- Email: [email protected]
Synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems using nonlinear active control
Chaos, Solitons & Fractals 69 (2014) 31–39
Contents lists available at ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems using nonlinear active control Piyush Pratap Singh ⇑, Jay Prakash Singh, B. K. Roy NIT Silchar, Silchar, Assam 788 010, India
a r t i c l e
i n f o
Article history: Received 5 April 2014 Accepted 7 September 2014
a b s t r a c t This paper aims at synchronization and anti-synchronization between Lu chaotic system, a member of unified chaotic system, and recently developed Bhalekar–Gejji chaotic system, a system which cannot be derived from the member of unified chaotic system. These synchronization and anti-synchronization have been achieved by using nonlinear active control since the parameters of both the systems are known. Lyapunov stability theory is used and required condition is derived to ensure the stability of error dynamics. Controller is designed by using the sum of relevant variables in chaotic systems. Simulation results suggest that proposed scheme is working satisfactorily. 2014 Elsevier Ltd. All rights reserved.
1. Introduction The important properties available in any chaotic dynamical system are sensitive to initial condition, aperiodic long time behavior, and are deterministic in nature [1]. Chaotic system has potential applications in physics, neurobiology, earth science, electrical engineering, and many other fields [2]. Motivated by stated applications, chaotic systems are also being applied in field of communication with the use of synchronization concept [2]. Secure communication is a major issue today. It is expected that synchronization of two chaotic systems may help towards secure communication. Motivated by such practical requirement, authors are motivated to consider synchronization and anti-synchronization between well-known Lu chaotic system, a member of unified chaotic system, and Bhalekar–Gejji chaotic systems developed during 2012. There are two methods for synchronization of chaotic systems known as drive-response scheme and coupling ⇑ Corresponding author. E-mail addresses: [email protected] (P. P. Singh), [email protected] (J. P. Singh), [email protected] (B. K. Roy). http://dx.doi.org/10.1016/j.chaos.2014.09.005 0960-0779/ 2014 Elsevier Ltd. All rights reserved.
scheme. Drive-response is also called as master–slave system which is widely used by the researchers. In this scheme the output of master (drive) system is used to control the slave system. Synchronization means to achieve zero error between states of master and slave systems. Firstly, the phenomenon of chaotic synchronization is illustrated by Yamada and Fujisaka [3]. But interest of the researchers increases after the pioneer work of Pecora and Carroll [4,5], where, two chaotic systems with different initial conditions are synchronized. Various control schemes have been developed for synchronization of chaotic system in the last two decades such as PC method [4], OGY method [6], active control [7–9], adaptive control [10,11], backstepping control [12,13], observer design [14], time delay feedback control [15], sliding mode control [16,17], etc. Many more techniques are also available related to synchronization like Complete synchronization [18], Lag synchronization [19], Phase synchronization [20], generalized synchronization [21], projective synchronization [22], generalized projective [23], QS synchronization [24], time scale synchronization [25], exponential synchronization [26], etc.
