Simulation Diyi Chen · Chengfu Liu · Cong Wu · Yongjian Liu · Xiaoyi Ma · Yujing You
Received: 16 August 2011 / Revised: 19 March 2012 / Published online: 15 May 2012 © Springer Science+Business Media, LLC 2012
Abstract A new fractional-order chaotic system is proposed in this paper, and a list of state trajectories is presented with fractional derivative of different areas. Furthermore, a circuit diagram is studied to realize the fractional-order chaotic system. The new fractional-order chaotic system can be controlled to reach synchronization based on the nonlinear control theory, and the results between numerical emulation and circuit simulation are in agreement with each other. Keywords Fractional-order · Chaos · Synchronization · Circuit simulation 1 Introduction Fractional calculus is a mathematical topic with more than 300 year old history, but its application to physics and engineering has attracted lots of attention only in the recent D. Chen · C. Liu · C. Wu · X. Ma () · Y. You Department of Electrical Engineering, Northwest A&F University, Shaanxi Yangling 712100, P.R. China e-mail: [email protected] D. Chen e-mail: [email protected] C. Liu e-mail: [email protected] C. Wu e-mail: [email protected] Y. You e-mail: [email protected] Y. Liu School of Mathematics and Information Science, Yulin Normal University, Guangxi Yulin 537000, P.R. China e-mail: [email protected]
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years. It has been found that many systems can be described by fractional differential equations, for example, in hydrology, astronomy, archaeology, neural networks [14, 17, 22, 26]. It is known that some fractional differential systems behave chaotically, e.g., the fractional-order Chen’s system [2], fractional-order Chua’s system [19, 28], fractional-order Rössler system [8], fractional-order Lü system [11], fractional-order Liu system [23], fractional-order unified system [3]. Recently, studying fractional differential systems has become an active research field. Synchronization of chaotic fractional differential systems starts to attract increasing attention due to its potential applications in secure communication and control processing. Many methods of synchronization have been reported, for instance, as early as 2005 and 2006, the fractional Lü system and the fractional Chua system were controlled to reach synchronization in [6, 18]. In [25], a new method was proposed and applied to “generalized projective synchronization” for a class of fractional-order chaotic system via a transmitted signal. The author presented an adaptive feedback control scheme for the synchronization of two coupled chaotic fractional order system with different fractional orders in [15]. In [24], the synchronized motions in a star network of coupled fractional-order systems in which the major element is coupled to each of the non-interacting individual elements were studied. In [16, 27], linear control method was presented. In [7], the authors presented a direct approach to design a stabilizing controller based on a special matrix structure to synchronize chaotic system and extended the approach to synchronize fractional chaotic systems. The Qi oscillator with fractional order was controlled to synchronization by applying three different kinds of methods in [20]. A circuit should be and must be an important manifestation of chaotic systems. A few papers have been published, which have made great contribution to this field of research [4, 9]. For example, in [10] a novel circuit diagram is designed for hardware implementation of a fractional-order hyperchaotic Liu system. In [13], stability analysis of the fractional-order modified autonomous van der Pol–Duffing circuit is studied using the fractional Routh–Hurwitz criteria. In [23], the authors deal with the memristor-based Chua’s circuit. In [21], a particularly elegant circuit is presented whose operation is accurately described by a simple variant of that equation in which the required nonlinearity is provided by a single diode and for which the analysis is particularly straightforward. In [5], circuit simulation of a new double-wing chaos and its synchronization are presented. But there are few papers which have synchronized circuit of fractional-order chaos system, although it is very important in theory and practice. In this paper, a new fractional-order chaos and its experimental circuit simulation are presented. Our aim is to synchronize two chaotic fractional-order systems. To achieve this goal, we have used nonlinear control methods, and the results between numerical emulation and circuit experimental simulation are in agreement with each other. This paper is organized as follows. In Sect. 2, the system model is presented. This section also includes the basic definition of the fractional-order chaos and the analysis of the new fractional-order chaos. Section 3 comprises two main parts: the basic circuit unit and the circuit simulation are presented. The results of numerical
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simulation and circuit simulation are given in Sect. 4 to illustrate the effectiveness of the proposed controller for system synchronization. Conclusions and discussions in Sect. 5 complete the paper.
