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Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control
Commun Nonlinear Sci Numer Simulat 14 (2009) 3351–3357
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control Xing-Yuan Wang *, Jun-Mei Song School of Electronic & Information Engineering, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history: Received 26 March 2008 Received in revised form 11 January 2009 Accepted 13 January 2009 Available online 21 January 2009 PACS: 05.45.Xt 05.45.Pq 05.45.Jn
a b s t r a c t Based on the stability theory of fractional order systems, this paper analyses the synchronization conditions of the fractional order chaotic systems with activation feedback method. And the synchronization of commensurate order hyperchaotic Lorenz system of the base order 0.98 is implemented based on this method. Numerical simulations show the effectiveness of this method in a class of fractional order chaotic systems. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Fractional order hyperchaos Lorenz system Activation feedback control Chaotic synchronization
1. Introduction The fractional differential calculus dates from 17th century, but until recent 10 years that it was applied to physics and engineering [1,2]. It was found that many systems in interdisciplinary fields could be described by the fractional differential equations, such as viscoelastic systems, dielectric polarization, electrode–electrolyte polarization and electromagnetic waves [3–6]. There are essential differences between ordinary differential equation (ODE) systems and fractional-order differential systems. Most properties and conclusions of ODE systems cannot be simply extended to that of the fractional-order differential systems. Nowadays, fractional-order systems have attracted more and more people’s attention. It is known that many fractional-order differential systems behave chaotically, and synchronization of these chaotic systems have been given much concern and study. For example, Grigorenko et al. proposed the fractional Lorenz system [7]; Hartley et al. proposed the fractional Chua system [8]; Li Chunguang et al. designed the linear feedback control and implemented the self-synchronization of the fractional Chen system [9]; Li Changpin et al. utilized the reliable numerical algorithm to compute the chaotic attractor of the fractional Chen system [10] and adopted combination of active–passive decomposition (APD) and one-way coupling methods, Pecora-Carroll method, bidirectional coupling method to research chaos synchronization of the Duffing, Lorenz and Rössler systems with fractional orders [11]; Tavazoei et al. researched the unreliability of frequency-domain approximation in recognising chaos in fractional-order systems [12]. And Deng et al. achieved the fractional Lü system synchronization with PC method and one-way coupling method [13], etc. On the basis of the above researches, activation feedback method is
* Corresponding author. E-mail addresses: [email protected] (X.-Y. Wang), [email protected] (J.-M. Song). 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.01.010
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adopted and the synchronization conditions of a class of fractional order chaotic systems are analyzed. And the self-synchronization of commensurate order hyperchaotic Lorenz systems of the base order 0.98 is achieved with this method. The authors have approved the effectiveness of the activation feedback method in the ordinary chaotic systems [14]. Numerical simulations show this method is available for the fractional order chaotic system. 2. Fractional differential and fractional-order system’s synchronization theory At present, there are several definitions of the fractional-order differential system. In the following, we present the most common one of them:
Da xðtÞ ¼ J ma xðmÞ ðtÞ ða > 0Þ; where m is the least integer which is not less than a, x(m) is the m-order derivative in usual sense, and Jb (b > 0) is the b-order Reimann–Liouville integral operator which satisfies
J b yðtÞ ¼
1 CðbÞ
Z
t
ðt sÞb1 yðsÞds;
0
here C stands for Gamma function and the operator Da is generally called ‘‘a-order Caputo differential operator” [15]. Given the fractional order chaotic system, i.e., the drive system is q
d X q ¼ AX þ BFðCXÞ þ D: dt
ð1Þ
Here, X 2 Rn1, A 2 Rnn, B 2 Rnm, C 2 R1n and D 2 Rn1 are continuous matrixes, and CX 2 R1; F:R1 ? Rm is nonlinear mapping; AX is the linear part and BF(CX) is the nonlinear part. 0 < q 6 1 and the component of X, i.e., xi, its fractional order qi may be unequal. Let the corresponding response system be q
d X q ¼ AX þ BFðCX Þ þ D þ uðtÞ; dt
ð2Þ
where X* 2 Rn*1, u(t) is the active control function. Define the error of the systems is e(t) = X(t) X*(t). And with Eqs. (1) and (2), the fractional error equation can be described as q
d e q ¼ AeðtÞ þ BEðX; X Þ þ uðtÞ: dt
ð3Þ
E(X, X*) is determined directly from Eqs. (2) and (3). Here, let the control function be
uðtÞ ¼ MðX; X Þ þ NeðtÞ; where
M(X, X*)
is determined by
ð4Þ BE(X, X*),
and N is the control parameter matrix.