32
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
Another important phenomenon absorbed between two dynamical systems is anti-synchronization. The purpose of anti-synchronization is to use the output of master system to control the slave system. During anti-synchronization state of both master and slave systems shows the same amplitude but opposite in phase. Anti-synchronization causes the sum of state of master and slave systems converge asymptotically to zero. Active control is used as tool for anti-synchronization [27–31] by last few years. Synchronization of chaotic system using nonlinear active control was first reported by Bai and Lonngren [32]. So, many papers are available in literature based on use of nonlinear active control as a tool for synchronization and anti-synchronization of chaotic systems in last few years. In most of the reported papers of synchronization and anti-synchronization between chaotic systems involving Lu system [33,34] the other system is chosen from a set of chaotic systems which are derived from the members of unified chaotic system. But Bhalekar–Gejji chaotic [35,36] system does not fit in unified chaotic system. Very few papers have been reported with Lu and other chaotic system which cannot be derived from the members of unified chaotic system [37] and present paper is one of them. In this paper we are considering the problem of synchronization and antisynchronization both. Further, Bhalekar–Gejji chaotic system is newly developed chaotic system in 2012. Global synchronization and anti-synchronization of Bhalekar–Gejji chaotic system with any other chaotic system using nonlinear active control is reported for first time which reflects the novelty of this paper. In first part of the paper active control structure is designed based upon the error dynamics of master and slave systems. In second part, global synchronization is achieved by using the Lyapunov’s stability theory [38]. The organization of paper is as follows. In Sections 2 and 3, problem statement for synchronization and antisynchronization of two chaotic systems is discussed. Global synchronization of Lu and Bhalekar–Gejji chaotic system is discussed in Section 4 followed by global antisynchronization in Section 5. In Section 6, simulation results are shown for validation of synchronization and anti-synchronization. Finally, conclusions of paper are presented in Section 7. 2. Problem statement for synchronization of chaotic system Consider a chaotic system as a drive system having state vector X m 2 Rn , and P 2 Rnn is system matrix, given by
X_ m ¼ PX m þ f ðX m Þ n
ð1Þ n
where f ðX m Þ : R ! R is nonlinear part of the system. Consider another chaotic system as a slave (or response) system having state vector Y s 2 Rn , and Q 2 Rnn is the system matrix given as in (2).
Y_ s ¼ Q Y s þ gðY s Þ þ U
ð2Þ
where, g(Ys) is nonlinear part of the slave system, and U is nonlinear active controller added in (2) for synchroniza-
tion. Synchronization error e 2 Rn between states X m and Y s is defined as:
e ¼ Y s Xm
ð3Þ
From (1)–(3), we can write error dynamics as:
e_ ¼ Q Y s þ gðY s Þ PX m f ðX m Þ þ U
ð4Þ
Therefore, the synchronization problem is to determine nonlinear controller U, so that
limkeðtÞk ¼ 0;
t!1
8eðtÞ 2 Rn
ð5Þ
Consider a positive definite Lyapunov function [33] as:
VðeÞ ¼
n 1X e2 2 k¼1 k
ð6Þ
where, ek is the kth state error. Assuming that first partial time derivative of ek exists, then
_ VðeÞ ¼
n X ek e_ k
ð7Þ
i¼1
_ is negative defiHence, find U in such a way that VðeÞ nite and Vðe; tÞ ! 1 as eðtÞ ! 1, then the error eðtÞ (states X m and Y s are) is globally asymptotically stable in large. As a result, states of drive and response system are globally asymptotically synchronized. In next Section, problem for anti-synchronization of chaotic system is discussed. 3. Problem statement for anti-synchronization of chaotic system Error eas 2 Rn between two states X m and Y s for antisynchronization of states of drive system (1) and response system (2) using the active control U is defined as:
eas ¼ Y s þ X m
ð8Þ
From (1), (2) and (8) we can write error dynamics as:
e_ as ¼ QY s þ gðY s Þ þ PX m þ f ðX m Þ þ U
ð9Þ
Thus, anti-synchronization problem is to determine the nonlinear controller U, so that it satisfies limt!1 keas ðtÞk ¼ 0; 8eas ðtÞ 2 Rn . Consider a positive definite Lyapunov function,
Vðeas Þ ¼
n 1X e2 2 k¼1 ask
with the assumption that parameters of drive and response systems are known and states are measurable. Hence, problem is to find U so that time derivative of Vðeas Þ exists and must be negative definite as
_ as Þ ¼ Vðe
n X
eask e_ ask
k¼1
along with V(eas) ? 1 as ||eas(t)|| ? 1, then eas is (states, Xm and Ys are) globally asymptotically stable in large, and the states of drive and response systems are globally asymptotically synchronized at the large. Based on the
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
33
given outline for synchronization in Section 2, controller is designed in next section.
