- Email: [email protected]
Full state hybrid projective synchronization in hyperchaotic systems
Chaos, Solitons and Fractals 42 (2009) 1502–1510
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Full state hybrid projective synchronization in hyperchaotic systems Yan-Dong Chu, Ying-Xiang Chang *, Jian-Gang Zhang *, Xian-Feng Li, Xin-Lei An School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou 730070, People’s Republic of China
a r t i c l e
i n f o
Article history: Accepted 11 March 2009
a b s t r a c t In this letter, we investigate the full state hybrid projective synchronization (FSHPS) which includes complete synchronization, anti-synchronization and projective synchronization as its special items. Based on Lyapunov stability theory a controller can be designed for achieving the FSHPS of hyperchaotic systems. Numerical simulations are provided to verify the effectiveness of the proposed scheme. Ó 2009 Published by Elsevier Ltd.
1. Introduction Chaos has become a focal point for nonlinear problems in subjects ranging from physics and chemistry [1–3] to biology and economics [4–8]. Chaos synchronization has received a great deal of interest among scientists from various research fields [10–19] since Pecora and Carroll [9] introduced a method to synchronization two identical chaotic systems with different initial conditions. Many different techniques and methods have been proposed to achieve chaos control and chaos synchronization such as linear and non-linear feedback approach [20–23], PC method [9], OGY method [24], adaptive method [25], time-delay feedback approach [11], etc. On the other hand, when the chaotic systems have some uncertain parameters it is generally difficult to control the systems. Therefore, the derivation of an adaptive controller for the control and synchronization of chaotic systems in the presence of unknown system parameters is an important issue [26–28]. In this regard, the adaptive feedback synchronization for several chaotic systems has been investigated by Wang et al. [25], Han et al. [27], Elabbasy et al. [28], Lu et al. [29], and Park [14]. Recently, Wen [30] presented a full-state projective synchronization between two dynamical systems. Consider two dynamical systems
_ xðtÞ ¼ FðxÞ drive system _ yðtÞ ¼ GðyÞ þ uðx; yÞ response system
ðaÞ ðbÞ
where x ¼ ðx1 ; x2 . . . xn ÞT ; y ¼ ðy1 ; y2 . . . yn ÞT 2 Rn are state variables of the drive system (a) and response system (b), respectively, uðx; yÞ is a controller. If there exists a nonzero constant a such that limt!1 ky axk ¼ 0 that is, limt!1 kyi axi k ¼ 0ði ¼ 1; 2 . . . nÞ, then we call them full state projective synchronization (FSHPS). The novelty feature of this synchronization phenomenon is that the scaling factors can be arbitrarily designed to different state variables by means of control. Projective synchronization attracted lots of attention [31–40] because of its proportionality between the synchronized dynamical states. In applications to secure communications [39–41], this feature can be used to extend binary digital to M-nary digital communication [40] for achieving fast communication. Motivated by the above discussion, the aim of this letter is study the FSHPS of drive and response chaotic (hyperchaotic) systems, based on Lyapunov stability theorem. A general controller law is proposed for the FSHPS of chaotic systems. This letter is organized as follows: In Section 2, the FSHPS scheme is presented briefly. In Section 3, the scheme is applied to investigate modified unified system and hyperchaotic Lü system. Numerical simulations are used to verify the effectiveness of the proposed scheme. Finally, conclusions end the paper.