Theorem 1. For n-dimensional fractional system, if all the eigenvalues (k1, k2, . . . , kn) of the Jacobian matrix of some equilibrium point satisfy
j argðki Þj > ap=2;
a ¼ maxðq1 ; q2 ; . . . ; qn Þði ¼ 1; 2; . . . ; nÞ;
ð5Þ
then the fractional-order system is asymptotically steady at the equilibrium [16,17]. Denis Matignon researched stability results for fractional differential equations with application to control processing and proposed that stabilities are guaranteed if the roots of some polynomial (the eigenvalues of the matrix of dynamics or the poles of the transfer matrix) lie outside the closed angular sector [16]:
j argðkÞj 6 ap=2: So, if jarg(ki)j > ap/2, a = max(q1, q2, . . . , qn)(i = 1, 2, . . . , n) is satisfied, the eigenvalues of the matrix of dynamics must lie outside the closed angular sector jarg(k)j 6 ap/2, then the stabilities are guaranteed. Theorem 1 can be proved. Fig. 1 illustrates Theorem 1. Obviously, if one equilibrium is stable, the fractional-order system will be steady at one point; only when none equilibrium is stable, the fractional-order system is in chaos. Obviously, zero point is an equilibrium of Eq. (3). Here, given the fractional order always less than 1. If the value of N can satisfy all latent root (k1, k2, . . . , kn) of Jacobi matrix of Eq. (3) at the zero are 1, the condition (5) can be satisfied. So, Eq. (3) is asymptotic stable according to Theorem 1. The error system (3) can be steady finally, i.e., the drive system (1) and the response system (2) can be achieved asymptotic synchronization. 3. The fractional order hyperchaos Lorenz systems description Lorenz system can be described as follows [18]:
X.-Y. Wang, J.-M. Song / Commun Nonlinear Sci Numer Simulat 14 (2009) 3351–3357
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Fig. 1. Stability region of the fractional-order system.
8 > < x_ ¼ aðy xÞ; y_ ¼ cx y xz; > :_ z ¼ xy bz:
ð6Þ
When a = 10, b = 8/3 and c = 28, Lorenz system exhibits a chaotic behavior. The author [19] adds a nonlinear controller w to _ ¼ yz þ rw, and then obtains a new system the first equation of system (6), let w
8 x_ ¼ aðy xÞ þ w; > > > < y_ ¼ cx y xz; > z_ ¼ xy bz; > > : _ ¼ yz þ rw: w
ð7Þ
When a = 10, b = 8/3, c = 28 and r = 1, system (7) exhibits a hyperchaotic behavior [19]. Four-dimensional fractional order Lorenz system can be described as
8 dq1 x ¼ aðy xÞ þ w; > > dtq1 > > dq2 y > < q ¼ cx y xz; dt 2 > dqq3 z ¼ xy bz; > > dt 3 > > : dq4 w ¼ yz þ rw; dtq4
ð8Þ
here, select the parameters as a = 10, b = 8/3, c = 28 and r = 1. As was demonstrated recently by Petras in case of Chua’s system [20], if one derives the equations properly, not all equations are of fractional order. So, q1, q2, q3, q4 are not always equal and not all fractions. But in this paper, the dynamic behaviors of four-dimensional fractional order Lorenz system are not our research purpose, so we just select qi = 0.98, i = 1, 2, 3, 4 as an example to research the synchronization of the fractional order hyperchaotic Lorenz systems. According to the method proposed by Ramasubramanian et al. [21], we can obtain when qi = 0.98 (i = 1, 2, 3, 4), the Lyapunov exponents of system (8): k1 = 0.3362, k2 = 0.1568, k3 = 0 and k4 = 15.1724. It is obvious that system (8) exhibits a hyperchaotic behavior when qi = 0.98 (i = 1, 2, 3, 4). The projections of the attractor are shown in Fig. 2. 4. The activation feedback control method Supposed system (8) is the drive system, according to Eq. (3), the response system can be described as follows:
8 dq1 x ¼ aðy x Þ þ w þ u1 ðtÞ; > > dtq1 > > dq2 y > < q ¼ cx y x z þ u2 ðtÞ; dt 2 > dq3qz ¼ x y bz þ u3 ðtÞ; > > dt 3 > > : dq4 w ¼ y z þ rw þ u4 ðtÞ; dtq4
ð9Þ
ui(t) (i = 1, 2, 3, 4) is the control function. Define the error variables are ex = x x*, ey = y y*, ez = z z* and ew = w w*, and then the error system can be obtained as
8 dq1 ex ¼ aðey ex Þ þ ew u1 ðtÞ; q > > > dtq 1 > > 2 ey d < q ¼ cex ey ðxz x z Þ u2 ðtÞ; dt 2 dq3 ez > > ¼ bez þ xy x y u3 ðtÞ; > dt q3 > > : dq4 ew ¼ rew yz þ y z u4 ðtÞ: dt q4
ð10Þ
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Fig. 2. When q = 0.98, the projections of attractor of system (8).
Let the activation feedback control function be
8 u1 ðtÞ ¼ 1 ðtÞ; > > > < u2 ðtÞ ¼ xz þ x z þ 2 ðtÞ; > > u3 ðtÞ ¼ xy x y þ 3 ðtÞ; > : u4 ðtÞ ¼ y z yz þ 4 ðtÞ;
ð11Þ
and 1(t), 2(t), 3(t) and 4(t) are the control inputs. Select
1
ex
1
B ðtÞ C Be B 2 C B y C ¼ NB B @ 3 ðtÞ A @ ez
C C C; A
0
1 ðtÞ
0
ð12Þ
ew
4 ðtÞ where
0 B B N¼B @
1
1a
a
0
1
c
0
0
0
0 b þ 1
0 C C C: 0 A
0
0
0
ð13Þ
rþ1
According to Eqs. (10)–(13), obviously, the value of N can satisfy all latent root (k1, k2, k3, k4) of Jacobi matrix of Eq. (10) at the zero are 1. Based on Theorem 1 and the stability theory of fractional-order systems, the drive system (8) and the response system (9) can achieve chaotic synchronization. Conclusion. Let the activation feedback control function be Eq. (11), and select the control input as Eq. (12), thus the drive system (8) and the response system (9) can achieve chaotic synchronization. 5. Numerical simulation In the MATLAB simulations, the fractional order qi and the time step are always chosen as qi = 0.98 (i = 1, 2, 3, 4) and s = 0.01 s, respectively. The above activation feedback scheme in Section 4 is used to solve the differential equations of the systems (8) and (9). The initial states of the drive system (8) and the response system (9) are taken as x(0) = 12, y(0) = 22, z(0) = 31, w(0) = 4 and x*(0) = 22, y*(0) = 32, z*(0) = 40, w*(0) = 11, respectively. Figs. 3 and 4 show the numerical simulation results. The time response of the states of the drive and response systems (8) and (9), respectively, is displayed in Fig. 3. In Fig. 3, the solid line is the states of the drive system (8) and the dashed line is
X.-Y. Wang, J.-M. Song / Commun Nonlinear Sci Numer Simulat 14 (2009) 3351–3357
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Fig. 3. The time response of the states of the drive and response systems (8) and (9).
Fig. 4. Errors state of the driving system (8) and the response system (9).
the states of the response system (9). As Fig. 3 shows, the synchronization of the drive system (8) and the response system (9) is achieved after a short transient. Fig. 4 shows the time response of synchronization errors between system (8) and system (9). From Fig. 4, ex(t), ey (t), ez(t) and ew(t) are stable after a short moment with the effect of the control, respectively. When qi (i = 1, 2, 3, 4) selects other values which can make system (8) is hyperchaotic, the above method is effective, too. Wang Yihong et al. proposed the fractional Brusselator with efficient dimension less than 1 have a limit cycle [22]. Does the four-dimensional fractional order Lorenz system have a limit cycle? So we have made a large of simulations and we find
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Fig. 5. Limit cycle of four-dimentsional fractional order Lorenz system.