systems within unified chaotic system [37]. The dynamics of Bhalekar–Gejji chaotic system is given as:
4. Synchronization of Lu and Bhalekar–Gejji chaotic system
y_ 1 ¼ wy1 y22 þ u1 y_ 2 ¼ lðy3 y2 Þ þ u2 y_ 3 ¼ ay2 by3 þ y1 y2 þ u3
Here, Lu system and Bhalekar–Gejji systems are considered as master and slave respectively. Lu and Chen found in 2002, a critical new chaotic system [33,34] which represents a bridge between the Lorenz and Chen attractors and this new chaotic system is known as Lu system. A unified chaotic system [37] is defined involving Lorenz, Chen and Lu family. Many other chaotic systems like Liu, Pan, Bao, Li are reported. All these systems can be derived from the member of unified chaotic system. Lu system dynamics is given as:
x_ 1 ¼ aðx2 x1 Þ x_ 2 ¼ cx2 x1 x3 x_ 3 ¼ bx3 þ x1 x2
ð10Þ
where x1, x2, x3 are states and a, b, c are the positive parameters of the system (10). The phase plane behaviors for chaotic Lu system are given in the Fig. 1. Chaotic behavior of Lu system is obtained with a = 36, b = 30, c = 20. There are three equilibrium points. The equilibrium points are calculated as E1(0, 0, 0), E2(7.746, 7.746, 20), and E3(7.746, 7.746, 20). Lyapunov exponents are (1.5046, 0, 22.5044) at (0, 0, 0) equilibrium points and thus showing chaotic behavior. The slave system is chosen as a Bhalekar–Gejji chaotic system reported during 2012. This system is neither diffeomorphic nor topologically equivalent with the chaotic
ð11Þ
where, y1, y2, y3 are the states and w < 0, and l, a, b are the positive parameters of the system (11). This system shows two-scroll butterfly-shaped attractor for certain values of parameters. The two-scroll attractor in this system is formed from two one-scroll attractors. Detail forming mechanism of Bhalekar–Gejji using the control parameter is given in [36]. Parameters for chaotic behavior are selected as a = 27.3, b = 1, l = 10, w = 2.667. It has three equilibrium points as E1(0, 0, 0), E2(26.3, 8.3751, 8.3751), and E3(26.3, 8.3751, 8.3751). The Lyapunov exponents are (0.9694, 0.008, 14.6452) at (0, 0, 0) equilibrium point and thus showing chaotic behavior. The phase plane for system (11) with ui = 0, "i are given in Fig. 2. u1, u2, u3 are the active nonlinear controllers to be designed. The synchronization error e 2 R3 is defined as:
ei ¼ yi xi
where i ¼ 1; 2; 3
ð12Þ
The error dynamics equations are obtained as follows:
e_ 1 ¼ we1 þ aðx1 x2 Þ y22 þ wx1 þ u1 e_ 2 ¼ le2 þ lðy3 x2 Þ þ x1 x3 cx2 þ u2 e_ 3 ¼ be3 þ ay2 bx3 þ bx3 þ y1 y2 x1 x2 þ u3
ð13Þ
We need to find the nonlinear active control law for ui, "i in such a manner that the error dynamics of (13) is globally asymptotically stable in the large. Let,
Fig. 1. The phase portraits for chaotic Lu system, (a) on (x1, x2) plane, (b) on (x1, x3) plane, (c) on (x2, x3) plane, (d) on (x1, x2, x3) space.
34
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
Fig. 2. Phase portraits of Bhalekhar–Gejji chaotic system (a) on (y1, y2) plane, (b) on (y1, y3) plane, (c) on (y2, y3) plane, (d) on (y1, y2, y3) space.