* Corresponding authors. Tel./fax: +86 931 4938401 (Y.X. Chang). E-mail addresses: [email protected] (Y.-D. Chu), [email protected] (Y.-X. Chang), [email protected] (J.-G. Zhang), [email protected] (X.-F. Li). 0960-0779/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.chaos.2009.03.049
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2. General FSHPS scheme Consider an n-dimensional hyperchaotic system in the form of
x_ ¼ FðxÞ
ð1Þ T
T
n
n
where x ¼ ðx1 ; x2 . . . xn Þ 2 R is the state vector, and FðxÞ ¼ ðF 1 ðxÞ; F 2 ðxÞ . . . F n ðxÞÞ 2 R is a continuous nonlinear vector function. We refer to (1) as the drive system. The response system is defined by
y_ ¼ FðyÞ þ u
ð2Þ T
n
n
T
n
where y ¼ ðy1 ; y2 ; . . . yn Þ 2 R ; FðyÞ ¼ ðF 1 ðyÞ; F 2 ðyÞ . . . F n ðyÞÞ 2 R ; u ¼ ðu1 ðx; yÞ; u2 ðx; yÞ . . . un ðx; yÞÞ 2 R is the controller to be determined for the purpose of FSHPS. Let the vector error state be eðtÞ ¼ yðtÞ axðtÞ, where a is a n-order diagonal matrix, that is, a ¼ diagða1 ; a2 . . . an Þ; ai 2 R; i ¼ 1; 2 . . . n. Thus the error dynamical system between the drive system (1) and response system (2) is
_ eðtÞ ¼ y_ ax_ ¼ e F ðx; yÞ þ u
ð3Þ
where e F ðx; yÞ ¼ FðyÞ aFðxÞ ¼ ðF 1 ðyÞ a1 F 1 ðxÞ; F 2 ðyÞ a2 F 2 ðxÞ; . . . ; F n ðyÞ an F n ðxÞÞT . Hence the FSHPS problem becomes the stability of error dynamics (3). If it is globally stabilized at the origin, the FSHPS of drive system (1) and response system (2) can be globally realized. In the following, we will give a principle to find suitable feedback controller such that the two chaotic (hyperchaotic) systems are FSHPS. Construct a dynamical Lyapunov function
V¼
1 T e Pe 2
ð4Þ
where P is a positive definite constant matrix. One may choose P as the corresponding definite matrix in most case. The time derivative of V along the trajectories of Eq. (3) is
V_ ¼ eT Pðu þ e FÞ
ð5Þ
Suppose we select an appropriate controller u such that V_ < 0, that is V_ is negative definite. Then according to the Lyapunov stability theorem, the FSHPS of chaotic system (1) and (2) is achieved under the certain chosen feedback controller u. 3. Applications of the FSHPS scheme In this section, we will choose modified unified system and hyperchaotic Lü system to illustrate the effectiveness of the proposed scheme. 3.1. FSHPS of modified unified system The nonlinear differential equations that describe the unified system are
8 > < x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ x_ 2 ¼ ð28 32aÞx1 x1 x3 þ ð29a 1Þx2 > :_ x3 ¼ x1 x2 8þ3 a x3
ð6Þ
where the constant a 2 ½0; 1, This system can be reformulated in the following canonical form [42]
2
3 2 a11 x_ 1 6_ 7 6 4 x2 5 ¼ 4 a21 0 x_ 3
a12 a22 0
3 2 32 3 0 0 0 x1 x1 76 7 6 76 7 0 54 x2 5 þ x1 4 0 0 1 54 x2 5 0 0
32
x3
0 1
0
x3
According to the canonical-form criterion formulated in [43], if system (6) satisfies the conditions a12 a21 > 0, a12 a21 < 0 or a12 a21 ¼ 0, then it is referred to as the generalized Lorenz,Chen or Lü system, respectively. In order to have a better view, we can modify the system (6) as follows
8 > < x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ x_ 2 ¼ ð17:5a þ 10:5Þx1 signðaÞx1 x3 þ ð13:3 14aÞx2 > :_ x3 ¼ signðaÞx1 x2 83 x3
ð7Þ
1 uP0 where a ¼ cosðwtÞ 2 ½1; 1, and signðuÞ ¼ . When w ¼ 5:3, the system (7) has a hyperchaotic attractor as 1 u < 0 shown in Figs. 1 and 2. In order to observe the FSHPS of the modified unified system, we define the response system of (7) as follows
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Fig. 1. The hyperchaotic attractors in R3 .
8 > < y_ 1 ¼ ð25a þ 10Þðy2 y1 Þ þ u1 y_ 2 ¼ ð17:5a þ 10:5Þy1 signðaÞy1 y3 þ ð13:3 14aÞy2 þ u2 > :_ y3 ¼ signðaÞy1 y2 83 y3 þ u3
ð8Þ
where u ¼ ðu1 ; u2 ; u3 ÞT is the nonlinear controller to be designed for FSHPS of two unified systems with the initial parameters in spite of the differences in initial conditions. Define the FSHPS error signal as e ¼ y ax that is,
8 > < e1 ðtÞ ¼ y1 a1 x1 e2 ðtÞ ¼ y2 a2 x2 > : e3 ðtÞ ¼ y3 a3 x3
ð9Þ
where a ¼ diagða1 ; a2 ; a3 Þ and a1 ; a2 ; a3 are different desired in advance scaling factors for FSHPS. Subtracting Eq. (7) from multiplies Eq. (8), the error dynamical system can be written as
8 e_ 1 ðtÞ ¼ ð25a þ 10Þðy2 y1 Þ a1 ½ð25a þ 10Þðx2 x1 Þ þ u1 > > > < e_ ðtÞ ¼ ð17:5a þ 10:5Þy signðaÞy y þ ð13:3 14aÞy a ½ð17:5a þ 10:5Þx 2 2 1 1 1 3 2 > signðaÞx1 x3 þ ð13:3 14aÞx2 þ u2 > > : e_ 3 ðtÞ ¼ signðaÞy1 y2 83 y3 a3 ½signðaÞx1 x2 83 x3 þ u3
ð10Þ
The goal of control study is to find a controller for system (10) such that system (7) and system (8) are in FSHPS. Let us now choose the control functions as follows:
8 u ¼ a1 ½ð25a þ 10Þðx2 x1 Þð25a þ 10Þðy2 y1 Þ e1 > > > 1 < u2 ¼ signðaÞy1 y3 ð17:5a þ 10:5Þy1 ð13:3 14aÞy2 þ a2 ½ð17:5a þ 10:5Þx1 signðaÞx1 x3 þ ð13:3 14aÞx2 e2 > > > : u ¼ a signðaÞx x 8 x signðaÞy y þ 8 y e 3 3 1 2 3 1 2 3 3 3 3
Fig. 2. The attractors in R2 with the scaling factors a1 ¼ 1; a2 ¼ 1; a3 ¼ 1.