Fig. 6. The states of the dynamical system (8) and the response system (9).
the four-dimensional fractional order Lorenz system exists a limit cycle when the parameters are a = 10, b = 8/3, c = 18, r = 1.5 and qi = 0.98 (i = 1, 2, 3, 4). The simulation results are displayed in Fig. 5, the first 1000 points are removed. Here, initial value is x(0) = 12, y(0) = 22, z(0) = 31, w(0) = 4. Li Changpin et al. first proposed and studied the synchronization of limit sets for general ODEs [23]. Here, given the drive system (8) and the response system (9) satisfy a = 10, b = 8/3, c = 18, r = 1.5 and qi = 0.98 (i = 1, 2, 3, 4). Then the drive system (8) and the response system (9) have limit cycle. The above activation feedback scheme in Section 4 is used to solve the differential equations of the systems (8) and (9). The initial states of the drive system (8) and the response system (9) are taken as x(0) = 12, y(0) = 22, z(0) = 31, w(0) = 4 and x*(0) = 22, y*(0) = 32, z*(0) = 40, w*(0) = 11, respectively. From Figs. 6 and 7, the activation feedback scheme is effective to get the limit cycle synchronization of fractional systems. In Fig. 6, the solid line is the states of the drive system (8) and the dashed line is the states of the response system (9). As Fig. 6 shows, the synchronization of the drive system (8) and the response system (9) is achieved after a short transient. Fig. 7 shows the time response of synchronization errors between system (8) and system (9). From Fig. 7, ex(t), ey(t), ez(t) and ew(t) are stable after a short moment with the effect of the control, respectively. 6. Conclusions Based on the stability theory of fractional-order systems, this paper analyses the synchronization conditions of a class of fractional order chaotic systems with activation feedback method. The method is always effective in the fractional order chaotic systems theoretically. The commensurate hyperchaotic Lorenz system of the base order 0.98 is taken as an example and
X.-Y. Wang, J.-M. Song / Commun Nonlinear Sci Numer Simulat 14 (2009) 3351–3357
3357
Fig. 7. Error state of the driving system (8) and the response system (9).
the self-synchronization is implemented based on activation feedback method. Numerical simulations show the activation feedback method can be applied in chaos synchronization, hyperchaos synchronization and limit cycle synchronization of fractional systems. Acknowledgements This research is supported by the National Natural Science Foundation of China (No. 60573172), the Superior University doctor subject special scientific research foundation of China (No. 20070141014) and the Natural Science Foundation of Liaoning province (No. 20082165). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Podlubny I. Fractional differential equations. New York: Academic Press; 1999. Hilfer R. Applications of fractional calculus in physics. New Jersey: World Scientific; 2001. Bagley RL, Calico RA. Fractional order state equations for the control of viscoelastically damped structures. J Guidance Control Dyn 1991;14(2):304–11. Koeller RC. Application of fractional calculus to the theory of viscoelasticity. J Appl Mech 1984;51(2):294–8. Koeller RC. Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics. Acta Mech 1986;58(3–4):251–64. Heaviside O. Electromagnetic theory. New York: Chelsea; 1971. Grigorenko I, Grigorenko E. Chaotic dynamics of the fractional Lorenz system. Phys Rev Lett 2003;91(3):034101. Hartley TT, Lorenzo CF, Qammer HK. Chaos in a fractional order Chua’s system. IEEE Trans CAS-I 1995;42(8):485–90. Li CG, Chen GR. Chaos in the fractional order Chen system and its control. Chaos Solitons Fract 2004;22(3):549–54. Li CP, Peng GJ. Chaos in Chen’s system with a fractional order. Chaos Solitons Fract 2004;22(2):443–50. Li CP, Deng WH. Chaos synchronization of fractional-order differential system. Int J Mod Phys B 2006;20(7):791–803. Tavazoei MS, Haeri M. Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems. IET Signal Processing 2007;1(4):171–81. Deng WH, Li CP. Chaos synchronization of the fractional Lü system. Physica A 2005;353:61–72. Wang XY, Song JM. Synchronization of the unified chaotic system. Nonlinear Anal: Theory Meth Appl 2008;69(10):3409–16. Caputo M. Linear models of dissipation whose Q is almost frequency independent. Geophys J Roy Astronom Soc 1967;13:529–39. Matignon D. Stability results for fractional differential equations with application to control processing. Computational engineering system application, vol. 2. Lille (France): IMACS-SMC; 1996. p. 963–8. Mohammad ST, Mohammad H. A note on the stability of fractional order systems. Math Comput Simulat 2007. doi:10.1016/j.matcom.2008.07.003. Lorenz EN. Deterministic nonperodic flow. J Atmos Sci 1963;20:130–41. Wang XY, Wang MJ. A hyperchaos generated from Lorenz system. Physica A 2008;387(14):3751–8. Petras I. A note on the fractional-order Chua’s system. Chaos Solitons Fract 2008;38(1):140–7. Ramasubramanian K, Sriram MS. A comparative study of computation of Lyapunov spectra with different algorithms. Physica D 2000;139(1, 2):72–86. Wang YH, Li CP. Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle? Phys Lett A 2007;363(5&6):414–9. Li CP, Deng WH. Synchronization of limit sets. Mod Phys Lett B 2007;21(9):551–8.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control Xing-Yuan Wang *, Jun-Mei Song School of Electronic & Information Engineering, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history: Received 26 March 2008 Received in revised form 11 January 2009 Accepted 13 January 2009 Available online 21 January 2009 PACS: 05.45.Xt 05.45.Pq 05.45.Jn
a b s t r a c t Based on the stability theory of fractional order systems, this paper analyses the synchronization conditions of the fractional order chaotic systems with activation feedback method. And the synchronization of commensurate order hyperchaotic Lorenz system of the base order 0.98 is implemented based on this method. Numerical simulations show the effectiveness of this method in a class of fractional order chaotic systems. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Fractional order hyperchaos Lorenz system Activation feedback control Chaotic synchronization
1. Introduction The fractional differential calculus dates from 17th century, but until recent 10 years that it was applied to physics and engineering [1,2]. It was found that many systems in interdisciplinary fields could be described by the fractional differential equations, such as viscoelastic systems, dielectric polarization, electrode–electrolyte polarization and electromagnetic waves [3–6]. There are essential differences between ordinary differential equation (ODE) systems and fractional-order differential systems. Most properties and conclusions of ODE systems cannot be simply extended to that of the fractional-order differential systems. Nowadays, fractional-order systems have attracted more and more people’s attention. It is known that many fractional-order differential systems behave chaotically, and synchronization of these chaotic systems have been given much concern and study. For example, Grigorenko et al. proposed the fractional Lorenz system [7]; Hartley et al. proposed the fractional Chua system [8]; Li Chunguang et al. designed the linear feedback control and implemented the self-synchronization of the fractional Chen system [9]; Li Changpin et al. utilized the reliable numerical algorithm to compute the chaotic attractor of the fractional Chen system [10] and adopted combination of active–passive decomposition (APD) and one-way coupling methods, Pecora-Carroll method, bidirectional coupling method to research chaos synchronization of the Duffing, Lorenz and Rössler systems with fractional orders [11]; Tavazoei et al. researched the unreliability of frequency-domain approximation in recognising chaos in fractional-order systems [12]. And Deng et al. achieved the fractional Lü system synchronization with PC method and one-way coupling method [13], etc. On the basis of the above researches, activation feedback method is