u1 ¼ aðx1 x2 Þ þ y22 wx1 u2 ¼ lðy3 x2 Þ x1 x3 þ cx2
ð14Þ
u3 ¼ ay2 þ bx3 bx3 y1 y2 þ x1 x2 Substituting the controller dynamics (14) in error dynamics (13), we have error dynamics as:
e_ 1 ¼ we1 e_ 2 ¼ le2 e_ 3 ¼ be3
ð15Þ
Now, considering a positive definite Lyapunov function as:
anti-synchronization in Section 3, numerical details are given in the next section. 5. Anti-synchronization of Lu and Bhalekar–Gejji chaotic system In this section we discuss anti-synchronization of nonidentical Lu system dynamics (10) and Bhalekar–Gejji system dynamics (11). The anti-synchronization error eas 2 R3 is defined as:
easi ¼ yi þ xi
where i ¼ 1; 2; 3
ð19Þ
The error dynamics for anti-synchronization is given as:
1 VðeÞ ¼ e21 þ e22 þ e23 2
ð16Þ
Assuming first order partial derivatives of (16) exists, we obtained
_ VðeÞ ¼ e1 e_ 1 þ e2 e_ 2 þ e3 e_ 3
_ VðeÞ ¼
e22
l
be23
e_ as3 ¼ y_ 3 þ x_ 3 ¼ beas3 þ bx3 bx3 þ ay2 þ y1 y2 þ x1 x2 þ u3 ð20Þ
ð17Þ
Substituting (15) in (17), we obtained
we21
e_ as1 ¼ y_ 1 þ x_ 1 ¼ weas1 wx1 þ aðx2 x1 Þ y22 þ u1 e_ as2 ¼ y_ 2 þ x_ 2 ¼ leas2 þ lðy3 þ x2 Þ þ cx2 x1 x3 þ u2
Let, the nonlinear control law is defined as
u1 ¼ wx1 aðx2 x1 Þ þ y22 ð18Þ
_ Since w < 0, VðeÞ is a negative definite. Now, as per Lyapunov stability theory [38] error dynamics (15) is globally asymptotically stable in the large for equilibrium state at origin i.e. error dynamics will converge to zero as t ? 1 with the control law in (14). Thus, two chaotic systems Lu (10) and Bhalekar–Gejji (11) are globally asymptotically synchronized for any initial condition. Based on the outline developed for
u2 ¼ lðy3 þ x2 Þ cx2 þ x1 x3
ð21Þ
u3 ¼ bx3 þ bx3 ay2 y1 y2 x1 x2 Then, error dynamics is given as
e_ as1 ¼ weas1 e_ as2 ¼ leas2 e_ as3 ¼ beas3
ð22Þ
Now, consider a positive definite Lyapunov function as:
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
35
Fig. 3. Time response plots of synchronized states of master and slave. (a) First state of master and slave. (b) Second state of master and slave. (c) Third state of master and slave.
Fig. 4. Time response plots of the control input for synchronization between each states of master–slave (a) first control input. (b) Second control input. (c) Third control input.
36
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
Fig. 5. Synchronization error between the states of master and slave systems.
Fig. 6. Time response plots of anti-synchronized states of master and slave. (a) First state of master and slave. (b) Second state of master and slave. (c) Third state of master and slave.
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
37
Fig. 7. Time response plots of control inputs for anti-synchronization between each states of master–slave (a) first control input. (b) Second control input. (c) Third control input.
Fig. 8. Anti-synchronization error between the states of master and slave systems.