ð11Þ
Y.-D. Chu et al. / Chaos, Solitons and Fractals 42 (2009) 1502–1510
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Construct the Lapunov function
V¼
1 2 ðe þ e22 þ e23 Þ 2 1
Then the time derivative of V along the solutions of Eq. (9) is
V_ ¼ e21 e22 e23 Since V is a positive define function and V_ is a negative definite function, according to the Lapunov’s direct method, the error variables become zero as time t tends to infinity, that is, limt!1 ky1 a1 x1 k ¼ limt!1 ky2 a2 x2 k ¼ limt!1 ky3 a3 x3 k ¼ 0. This means that two LFRBM systems are in FSHPS under the controller (11). For the numerical simulations, fourth-order Runge-Kutta method is used to solve the systems of differential Eqs. (7) and (8),with time step size 0.001.In all numerical simulations, the system parameters are chosen to be w ¼ 5:3 so that the two unified systems both have a hyperchaotic attractor. The initial states of the drive system and response system are x1 ð0Þ ¼ 0:3; x2 ð0Þ ¼ 0; x3 ð0Þ ¼ 0:18 and y1 ð0Þ ¼ 0:325; y2 ð0Þ ¼ 0:1; y3 ð0Þ ¼ 0:18, respectively. This simulation results are shown from Figs. 2–4. 3.2. FSHPS of hyperchaotic Lü system For further illustrate the effectiveness of the proposed scheme, in this subsection, we select the hyperchaotic Lü system [22], which is described by the following equations
8 x_ 1 ðtÞ ¼ aðx2 x1 Þ þ x4 > > > < x_ ðtÞ ¼ bx x x 2 2 1 3 > _ 3 ðtÞ ¼ cx3 þ x1 x2 x > > : x_ 4 ðtÞ ¼ dx4 þ x1 x3 This sysytem is hyperchaotic for a ¼ 26; b ¼ 20; c ¼ 3; d ¼ 1:3.
Fig. 3. Steady-state plane trajectories and error with scaling factors a1 ¼ 1; a2 ¼ 1; a3 ¼ 1.
ð12Þ
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Fig. 4. The attractors in R2 and errors with the scaling factors a1 ¼ 2; a2 ¼ 1; a3 ¼ 1:5.