2 System Model 2.1 System Description The new fractional-order chaotic system is described as follows: ⎧ q d x ⎪ ⎪ = a(y − x) + byz + w ⎪ ⎪ dt q ⎪ ⎪ ⎪ ⎪ dq y ⎪ ⎪ ⎨ q = −cx − dxz2 + gy dt (1) qz ⎪ d ⎪ 2 ⎪ = y − kz ⎪ ⎪ dt q ⎪ ⎪ ⎪ dq w ⎪ ⎪ ⎩ = by − w dt q where x, y, z, w are state variables and q is fractional-order which satisfies 0 < q < 1; also, a, b, c, d, g, and k are known non-negative constants. One sets q = 0.80. Thus, the simulation is done with the initial value [x, y, z, w]T = [0.1, 0.1, 2.1, 0.1]T . When (a, b, c, d, g, k) = (35, 2.5, 7, 4, 28, 1/3), system (1) demonstrates a chaotic attractor. State trajectories of the four-dimensional fractionalorder system are shown in Fig. 1. State trajectories of system (1) for a different q are shown in Fig. 2. 2.2 Stability Theory and Local Stability Some stability theorems on fractional-order systems and their related results are introduced. Theorem 1 Consider the following linear system [12]: dq x = Ax, dt q
x(0) = x0
(2)
where, 0 < q < 1, x ∈ R n , and A ∈ R n×n . The system is asymptotically stable if and only if |arg(λ)| > qπ/2 is satisfied for all eigenvalues of matrix A. Also, this system is stable if and only if |arg(λ)| ≥ qπ/2 is satisfied for all eigenvalues of matrix A, and those critical eigenvalues that satisfy |arg(λ)| = qπ/2 have geometric multiplicity one. The geometric multiplicity of an eigenvalue λ of the matrix A is the dimension of the subspace of vectors v for which Av = λv is satisfied. Theorem 2 Consider the following commensurate fractional-order system [1]: dq x = f (x) dt q
(3)
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Fig. 1 State trajectories of system (6) with q = 0.8
where, 0 < q < 1, x ∈ R n . The equilibrium points of system (3) are calculated by solving the following equation: f (x) = 0. These points are locally asymptotically stable if all eigenvalues λi of the Jacobian matrix J = ∂f/∂x evaluated at the equilibrium points satisfy |arg(λ)| > qπ/2. We can use Theorem 1 to analyze the stability of system (1) at its equilibrium points. Let λi the be eigenvalues of the Jacobian matrix at a saddle point in the set-
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Fig. 2 State trajectories of the system with different q
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ting of Theorem 2. Therefore, the instability region of these saddle points can be determined by the following condition: q>
|Im(λ)| 2 arctan . π Re(λ)
(4)
We can use Theorem 2 for the fractional-order system (1) which has seven equilibrium points and their corresponding eigenvalues are as given below: Equilibrium points
Their corresponding eigenvalues
E0 (0, 0, 0, 0)
λ1 = −30.847, λ2 = 23.823, λ3 = −0.97638, λ4 = −0.33333
E1 (−2.0430, −0.9925, 0.81719, 2.0034)
λ1 = −29.546, λ2 = 21.738, λ3 = 0.4585, λ4 = −0.98297
E2 (2.0430, 0.9925, 0.81719, 2.0034)
λ1 = −28.786, λ2 = 20.924, λ3 = 0.51142, λ4 = −0.98249
E3 (−2.2653i, −0.81142i, −0.90611i, −2.4631)
λ1 = −32.849 − 0.14436i, λ2 = 26.323 + 0.13977i, λ3 = −0.85676 + 0.0077466i, λ4 = −0.95049 − 0.0031571i
E4 (2.2653i, 0.81142i, 0.90611i, −2.4631)
λ1 = −32.849 + 0.14436i, λ2 = 26.323 − 0.13977i, λ3 = −0.85676 − 0.0077466i, λ4 = −0.95049 + 0.0031571i
E5 (−5.5038i, −0.72293e − 1i, −2.2015i, −14.540)
λ1 = −41.265 + 0.90343i, λ2 = 34.349 − 0.89941i, λ3 = −1.0238 − 0.0012513i, λ4 = −0.39305 − 0.0027599i
E6 (5.5038i, 0.72293e − 1i, 2.2015i, −14.540)
λ1 = −41.265 − 0.90343i, λ2 = 34.349 + 0.89941i, λ3 = −1.0238 + 0.0012513i, λ4 = −0.39305 + 0.0027599i
Therefore, the necessary condition for the appearance of chaos in that fractionalorder system is q>
|Im(λ)| 2 0.13977 2 arctan > arctan . π Re(λ) π 26.323
The aforementioned condition is a necessary but not a sufficient condition, it does not warrant chaos itself. Obviously, the theory results based on Theorem 2 are consistent with Fig. 2.