38
Vðeas Þ ¼
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
1 2 eas1 þ e2as2 þ e2as3 2
ð23Þ
Assuming first order partial derivatives exist, we obtained
_ as Þ ¼ eas e_ as þ eas e_ as þ eas e_ as Vðe 1 1 2 2 3 3
ð24Þ
Hence, substituting error dynamics (22) in (24), we obtained
_ as Þ ¼ we2 le2 be2 < 0 Vðe as1 as2 as3
ð25Þ
Hence, error dynamics (22) is globally asymptotically stable at the large for equilibrium state at origin i.e. error dynamics will converge to zero as t ? 1 with the proposed control law in (21). Thus, two chaotic systems Lu (10) and Bhalekar–Gejji (11) are globally asymptotically anti-synchronized for any initial condition. Based on the theoretical explanation developed in Sections 4 and 5, corresponding simulation results are given and discussed in next section. 6. Results and discussion Forth-order Runge–Kutta method is used for solving the dynamics (10) and (11) with nonlinear active control law (14) for synchronization, and nonlinear active control law (21) for anti-synchronization. Simulation is done with time step h = 0.005. For chaotic behavior of Lu system (10) and Bhalekar–Gejji system (11), parameters are selected as a = 36, b = 3, c = 20 and a = 27.3, b = 1, l = 10, w = 2.667 respectively for chaotic behavior. 6.1. Synchronization of Lu and Bhalekar–Gejji chaotic dynamics The initial condition for simulation is assumed arbitrarily as xð0Þ ¼ ½ 10 20 30 T and yð0Þ ¼ ½ 17 22 9 T for synchronization. Time response plots of master and slave synchronization of states are given in Fig. 3. Time responses of the control input for synchronization between each states of master–slave are given in Fig. 4. Synchronization errors between the states of master and slave systems are given in Fig. 5. It is apparent from Figs. 3 and 5 that the expected synchronization between master and slave systems is achieved. 6.2. Anti-synchronization of Lu and Bhalekar–Gejji chaotic dynamics The initial condition for simulation is considered as xð0Þ ¼ ½ 10 20 17 T and yð0Þ ¼ ½ 7 7 9 T , for anti-synchronization. Time responses between master and slave anti-synchronized states are given in Fig. 6. Time responses of the control input for anti-synchronization between each states of master–slave are given in Fig. 7. Anti-synchronization errors between the states of master and slave systems are given in Fig. 8. Anti-synchronization of states between master and slave is successfully achieved as seen in the Figs. 6 and 8.
7. Conclusions In this paper global synchronization and anti-synchronization schemes are presented to investigate synchronization and anti-synchronization of chaotic Lu from unified chaotic system and newly developed Bhalekar– Gejji systems. In most of the reported paper of synchronization and anti-synchronization between chaotic systems involving Lu and other system is chosen from the members of unified chaotic system. But Bhalekar–Gejji does not fit in the unified chaotic systems. This paper deals with Lu and Bhalekar–Gejji, a combination which is not reported earlier. Nonlinear active controllers have been proposed to guarantee global asymptotically stability at the large. It has been shown that the master and slave systems are synchronized and anti-synchronized by proper choice of control inputs. Finally, simulations are presented to demonstrate effectiveness of the proposed synchronization and anti-synchronization schemes. The simulation results establish feasibility and effectiveness of proposed theoretical design. Synchronization and anti-synchronization of chaotic system can be used for secure communication and complex dynamical networks. Circuit implementation is in progress and we will be reported in future. Apart from these issues, other control and stabilization scheme can be used for synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems. References [1] Lorenz EN. Deterministic non-periodic flow. J Atmos Sci 1963;20:130–41. [2] Andrievskii BR, Fradkov AL. Control of chaos: methods and applications. II. Applications. Autom Remote Control 2004;65:505–33. [3] Fujisaka H, Yamada T. Stability theory of synchronized motion in coupled-oscillator systems. Prog Theor Phys 1983;69:32–47. [4] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. [5] Pecora LM, Carroll TL. Synchronizing chaotic circuits. IEEE Trans Circuits Syst 1991;38:453–6. [6] Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. [7] Chen HK. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos Solitons Fractals 2005;23:1245–51. [8] Chen Hsien-Keng. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos Solitons Fractals 2005;23: 1245–50. [9] Sundarapandian V. Global chaos synchronization of Lorenz and Pehlivan chaotic systems by nonlinear control. Int J Adv Sci Technol 2011;2:19–28. [10] Liao TL, Tsai SH. Adaptive synchronization of chaotic systems and its applications to secure communications. Chaos Solitons Fractals 2000;11:1387–96. [11] Sundarapandian V. Adaptive synchronization of hyperchaotic Lorenz and hyperchaotic Liu systems. Int J Instrum Control Syst 2011;1:1–18. [12] Yu YG, Zhang SC. Adaptive backstepping synchronization of uncertain chaotic systems. Chaos Solitons Fractals 2004;21:643–9. [13] Tan X, Zhang J, Yang Y. Synchronizing chaotic systems using backstepping design. Chaos Solitons Fractals 2003;16:37–45. [14] P.P. Singh, B.K. Roy, H. Handa, Observer based synchronization of 4D modified Lorenz-Stenflo chaotic system. In: 2013 Annual IEEE india conference (INDICON). (2013). p. 1–6. [15] Guo H, Zhong S. Synchronization criteria of time-delay feedback control system with sector bounded nonlinearity. Appl Math Comput 2009;191:550–7. [16] Yan J, Hung M, Chiang T, Yang Y. Robust synchronization of chaotic systems via adaptive sliding mode control. Phys Lett A 2006;356:220–5.