* Corresponding author. E-mail addresses: [email protected] (X.-Y. Wang), [email protected] (J.-M. Song). 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.01.010
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X.-Y. Wang, J.-M. Song / Commun Nonlinear Sci Numer Simulat 14 (2009) 3351–3357
adopted and the synchronization conditions of a class of fractional order chaotic systems are analyzed. And the self-synchronization of commensurate order hyperchaotic Lorenz systems of the base order 0.98 is achieved with this method. The authors have approved the effectiveness of the activation feedback method in the ordinary chaotic systems [14]. Numerical simulations show this method is available for the fractional order chaotic system. 2. Fractional differential and fractional-order system’s synchronization theory At present, there are several definitions of the fractional-order differential system. In the following, we present the most common one of them:
Da xðtÞ ¼ J ma xðmÞ ðtÞ ða > 0Þ; where m is the least integer which is not less than a, x(m) is the m-order derivative in usual sense, and Jb (b > 0) is the b-order Reimann–Liouville integral operator which satisfies
J b yðtÞ ¼
1 CðbÞ
Z
t
ðt sÞb1 yðsÞds;
0
here C stands for Gamma function and the operator Da is generally called ‘‘a-order Caputo differential operator” [15]. Given the fractional order chaotic system, i.e., the drive system is q
d X q ¼ AX þ BFðCXÞ þ D: dt
ð1Þ
Here, X 2 Rn1, A 2 Rnn, B 2 Rnm, C 2 R1n and D 2 Rn1 are continuous matrixes, and CX 2 R1; F:R1 ? Rm is nonlinear mapping; AX is the linear part and BF(CX) is the nonlinear part. 0 < q 6 1 and the component of X, i.e., xi, its fractional order qi may be unequal. Let the corresponding response system be q
d X q ¼ AX þ BFðCX Þ þ D þ uðtÞ; dt
ð2Þ
where X* 2 Rn*1, u(t) is the active control function. Define the error of the systems is e(t) = X(t) X*(t). And with Eqs. (1) and (2), the fractional error equation can be described as q
d e q ¼ AeðtÞ þ BEðX; X Þ þ uðtÞ: dt
ð3Þ
E(X, X*) is determined directly from Eqs. (2) and (3). Here, let the control function be
uðtÞ ¼ MðX; X Þ þ NeðtÞ; where
M(X, X*)
is determined by
ð4Þ BE(X, X*),
and N is the control parameter matrix.
Theorem 1. For n-dimensional fractional system, if all the eigenvalues (k1, k2, . . . , kn) of the Jacobian matrix of some equilibrium point satisfy
j argðki Þj > ap=2;
a ¼ maxðq1 ; q2 ; . . . ; qn Þði ¼ 1; 2; . . . ; nÞ;
ð5Þ
then the fractional-order system is asymptotically steady at the equilibrium [16,17]. Denis Matignon researched stability results for fractional differential equations with application to control processing and proposed that stabilities are guaranteed if the roots of some polynomial (the eigenvalues of the matrix of dynamics or the poles of the transfer matrix) lie outside the closed angular sector [16]:
j argðkÞj 6 ap=2: So, if jarg(ki)j > ap/2, a = max(q1, q2, . . . , qn)(i = 1, 2, . . . , n) is satisfied, the eigenvalues of the matrix of dynamics must lie outside the closed angular sector jarg(k)j 6 ap/2, then the stabilities are guaranteed. Theorem 1 can be proved. Fig. 1 illustrates Theorem 1. Obviously, if one equilibrium is stable, the fractional-order system will be steady at one point; only when none equilibrium is stable, the fractional-order system is in chaos. Obviously, zero point is an equilibrium of Eq. (3). Here, given the fractional order always less than 1. If the value of N can satisfy all latent root (k1, k2, . . . , kn) of Jacobi matrix of Eq. (3) at the zero are 1, the condition (5) can be satisfied. So, Eq. (3) is asymptotic stable according to Theorem 1. The error system (3) can be steady finally, i.e., the drive system (1) and the response system (2) can be achieved asymptotic synchronization. 3. The fractional order hyperchaos Lorenz systems description Lorenz system can be described as follows [18]:
X.-Y. Wang, J.-M. Song / Commun Nonlinear Sci Numer Simulat 14 (2009) 3351–3357
3353
Fig. 1. Stability region of the fractional-order system.