3 Circuit Diagram 3.1 System Numerical Simulation We derived the circuit diagram of integer-order chaotic system in detail, and the mathematical model of integer-order chaotic circuit system is as following:
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Fig. 3 Circuit diagram for a fractional-order cell
R5 R3 dx R3 R3 R3 =− x+ y+ yz + w, dt R4 R1 R2 C 1 R6 R2 C 1 R16 R2 C1 R25 R2 C1 R5 R9 dy R9 R5 R9 =− x− xz2 + y, dt R4 R7 R8 C 2 R4 R13 R8 C2 R10 R8 C2
(5)
R17 dz R17 R20 = y2 − z, dt R14 R15 C3 R19 R14 R15 C3 dw R39 R42 R39 y− w. = dt R37 R38 C4 R41 R38 R40 C4 The circuit parameters are: R1 = R6 = 2 K, R10 = 2.5 K,
R2 = R8 = R15 = 100 K, R13 = 17.5 K,
R4 = R5 = R7 = R14 = R19 = R20 = 10 K, R17 = 1 K, R38 = 100 K,
R3 = R9 = 7 K,
R29 = 70 K, R39 = 1 K,
R16 = 28 K,
R37 = 4 K, R40 = 10 K,
R41 = R42 = 10 K,
C1 = C2 = C3 = C4 = 1 µF. We can get the fractional order chaotic circuit just by changing the capacitors C1 , C2 , C3 , and C4 into the circuit units [10], which are shown in Fig. 3. 3.2 Circuit Diagram and Simulation Observations A circuit diagram to realize the chaotic attractor of the new fractional-order system is shown in Fig. 4. Circuit simulation observations of the chaotic attractor in different planes through Multisim software are presented in Fig. 5. We compare Fig. 1(c) with Fig. 5(a), Fig. 1(d) with Fig. 5(b), Fig. 1(e) with Fig. 5(c), Fig. 1(f) with Fig. 5(d), respectively. And it is clear that the results of circuit simulation are basically in agreement with the results of numerical simulation. It indicates that the fractional-order system can be easily achieved by the circuit. With EWB and Multisim software, the simulation results should be basically in agree-
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Fig. 4 Circuit diagram to realize the chaotic attractor of the new fractional-order system
ment with the results of the actual circuit experiment. Therefore, the simulation results are valid.
4 Design of Synchronization Controller 4.1 Nonlinear Feedback Method Through a nonlinear feedback system, we can transform the nonlinear synchronization errors of a nonlinear fractional-order chaotic system into linear errors, which will change the complex synchronization problem of a nonlinear chaotic system into a synchronization problem of a linear system. Therefore, we can achieve synchronization of the nonlinear fractional-order system easily. This method will be used in this section.
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Fig. 5 Circuit simulation observations of the chaotic attractor in different planes
The drive system is described by: ⎛ dq x ⎞ m ⎛ ⎞ ⎞ ⎛ ⎞⎛ q 2.5ym zm −35 35 0 1 xm ⎜ ddt ⎟ qy ⎜ qm ⎟ ⎜ 2 ⎟ ⎟ ⎜ ⎜ 0 0 ⎟ ⎜ dt ⎟ ⎜ −7 28 ⎟ ⎜ ym ⎟ + ⎜ −4xm zm ⎟ . ⎜ dq z ⎟ = ⎝ 2 0 0 −1/3 0 ⎠ ⎝ zm ⎠ ⎝ ym ⎠ ⎜ qm ⎟ ⎝ dt ⎠ wm 0 2.5 0 −1 0 q d w
(6)
m
dt q
The response system is defined by: ⎛ dq x ⎞ s ⎛ ⎞ ⎞⎛ ⎞ ⎛ q 2.5ys zs −35 35 0 1 xs ⎜ ddtq y ⎟ ⎜ qs ⎟ ⎜ 2⎟ ⎜ ⎟ ⎜ 0 0 ⎟ ⎜ dt ⎟ ⎜ −7 28 ⎟ ⎜ ys ⎟ + ⎜ −4xs zs ⎟ + u, ⎜ dq z ⎟ = ⎝ 2 ⎠ ⎝ ⎠ ⎝ 0 0 −1/3 0 zs ys ⎠ ⎜ qs ⎟ ⎝ dt ⎠ ws 0 2.5 0 −1 0 dq w dt q
s
(7)
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Fig. 6 Synchronization errors of system (11) and systems (6) and (7) with the initial values xm (0) = 2, ym (0) = 0.8, zm (0) = 3.8, wm (0) = 0.7; xs (0) = 0.1, ys (0) = 0.15, zs (0) = 5, and ws (0) = −1.1
where the subscripts m and s stand for the elements belonging to the drive and response oscillators, respectively, u is a nonlinear controller to be designed, and q = 0.8. The error system is defined as follows: ⎧ e1 = xs − xm , ⎪ ⎪ ⎪ ⎨e = y − y , 2 s m ⎪ e = z − z 3 s m, ⎪ ⎪ ⎩ e4 = ws − wm .