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39 [17] Chen D, Zhang R, Ma X, Liu S. Chaotic synchronization and antisynchronization for a novel class of multiple chaotic systems via a sliding mode control scheme. Nonlinear Dyn 2012;69:35–55. http:// dx.doi.org/10.1007/s11071- 011-0244-7. [18] Fabiny L, Wiesenfield K. Clustering behaviour of oscillator arrays. Phys Rev A 1991;43:2640–8. [19] Rosenblum MG, Pikovsky AS, Kurths J. From phase to lag synchronization in coupled chaotic oscillators. Phys Rev Lett 1997;78:4193–6. [20] Ge ZM, Chen CC. Phase synchronization of coupled chaotic multiple time scales systems. Chaos Solitons Fractals 2004;20:639–47. [21] Wang YW, Guan ZH. Generalized synchronization of continuous chaotic systems. Chaos Solitons Fractals 2006;27:97–101. [22] Qiang J. Projective synchronization of a new hyper chaotic Lorenz system. Phys Lett A 2007;370:40–5. [23] Jian-Ping Y, Chang-Pin L. Generalized projective synchronization for the chaotic Lorenz systems and the chaotic Chen system. J Shanghai Univ 2006;10:299–304. [24] Yan ZY. Q–S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic system-A symbolic–numeric computation approach. Chaos 2005;15:20–9. [25] Hramov AE, Koronovski AA. An approach to chaotic synchronization, chaos: an interdisciplinary. J Nonlinear Sci 2004;14:603–10. [26] Ju Park H. Exponential synchronization of the Genesio–Tesi chaotic system via a novel feedback control. Phys Scr 2007;76:617–22. [27] Zhang X, Zhu H. Anti-synchronization of two different hyperchaotic systems via active and adaptive control. Int J Nonlinear Sci 2008;6:216–23.
39
[28] Li GH. Synchronization and anti-synchronization of Colpitts oscillators using active control. Chaos Solitons Fractals 2005;26: 87–93. [29] Khan A, Prasad RP. Anti-synchronization of Pan and Lorenz–Lu–Liu– Cai chaotic systems by active nonlinear control. Int J Artif Life Res 2012;3:15–25. [30] Li W, Chen X, Zhiping S. Anti-synchronization of two different chaotic systems. Physica A 2008;387:3747–50. [31] Li GH, Zhou SP. Anti-synchronization in different chaotic systems. Chaos Solitons Fractals 2007;32:516–20. [32] Bai EW, Lonngren KE. Synchronization of two lorenz systems using active control. Chaos Solitons Fractals 1997;8(1):51–8. [33] Lu J, Chen G. A new chaotic attractor coined. Int J Bifurcation Chaos 2002;12:659–61. [34] Lu J, Chen G, Zhang S. Dynamical analysis of a new chaotic attractor. Int J Bifurcation Chaos 2002;12:1001–15. [35] S.B. Bhalekar, V. Daftardar-Gejji, A new chaotic dynamical system and its synchronization. In: Proceedings of the international conference on mathematical sciences in honor of Prof. A. M. Mathai, Palai, Kerla, India, 2011. [36] Bhalekar SB. Forming mechanizm of Bhalekar–Gejji chaotic dynamical system. Am J Comput Appl Math 2012;2:257–9. [37] Lu J, Chen G, Cheng D, Celikovsky S. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurcation Chaos 2002;12:2917–26. [38] Hahn W. The stability of motion. Berlin: Springer; 1967.