In order to observe the FSHPS of hyperchaotic Lü systems, we define the response system of (12) as follows:
8 y_ 1 ðtÞ ¼ aðy2 y1 Þ þ y4 þ u1 > > > < y_ 2 ðtÞ ¼ by2 y1 y3 þ u2 > > y_ 3 ðtÞ ¼ cy3 þ y1 y2 þ u3 > : y_ 4 ðtÞ ¼ dy4 þ y1 y3 þ u4
ð13Þ
where uT ¼ ðu1 ; u2 ; u3 ; u4 Þ is the controller to be designed for FSHPS of two hyperchaotic Lü systems with the same parameters a, b, c and d in spite of the differences in initial conditions. Define the FSHPS error signal as e ¼ y ax that is,
8 e1 ðtÞ ¼ y1 a1 x1 > > > < e ðtÞ ¼ y a x 2 2 2 2 > e ðtÞ ¼ y a 3 3 x3 > 3 > : e4 ðtÞ ¼ y4 a4 x4
ð14Þ
where a ¼ diagða1 ; a2 ; a3 ; a4 Þ and a1 ; a2 ; a3 and a4 are different designed in advance scaling factors for FSHPS. Subtracting Eq. (12) from a times Eq. (13), then the error dynamical system can be written as
8 e1 ðtÞ ¼ aðy2 y1 Þ þ y4 a1 ½aðx2 x1 Þ þ x4 þ u1 > > > < e2 ðtÞ ¼ by2 y1 y3 a2 ðbx2 x1 x3 Þ þ u2 > > e3 ðtÞ ¼ cy3 þ y1 y2 a3 ðcx3 þ x1 x2 Þ þ u3 > : e4 ðtÞ ¼ dy4 þ y1 y3 a4 ðdx4 þ x1 x3 Þ þ u4
ð15Þ
Y.-D. Chu et al. / Chaos, Solitons and Fractals 42 (2009) 1502–1510
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The goal of control study is to find a controller for system (15) such that systems (12) and (13) are in FSHPS. Let us now choose the control functions u1 ; u2 ; u3 ; u4 as follows
8 u1 ðtÞ ¼ aðy2 y1 Þ y4 þ a1 ½aðx2 x1 Þ þ x4 e1 > > > < u ðtÞ ¼ by þ y y þ a ðbx x x Þ e 2 2 2 1 3 2 2 1 3 > u ðtÞ ¼ cy y y þ a ðcx þ x x Þ e 3 3 3 1 2 3 > 3 1 2 > : u4 ðtÞ ¼ dy4 y1 y3 þ a4 ðdx4 þ x1 x3 Þ e4
ð16Þ
Construct the Lapunov function
V¼
1 2 ðe þ e22 þ e23 þ e24 Þ 2 1
Then the time derivative of V along the solutions of Eq. (11) is
V_ ¼ e21 e22 e23 e24 Since V is a positive define function and V_ is a negative definite function, according to the Lapunov’s direct method, the error variables become zero as time t tends to infinity, that is, limt!1 ky1 a1 x1 k ¼ limt!1 ky2 a2 x2 k ¼ limt!1 ky3 a3 x3 k ¼ limt!1 ky4 a4 x4 k ¼ 0. This means that two LFRBM systems are in FSHPS under the controller (16). For this numerical simulations, the system parameters are chosen to be a ¼ 36, d ¼ 20, c ¼ 3, d ¼ 1:3 such that the two hyperchaotic systems display hyperchaotic behavior. We assume that the initial states of the drive system and response system are x1 ð0Þ ¼ 0:3; x2 ð0Þ ¼ 0:25; x3 ð0Þ ¼ 2; x4 ð0Þ ¼ 1:5; y1 ð0Þ ¼ 4; y3 ð0Þ ¼ 2:5; y4 ð0Þ ¼ 1:8, respectively. The simulations results are shown as Figs. 5–7. The above numerical simulations show that hyperchaotic FSHPS can be well achieved by the proposed controller scheme. 4. Conclusions In this letter, FSHPS scheme has been proposed for hyperchaotic systems. The modified unified system and hyperchaotic Lü systems are chosen to illustrate the proposed scheme based on Lyapunov’s direct method.
Fig. 5. The attractors in R2 and errors with the scaling factors a1 ¼ 1; a2 ¼ 1; a3 ¼ 1; a4 ¼ 1.
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Fig. 6. The errors between two hyperchaotic Lü systems with a1 ¼ 1; a2 ¼ 1; a3 ¼ 1; a4 ¼ 1.
Because of the complete synchronization, anti-synchronization, projective synchronization are all included in FSHPS, our results contain and extend most existing works. But there exist many interesting and difficult problems left out for in-depth study about this new type of synchronization behavior, such as what is the FSHPS behavior between other dynamical systems, discrete hyperchaotic maps, and artificial chaotic neural networks. Continued research would be desirable. Acknowledgements The authors express their gratitude to Prof. Luis Fernando Mello in Universidade Federal de Itajubá for his valuable comments on the paper. The authors also gratefully acknowledge support from the National Natural Science Foundation of China
Y.-D. Chu et al. / Chaos, Solitons and Fractals 42 (2009) 1502–1510
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Fig. 7. The time evolution of the errors between two hyperchaotic Lü systems with the scaling factors a1 ¼ 0:5; a2 ¼ 1; a3 ¼ 2; a4 ¼ 3.