(8)
The controller was designed as follows: ⎞ ⎛ 2.5ym zm ⎞ ⎛ 2.5ys zs ⎞ u1 ⎜ 2 ⎟ 2⎟ ⎜ u2 ⎟ ⎜ −4xm zm ⎟ ⎜ −4xs zs ⎟ ⎟ ⎜ u=⎜ ⎟−⎜ ⎟ − Be ⎝ u3 ⎠ = ⎜ 2 2 ⎝ ym ⎠ ⎝ ys ⎠ u4 0 0 ⎛
(9)
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Fig. 7 Circuit diagram of synchronization of system (11) and systems (6) and (7)
where B ∈ R 4×4 , e = (e1 , e2 , e3 , e4 )T . Also ⎛ dq e ⎞ 1 ⎛ ⎞ q −35 35 0 1 ⎜ ddtq e ⎟ ⎜ q2 ⎟ ⎜ 0 0 ⎟ ⎜ dt ⎟ ⎜ −7 28 ⎟ e − Be = (A − B)e. ⎜ dq e ⎟ = ⎝ 0 0 −1/3 0 ⎠ ⎜ q3 ⎟ ⎝ dt ⎠ 0 2.5 0 −1 dq e 4
dt q
One sets
⎛
⎞ 0 −35 0 0 ⎜ 0 29 0 0 ⎟ ⎟. B =⎜ ⎝0 0 1 0⎠ 1 0 0 1
(10)
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Fig. 7 (Continued)
Thus, (A − B) has the all the eigenvalues λi (i = 1, 2, . . . , n) satisfying the condition | arg(λi )| > qπ/2. According to the principle of stability for a fractional-order linear system, the zero point of the error system is asymptotically stable. Using Theorem 1, we know that the synchronization errors ei (t) (i = 1, 2, 3, 4) converge to zero, which means that the chaotic synchronization of the systems (6) and (7) is achieved. The simulation results with the initial values xm (0) = 2, ym (0) = 0.8, zm (0) = 3.8, wm (0) = 0.7, xs (0) = 0.1, ys (0) = 0.15, zs (0) = 5, and ws (0) = −1.1 are shown in Fig. 6. 4.2 Circuit Diagram for Synchronization of Fractional-Order Systems The circuit diagram of these two chaotic systems was designed in detail. The mathematical model of the systems is given as follows: d q xs R48 R51 R48 R48 R48 =− xs + ym + ym zm + ws q dt R50 R52 R36 C0 R30 R36 C0 R49 R36 C0 R139 R36 C0
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Fig. 8 Circuit simulation observations of synchronization of system (11) and systems (6) and (7)
d q ys R55 R51 R55 2 =− xs − xm zm q dt R50 R53 R54 C0 R57 R54 C0 R55 R55 R96 R94 + ym − ys R95 R56 R54 C0 R93 R115 R54 C0 R60 R60 R63 d q zs 2 = ym − zs q dt R58 R59 C0 R62 R61 R59 C0 d q ww R81 R81 R84 = ys − ws . dt q R79 R80 C0 R83 R82 R80 C0
(11)
The parameters of system (11) are: R36 = R54 = R59 = R80 = 100 K, R49 = 28 K,
R30 = R48 = 2 K,
R57 = 17.5 K,
R50 = R51 = R53 = R58 = R62 = R63 = R82 = R83 = R84 = R93 = R94 = 10 K, R48 = R55 = 7 K,
R61 = 30 K,
R79 = 4 K,
R56 = R115 = R139 = 70 K, R60 = R81 = R95 = 1 K,
R96 = 29 K,
C0 = 1 uF,
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where C0 is the equivalent capacitance for the fractional-order cell circuit. Circuit diagrams for synchronization of system (6) and system (7) are shown in Fig. 7. And circuit simulation observations of synchronization of system (6) and system (7) are shown in Fig. 8, which are smooth lines with a 45◦ angle. And that means they achieve precise synchronization. As shown in the simulation results, the circuit simulation provided by Multisim software is in full agreement with the numerical simulation using MATLAB, which proves that system (6) has been synchronized with system (7) and indicates that synchronization can be achieved by the circuit diagram successfully.
5 Conclusions and Discussion In this paper, a new 4D fractional-order chaos was proposed and its local stability was studied. In addition, the circuit experimental simulation was presented. Furthermore, the new fractional-order chaotic system could be controlled to reach synchronization based on the nonlinear control theory, and the results between numerical emulation and circuit experimental simulation were in agreement with each other. Better synchronization methods of a fractional-order system and its circuit diagram need to be studied. This knowledge should be applied in engineering, e.g., in communications and so on. Acknowledgements The authors would like to thank the reviewers for their useful comments and suggestions on our manuscript. This work was supported by the scientific research foundation of National Natural Science Foundation (No. 51109180), (No. 11161051), Personnel Special Fund of North West A&F University (RCZX-2009-01), the Natural Science Foundation of Guangxi Province (Grant No. 2012GXNSFAA053014).
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