Contents lists available at ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems using nonlinear active control Piyush Pratap Singh ⇑, Jay Prakash Singh, B. K. Roy NIT Silchar, Silchar, Assam 788 010, India
a r t i c l e
i n f o
Article history: Received 5 April 2014 Accepted 7 September 2014
a b s t r a c t This paper aims at synchronization and anti-synchronization between Lu chaotic system, a member of unified chaotic system, and recently developed Bhalekar–Gejji chaotic system, a system which cannot be derived from the member of unified chaotic system. These synchronization and anti-synchronization have been achieved by using nonlinear active control since the parameters of both the systems are known. Lyapunov stability theory is used and required condition is derived to ensure the stability of error dynamics. Controller is designed by using the sum of relevant variables in chaotic systems. Simulation results suggest that proposed scheme is working satisfactorily. 2014 Elsevier Ltd. All rights reserved.
1. Introduction The important properties available in any chaotic dynamical system are sensitive to initial condition, aperiodic long time behavior, and are deterministic in nature [1]. Chaotic system has potential applications in physics, neurobiology, earth science, electrical engineering, and many other fields [2]. Motivated by stated applications, chaotic systems are also being applied in field of communication with the use of synchronization concept [2]. Secure communication is a major issue today. It is expected that synchronization of two chaotic systems may help towards secure communication. Motivated by such practical requirement, authors are motivated to consider synchronization and anti-synchronization between well-known Lu chaotic system, a member of unified chaotic system, and Bhalekar–Gejji chaotic systems developed during 2012. There are two methods for synchronization of chaotic systems known as drive-response scheme and coupling ⇑ Corresponding author. E-mail addresses: [email protected] (P. P. Singh), [email protected] (J. P. Singh), [email protected] (B. K. Roy). http://dx.doi.org/10.1016/j.chaos.2014.09.005 0960-0779/ 2014 Elsevier Ltd. All rights reserved.
scheme. Drive-response is also called as master–slave system which is widely used by the researchers. In this scheme the output of master (drive) system is used to control the slave system. Synchronization means to achieve zero error between states of master and slave systems. Firstly, the phenomenon of chaotic synchronization is illustrated by Yamada and Fujisaka [3]. But interest of the researchers increases after the pioneer work of Pecora and Carroll [4,5], where, two chaotic systems with different initial conditions are synchronized. Various control schemes have been developed for synchronization of chaotic system in the last two decades such as PC method [4], OGY method [6], active control [7–9], adaptive control [10,11], backstepping control [12,13], observer design [14], time delay feedback control [15], sliding mode control [16,17], etc. Many more techniques are also available related to synchronization like Complete synchronization [18], Lag synchronization [19], Phase synchronization [20], generalized synchronization [21], projective synchronization [22], generalized projective [23], QS synchronization [24], time scale synchronization [25], exponential synchronization [26], etc.