(Project No. 50475109), the Natural Science Foundation of Gansu Province, Government of China (Project No. 3ZS051-A25030, 3ZS-042-B25-049) and Scientific Research Foundations of Lanzhou Jiaotong University of China (DXS-2008-031, DXS2008-032). References [1] Henryk Merta. Characteristic time series and operation region of the system of two tank reactors (CSTR) with variable division of recirculation stream. Chaos, Solitons & Fractals 2006;27:279–85. [2] Henryk Merta. Chaotic dynamics of a cascade of plug flow tubular reactors (PFTRs) with division of recirculating stream. Chaos, Solitons & Fractals 2005;23:1211–9. [3] Tomasz Stachowiak, Toshio Okada. A numerical analysis of chaos in the double pendulum. Chaos, Solitons & Fractals 2006;29:417–22. [4] Sunita Gakkhar, Brahampal Singh. Complex dynamic behavior in a food web consisting of two preys and a predator. Chaos, Solitons & Fractals 2005;24:789–801. [5] Sunita Gakkhar, Ra’id Kamel Naji. 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Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Full state hybrid projective synchronization in hyperchaotic systems Yan-Dong Chu, Ying-Xiang Chang *, Jian-Gang Zhang *, Xian-Feng Li, Xin-Lei An School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou 730070, People’s Republic of China
a r t i c l e
i n f o
Article history: Accepted 11 March 2009
a b s t r a c t In this letter, we investigate the full state hybrid projective synchronization (FSHPS) which includes complete synchronization, anti-synchronization and projective synchronization as its special items. Based on Lyapunov stability theory a controller can be designed for achieving the FSHPS of hyperchaotic systems. Numerical simulations are provided to verify the effectiveness of the proposed scheme. Ó 2009 Published by Elsevier Ltd.
1. Introduction Chaos has become a focal point for nonlinear problems in subjects ranging from physics and chemistry [1–3] to biology and economics [4–8]. Chaos synchronization has received a great deal of interest among scientists from various research fields [10–19] since Pecora and Carroll [9] introduced a method to synchronization two identical chaotic systems with different initial conditions. Many different techniques and methods have been proposed to achieve chaos control and chaos synchronization such as linear and non-linear feedback approach [20–23], PC method [9], OGY method [24], adaptive method [25], time-delay feedback approach [11], etc. On the other hand, when the chaotic systems have some uncertain parameters it is generally difficult to control the systems. Therefore, the derivation of an adaptive controller for the control and synchronization of chaotic systems in the presence of unknown system parameters is an important issue [26–28]. In this regard, the adaptive feedback synchronization for several chaotic systems has been investigated by Wang et al. [25], Han et al. [27], Elabbasy et al. [28], Lu et al. [29], and Park [14]. Recently, Wen [30] presented a full-state projective synchronization between two dynamical systems. Consider two dynamical systems
_ xðtÞ ¼ FðxÞ drive system _ yðtÞ ¼ GðyÞ þ uðx; yÞ response system
ðaÞ ðbÞ
where x ¼ ðx1 ; x2 . . . xn ÞT ; y ¼ ðy1 ; y2 . . . yn ÞT 2 Rn are state variables of the drive system (a) and response system (b), respectively, uðx; yÞ is a controller. If there exists a nonzero constant a such that limt!1 ky axk ¼ 0 that is, limt!1 kyi axi k ¼ 0ði ¼ 1; 2 . . . nÞ, then we call them full state projective synchronization (FSHPS). The novelty feature of this synchronization phenomenon is that the scaling factors can be arbitrarily designed to different state variables by means of control. Projective synchronization attracted lots of attention [31–40] because of its proportionality between the synchronized dynamical states. In applications to secure communications [39–41], this feature can be used to extend binary digital to M-nary digital communication [40] for achieving fast communication. Motivated by the above discussion, the aim of this letter is study the FSHPS of drive and response chaotic (hyperchaotic) systems, based on Lyapunov stability theorem. A general controller law is proposed for the FSHPS of chaotic systems. This letter is organized as follows: In Section 2, the FSHPS scheme is presented briefly. In Section 3, the scheme is applied to investigate modified unified system and hyperchaotic Lü system. Numerical simulations are used to verify the effectiveness of the proposed scheme. Finally, conclusions end the paper.