8 > < x_ ¼ aðy xÞ; y_ ¼ cx y xz; > :_ z ¼ xy bz:
ð6Þ
When a = 10, b = 8/3 and c = 28, Lorenz system exhibits a chaotic behavior. The author [19] adds a nonlinear controller w to _ ¼ yz þ rw, and then obtains a new system the first equation of system (6), let w
8 x_ ¼ aðy xÞ þ w; > > > < y_ ¼ cx y xz; > z_ ¼ xy bz; > > : _ ¼ yz þ rw: w
ð7Þ
When a = 10, b = 8/3, c = 28 and r = 1, system (7) exhibits a hyperchaotic behavior [19]. Four-dimensional fractional order Lorenz system can be described as
8 dq1 x ¼ aðy xÞ þ w; > > dtq1 > > dq2 y > < q ¼ cx y xz; dt 2 > dqq3 z ¼ xy bz; > > dt 3 > > : dq4 w ¼ yz þ rw; dtq4
ð8Þ
here, select the parameters as a = 10, b = 8/3, c = 28 and r = 1. As was demonstrated recently by Petras in case of Chua’s system [20], if one derives the equations properly, not all equations are of fractional order. So, q1, q2, q3, q4 are not always equal and not all fractions. But in this paper, the dynamic behaviors of four-dimensional fractional order Lorenz system are not our research purpose, so we just select qi = 0.98, i = 1, 2, 3, 4 as an example to research the synchronization of the fractional order hyperchaotic Lorenz systems. According to the method proposed by Ramasubramanian et al. [21], we can obtain when qi = 0.98 (i = 1, 2, 3, 4), the Lyapunov exponents of system (8): k1 = 0.3362, k2 = 0.1568, k3 = 0 and k4 = 15.1724. It is obvious that system (8) exhibits a hyperchaotic behavior when qi = 0.98 (i = 1, 2, 3, 4). The projections of the attractor are shown in Fig. 2. 4. The activation feedback control method Supposed system (8) is the drive system, according to Eq. (3), the response system can be described as follows:
8 dq1 x ¼ aðy x Þ þ w þ u1 ðtÞ; > > dtq1 > > dq2 y > < q ¼ cx y x z þ u2 ðtÞ; dt 2 > dq3qz ¼ x y bz þ u3 ðtÞ; > > dt 3 > > : dq4 w ¼ y z þ rw þ u4 ðtÞ; dtq4
ð9Þ
ui(t) (i = 1, 2, 3, 4) is the control function. Define the error variables are ex = x x*, ey = y y*, ez = z z* and ew = w w*, and then the error system can be obtained as
8 dq1 ex ¼ aðey ex Þ þ ew u1 ðtÞ; q > > > dtq 1 > > 2 ey d < q ¼ cex ey ðxz x z Þ u2 ðtÞ; dt 2 dq3 ez > > ¼ bez þ xy x y u3 ðtÞ; > dt q3 > > : dq4 ew ¼ rew yz þ y z u4 ðtÞ: dt q4
ð10Þ
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Fig. 2. When q = 0.98, the projections of attractor of system (8).
Let the activation feedback control function be
8 u1 ðtÞ ¼ 1 ðtÞ; > > > < u2 ðtÞ ¼ xz þ x z þ 2 ðtÞ; > > u3 ðtÞ ¼ xy x y þ 3 ðtÞ; > : u4 ðtÞ ¼ y z yz þ 4 ðtÞ;
ð11Þ
and 1(t), 2(t), 3(t) and 4(t) are the control inputs. Select
1
ex
1
B ðtÞ C Be B 2 C B y C ¼ NB B @ 3 ðtÞ A @ ez
C C C; A
0
1 ðtÞ
0
ð12Þ
ew
4 ðtÞ where
0 B B N¼B @
1
1a
a
0
1
c
0
0
0
0 b þ 1
0 C C C: 0 A
0
0
0
ð13Þ
rþ1
According to Eqs. (10)–(13), obviously, the value of N can satisfy all latent root (k1, k2, k3, k4) of Jacobi matrix of Eq. (10) at the zero are 1. Based on Theorem 1 and the stability theory of fractional-order systems, the drive system (8) and the response system (9) can achieve chaotic synchronization. Conclusion. Let the activation feedback control function be Eq. (11), and select the control input as Eq. (12), thus the drive system (8) and the response system (9) can achieve chaotic synchronization. 5. Numerical simulation In the MATLAB simulations, the fractional order qi and the time step are always chosen as qi = 0.98 (i = 1, 2, 3, 4) and s = 0.01 s, respectively. The above activation feedback scheme in Section 4 is used to solve the differential equations of the systems (8) and (9). The initial states of the drive system (8) and the response system (9) are taken as x(0) = 12, y(0) = 22, z(0) = 31, w(0) = 4 and x*(0) = 22, y*(0) = 32, z*(0) = 40, w*(0) = 11, respectively. Figs. 3 and 4 show the numerical simulation results. The time response of the states of the drive and response systems (8) and (9), respectively, is displayed in Fig. 3. In Fig. 3, the solid line is the states of the drive system (8) and the dashed line is
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Fig. 3. The time response of the states of the drive and response systems (8) and (9).