* Corresponding authors. Tel./fax: +86 931 4938401 (Y.X. Chang). E-mail addresses: [email protected] (Y.-D. Chu), [email protected] (Y.-X. Chang), [email protected] (J.-G. Zhang), [email protected] (X.-F. Li). 0960-0779/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.chaos.2009.03.049
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2. General FSHPS scheme Consider an n-dimensional hyperchaotic system in the form of
x_ ¼ FðxÞ
ð1Þ T
T
n
n
where x ¼ ðx1 ; x2 . . . xn Þ 2 R is the state vector, and FðxÞ ¼ ðF 1 ðxÞ; F 2 ðxÞ . . . F n ðxÞÞ 2 R is a continuous nonlinear vector function. We refer to (1) as the drive system. The response system is defined by
y_ ¼ FðyÞ þ u
ð2Þ T
n
n
T
n
where y ¼ ðy1 ; y2 ; . . . yn Þ 2 R ; FðyÞ ¼ ðF 1 ðyÞ; F 2 ðyÞ . . . F n ðyÞÞ 2 R ; u ¼ ðu1 ðx; yÞ; u2 ðx; yÞ . . . un ðx; yÞÞ 2 R is the controller to be determined for the purpose of FSHPS. Let the vector error state be eðtÞ ¼ yðtÞ axðtÞ, where a is a n-order diagonal matrix, that is, a ¼ diagða1 ; a2 . . . an Þ; ai 2 R; i ¼ 1; 2 . . . n. Thus the error dynamical system between the drive system (1) and response system (2) is
_ eðtÞ ¼ y_ ax_ ¼ e F ðx; yÞ þ u
ð3Þ
where e F ðx; yÞ ¼ FðyÞ aFðxÞ ¼ ðF 1 ðyÞ a1 F 1 ðxÞ; F 2 ðyÞ a2 F 2 ðxÞ; . . . ; F n ðyÞ an F n ðxÞÞT . Hence the FSHPS problem becomes the stability of error dynamics (3). If it is globally stabilized at the origin, the FSHPS of drive system (1) and response system (2) can be globally realized. In the following, we will give a principle to find suitable feedback controller such that the two chaotic (hyperchaotic) systems are FSHPS. Construct a dynamical Lyapunov function
V¼
1 T e Pe 2
ð4Þ
where P is a positive definite constant matrix. One may choose P as the corresponding definite matrix in most case. The time derivative of V along the trajectories of Eq. (3) is
V_ ¼ eT Pðu þ e FÞ
ð5Þ
Suppose we select an appropriate controller u such that V_ < 0, that is V_ is negative definite. Then according to the Lyapunov stability theorem, the FSHPS of chaotic system (1) and (2) is achieved under the certain chosen feedback controller u. 3. Applications of the FSHPS scheme In this section, we will choose modified unified system and hyperchaotic Lü system to illustrate the effectiveness of the proposed scheme. 3.1. FSHPS of modified unified system The nonlinear differential equations that describe the unified system are
8 > < x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ x_ 2 ¼ ð28 32aÞx1 x1 x3 þ ð29a 1Þx2 > :_ x3 ¼ x1 x2 8þ3 a x3
ð6Þ
where the constant a 2 ½0; 1, This system can be reformulated in the following canonical form [42]
2
3 2 a11 x_ 1 6_ 7 6 4 x2 5 ¼ 4 a21 0 x_ 3
a12 a22 0
3 2 32 3 0 0 0 x1 x1 76 7 6 76 7 0 54 x2 5 þ x1 4 0 0 1 54 x2 5 0 0
32
x3
0 1
0
x3
According to the canonical-form criterion formulated in [43], if system (6) satisfies the conditions a12 a21 > 0, a12 a21 < 0 or a12 a21 ¼ 0, then it is referred to as the generalized Lorenz,Chen or Lü system, respectively. In order to have a better view, we can modify the system (6) as follows
8 > < x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ x_ 2 ¼ ð17:5a þ 10:5Þx1 signðaÞx1 x3 þ ð13:3 14aÞx2 > :_ x3 ¼ signðaÞx1 x2 83 x3
ð7Þ
1 uP0 where a ¼ cosðwtÞ 2 ½1; 1, and signðuÞ ¼ . When w ¼ 5:3, the system (7) has a hyperchaotic attractor as 1 u < 0 shown in Figs. 1 and 2. In order to observe the FSHPS of the modified unified system, we define the response system of (7) as follows
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Fig. 1. The hyperchaotic attractors in R3 .
8 > < y_ 1 ¼ ð25a þ 10Þðy2 y1 Þ þ u1 y_ 2 ¼ ð17:5a þ 10:5Þy1 signðaÞy1 y3 þ ð13:3 14aÞy2 þ u2 > :_ y3 ¼ signðaÞy1 y2 83 y3 þ u3
ð8Þ
where u ¼ ðu1 ; u2 ; u3 ÞT is the nonlinear controller to be designed for FSHPS of two unified systems with the initial parameters in spite of the differences in initial conditions. Define the FSHPS error signal as e ¼ y ax that is,
8 > < e1 ðtÞ ¼ y1 a1 x1 e2 ðtÞ ¼ y2 a2 x2 > : e3 ðtÞ ¼ y3 a3 x3
ð9Þ
where a ¼ diagða1 ; a2 ; a3 Þ and a1 ; a2 ; a3 are different desired in advance scaling factors for FSHPS. Subtracting Eq. (7) from multiplies Eq. (8), the error dynamical system can be written as
8 e_ 1 ðtÞ ¼ ð25a þ 10Þðy2 y1 Þ a1 ½ð25a þ 10Þðx2 x1 Þ þ u1 > > > < e_ ðtÞ ¼ ð17:5a þ 10:5Þy signðaÞy y þ ð13:3 14aÞy a ½ð17:5a þ 10:5Þx 2 2 1 1 1 3 2 > signðaÞx1 x3 þ ð13:3 14aÞx2 þ u2 > > : e_ 3 ðtÞ ¼ signðaÞy1 y2 83 y3 a3 ½signðaÞx1 x2 83 x3 þ u3
ð10Þ
The goal of control study is to find a controller for system (10) such that system (7) and system (8) are in FSHPS. Let us now choose the control functions as follows:
8 u ¼ a1 ½ð25a þ 10Þðx2 x1 Þð25a þ 10Þðy2 y1 Þ e1 > > > 1 < u2 ¼ signðaÞy1 y3 ð17:5a þ 10:5Þy1 ð13:3 14aÞy2 þ a2 ½ð17:5a þ 10:5Þx1 signðaÞx1 x3 þ ð13:3 14aÞx2 e2 > > > : u ¼ a signðaÞx x 8 x signðaÞy y þ 8 y e 3 3 1 2 3 1 2 3 3 3 3
Fig. 2. The attractors in R2 with the scaling factors a1 ¼ 1; a2 ¼ 1; a3 ¼ 1.
ð11Þ
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Construct the Lapunov function
V¼
1 2 ðe þ e22 þ e23 Þ 2 1
Then the time derivative of V along the solutions of Eq. (9) is
V_ ¼ e21 e22 e23 Since V is a positive define function and V_ is a negative definite function, according to the Lapunov’s direct method, the error variables become zero as time t tends to infinity, that is, limt!1 ky1 a1 x1 k ¼ limt!1 ky2 a2 x2 k ¼ limt!1 ky3 a3 x3 k ¼ 0. This means that two LFRBM systems are in FSHPS under the controller (11). For the numerical simulations, fourth-order Runge-Kutta method is used to solve the systems of differential Eqs. (7) and (8),with time step size 0.001.In all numerical simulations, the system parameters are chosen to be w ¼ 5:3 so that the two unified systems both have a hyperchaotic attractor. The initial states of the drive system and response system are x1 ð0Þ ¼ 0:3; x2 ð0Þ ¼ 0; x3 ð0Þ ¼ 0:18 and y1 ð0Þ ¼ 0:325; y2 ð0Þ ¼ 0:1; y3 ð0Þ ¼ 0:18, respectively. This simulation results are shown from Figs. 2–4. 3.2. FSHPS of hyperchaotic Lü system For further illustrate the effectiveness of the proposed scheme, in this subsection, we select the hyperchaotic Lü system [22], which is described by the following equations
8 x_ 1 ðtÞ ¼ aðx2 x1 Þ þ x4 > > > < x_ ðtÞ ¼ bx x x 2 2 1 3 > _ 3 ðtÞ ¼ cx3 þ x1 x2 x > > : x_ 4 ðtÞ ¼ dx4 þ x1 x3 This sysytem is hyperchaotic for a ¼ 26; b ¼ 20; c ¼ 3; d ¼ 1:3.
Fig. 3. Steady-state plane trajectories and error with scaling factors a1 ¼ 1; a2 ¼ 1; a3 ¼ 1.
ð12Þ
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Fig. 4. The attractors in R2 and errors with the scaling factors a1 ¼ 2; a2 ¼ 1; a3 ¼ 1:5.