Fig. 4. Errors state of the driving system (8) and the response system (9).
the states of the response system (9). As Fig. 3 shows, the synchronization of the drive system (8) and the response system (9) is achieved after a short transient. Fig. 4 shows the time response of synchronization errors between system (8) and system (9). From Fig. 4, ex(t), ey (t), ez(t) and ew(t) are stable after a short moment with the effect of the control, respectively. When qi (i = 1, 2, 3, 4) selects other values which can make system (8) is hyperchaotic, the above method is effective, too. Wang Yihong et al. proposed the fractional Brusselator with efficient dimension less than 1 have a limit cycle [22]. Does the four-dimensional fractional order Lorenz system have a limit cycle? So we have made a large of simulations and we find
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Fig. 5. Limit cycle of four-dimentsional fractional order Lorenz system.
Fig. 6. The states of the dynamical system (8) and the response system (9).
the four-dimensional fractional order Lorenz system exists a limit cycle when the parameters are a = 10, b = 8/3, c = 18, r = 1.5 and qi = 0.98 (i = 1, 2, 3, 4). The simulation results are displayed in Fig. 5, the first 1000 points are removed. Here, initial value is x(0) = 12, y(0) = 22, z(0) = 31, w(0) = 4. Li Changpin et al. first proposed and studied the synchronization of limit sets for general ODEs [23]. Here, given the drive system (8) and the response system (9) satisfy a = 10, b = 8/3, c = 18, r = 1.5 and qi = 0.98 (i = 1, 2, 3, 4). Then the drive system (8) and the response system (9) have limit cycle. The above activation feedback scheme in Section 4 is used to solve the differential equations of the systems (8) and (9). The initial states of the drive system (8) and the response system (9) are taken as x(0) = 12, y(0) = 22, z(0) = 31, w(0) = 4 and x*(0) = 22, y*(0) = 32, z*(0) = 40, w*(0) = 11, respectively. From Figs. 6 and 7, the activation feedback scheme is effective to get the limit cycle synchronization of fractional systems. In Fig. 6, the solid line is the states of the drive system (8) and the dashed line is the states of the response system (9). As Fig. 6 shows, the synchronization of the drive system (8) and the response system (9) is achieved after a short transient. Fig. 7 shows the time response of synchronization errors between system (8) and system (9). From Fig. 7, ex(t), ey(t), ez(t) and ew(t) are stable after a short moment with the effect of the control, respectively. 6. Conclusions Based on the stability theory of fractional-order systems, this paper analyses the synchronization conditions of a class of fractional order chaotic systems with activation feedback method. The method is always effective in the fractional order chaotic systems theoretically. The commensurate hyperchaotic Lorenz system of the base order 0.98 is taken as an example and
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Fig. 7. Error state of the driving system (8) and the response system (9).
the self-synchronization is implemented based on activation feedback method. Numerical simulations show the activation feedback method can be applied in chaos synchronization, hyperchaos synchronization and limit cycle synchronization of fractional systems. Acknowledgements This research is supported by the National Natural Science Foundation of China (No. 60573172), the Superior University doctor subject special scientific research foundation of China (No. 20070141014) and the Natural Science Foundation of Liaoning province (No. 20082165). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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