In order to observe the FSHPS of hyperchaotic Lü systems, we define the response system of (12) as follows:
8 y_ 1 ðtÞ ¼ aðy2 y1 Þ þ y4 þ u1 > > > < y_ 2 ðtÞ ¼ by2 y1 y3 þ u2 > > y_ 3 ðtÞ ¼ cy3 þ y1 y2 þ u3 > : y_ 4 ðtÞ ¼ dy4 þ y1 y3 þ u4
ð13Þ
where uT ¼ ðu1 ; u2 ; u3 ; u4 Þ is the controller to be designed for FSHPS of two hyperchaotic Lü systems with the same parameters a, b, c and d in spite of the differences in initial conditions. Define the FSHPS error signal as e ¼ y ax that is,
8 e1 ðtÞ ¼ y1 a1 x1 > > > < e ðtÞ ¼ y a x 2 2 2 2 > e ðtÞ ¼ y a 3 3 x3 > 3 > : e4 ðtÞ ¼ y4 a4 x4
ð14Þ
where a ¼ diagða1 ; a2 ; a3 ; a4 Þ and a1 ; a2 ; a3 and a4 are different designed in advance scaling factors for FSHPS. Subtracting Eq. (12) from a times Eq. (13), then the error dynamical system can be written as
8 e1 ðtÞ ¼ aðy2 y1 Þ þ y4 a1 ½aðx2 x1 Þ þ x4 þ u1 > > > < e2 ðtÞ ¼ by2 y1 y3 a2 ðbx2 x1 x3 Þ þ u2 > > e3 ðtÞ ¼ cy3 þ y1 y2 a3 ðcx3 þ x1 x2 Þ þ u3 > : e4 ðtÞ ¼ dy4 þ y1 y3 a4 ðdx4 þ x1 x3 Þ þ u4
ð15Þ
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The goal of control study is to find a controller for system (15) such that systems (12) and (13) are in FSHPS. Let us now choose the control functions u1 ; u2 ; u3 ; u4 as follows
8 u1 ðtÞ ¼ aðy2 y1 Þ y4 þ a1 ½aðx2 x1 Þ þ x4 e1 > > > < u ðtÞ ¼ by þ y y þ a ðbx x x Þ e 2 2 2 1 3 2 2 1 3 > u ðtÞ ¼ cy y y þ a ðcx þ x x Þ e 3 3 3 1 2 3 > 3 1 2 > : u4 ðtÞ ¼ dy4 y1 y3 þ a4 ðdx4 þ x1 x3 Þ e4
ð16Þ
Construct the Lapunov function
V¼
1 2 ðe þ e22 þ e23 þ e24 Þ 2 1
Then the time derivative of V along the solutions of Eq. (11) is
V_ ¼ e21 e22 e23 e24 Since V is a positive define function and V_ is a negative definite function, according to the Lapunov’s direct method, the error variables become zero as time t tends to infinity, that is, limt!1 ky1 a1 x1 k ¼ limt!1 ky2 a2 x2 k ¼ limt!1 ky3 a3 x3 k ¼ limt!1 ky4 a4 x4 k ¼ 0. This means that two LFRBM systems are in FSHPS under the controller (16). For this numerical simulations, the system parameters are chosen to be a ¼ 36, d ¼ 20, c ¼ 3, d ¼ 1:3 such that the two hyperchaotic systems display hyperchaotic behavior. We assume that the initial states of the drive system and response system are x1 ð0Þ ¼ 0:3; x2 ð0Þ ¼ 0:25; x3 ð0Þ ¼ 2; x4 ð0Þ ¼ 1:5; y1 ð0Þ ¼ 4; y3 ð0Þ ¼ 2:5; y4 ð0Þ ¼ 1:8, respectively. The simulations results are shown as Figs. 5–7. The above numerical simulations show that hyperchaotic FSHPS can be well achieved by the proposed controller scheme. 4. Conclusions In this letter, FSHPS scheme has been proposed for hyperchaotic systems. The modified unified system and hyperchaotic Lü systems are chosen to illustrate the proposed scheme based on Lyapunov’s direct method.
Fig. 5. The attractors in R2 and errors with the scaling factors a1 ¼ 1; a2 ¼ 1; a3 ¼ 1; a4 ¼ 1.
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Fig. 6. The errors between two hyperchaotic Lü systems with a1 ¼ 1; a2 ¼ 1; a3 ¼ 1; a4 ¼ 1.
Because of the complete synchronization, anti-synchronization, projective synchronization are all included in FSHPS, our results contain and extend most existing works. But there exist many interesting and difficult problems left out for in-depth study about this new type of synchronization behavior, such as what is the FSHPS behavior between other dynamical systems, discrete hyperchaotic maps, and artificial chaotic neural networks. Continued research would be desirable. Acknowledgements The authors express their gratitude to Prof. Luis Fernando Mello in Universidade Federal de Itajubá for his valuable comments on the paper. The authors also gratefully acknowledge support from the National Natural Science Foundation of China
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Fig. 7. The time evolution of the errors between two hyperchaotic Lü systems with the scaling factors a1 ¼ 0:5; a2 ¼ 1; a3 ¼ 2; a4 ¼ 3.
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