32
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
Another important phenomenon absorbed between two dynamical systems is anti-synchronization. The purpose of anti-synchronization is to use the output of master system to control the slave system. During anti-synchronization state of both master and slave systems shows the same amplitude but opposite in phase. Anti-synchronization causes the sum of state of master and slave systems converge asymptotically to zero. Active control is used as tool for anti-synchronization [27–31] by last few years. Synchronization of chaotic system using nonlinear active control was first reported by Bai and Lonngren [32]. So, many papers are available in literature based on use of nonlinear active control as a tool for synchronization and anti-synchronization of chaotic systems in last few years. In most of the reported papers of synchronization and anti-synchronization between chaotic systems involving Lu system [33,34] the other system is chosen from a set of chaotic systems which are derived from the members of unified chaotic system. But Bhalekar–Gejji chaotic [35,36] system does not fit in unified chaotic system. Very few papers have been reported with Lu and other chaotic system which cannot be derived from the members of unified chaotic system [37] and present paper is one of them. In this paper we are considering the problem of synchronization and antisynchronization both. Further, Bhalekar–Gejji chaotic system is newly developed chaotic system in 2012. Global synchronization and anti-synchronization of Bhalekar–Gejji chaotic system with any other chaotic system using nonlinear active control is reported for first time which reflects the novelty of this paper. In first part of the paper active control structure is designed based upon the error dynamics of master and slave systems. In second part, global synchronization is achieved by using the Lyapunov’s stability theory [38]. The organization of paper is as follows. In Sections 2 and 3, problem statement for synchronization and antisynchronization of two chaotic systems is discussed. Global synchronization of Lu and Bhalekar–Gejji chaotic system is discussed in Section 4 followed by global antisynchronization in Section 5. In Section 6, simulation results are shown for validation of synchronization and anti-synchronization. Finally, conclusions of paper are presented in Section 7. 2. Problem statement for synchronization of chaotic system Consider a chaotic system as a drive system having state vector X m 2 Rn , and P 2 Rnn is system matrix, given by
X_ m ¼ PX m þ f ðX m Þ n
ð1Þ n
where f ðX m Þ : R ! R is nonlinear part of the system. Consider another chaotic system as a slave (or response) system having state vector Y s 2 Rn , and Q 2 Rnn is the system matrix given as in (2).
Y_ s ¼ Q Y s þ gðY s Þ þ U
ð2Þ
where, g(Ys) is nonlinear part of the slave system, and U is nonlinear active controller added in (2) for synchroniza-
tion. Synchronization error e 2 Rn between states X m and Y s is defined as:
e ¼ Y s Xm
ð3Þ
From (1)–(3), we can write error dynamics as:
e_ ¼ Q Y s þ gðY s Þ PX m f ðX m Þ þ U
ð4Þ
Therefore, the synchronization problem is to determine nonlinear controller U, so that
limkeðtÞk ¼ 0;
t!1
8eðtÞ 2 Rn
ð5Þ
Consider a positive definite Lyapunov function [33] as:
VðeÞ ¼
n 1X e2 2 k¼1 k
ð6Þ
where, ek is the kth state error. Assuming that first partial time derivative of ek exists, then
_ VðeÞ ¼
n X ek e_ k
ð7Þ
i¼1
_ is negative defiHence, find U in such a way that VðeÞ nite and Vðe; tÞ ! 1 as eðtÞ ! 1, then the error eðtÞ (states X m and Y s are) is globally asymptotically stable in large. As a result, states of drive and response system are globally asymptotically synchronized. In next Section, problem for anti-synchronization of chaotic system is discussed. 3. Problem statement for anti-synchronization of chaotic system Error eas 2 Rn between two states X m and Y s for antisynchronization of states of drive system (1) and response system (2) using the active control U is defined as:
eas ¼ Y s þ X m
ð8Þ
From (1), (2) and (8) we can write error dynamics as:
e_ as ¼ QY s þ gðY s Þ þ PX m þ f ðX m Þ þ U
ð9Þ
Thus, anti-synchronization problem is to determine the nonlinear controller U, so that it satisfies limt!1 keas ðtÞk ¼ 0; 8eas ðtÞ 2 Rn . Consider a positive definite Lyapunov function,
Vðeas Þ ¼
n 1X e2 2 k¼1 ask
with the assumption that parameters of drive and response systems are known and states are measurable. Hence, problem is to find U so that time derivative of Vðeas Þ exists and must be negative definite as
_ as Þ ¼ Vðe
n X
eask e_ ask
k¼1
along with V(eas) ? 1 as ||eas(t)|| ? 1, then eas is (states, Xm and Ys are) globally asymptotically stable in large, and the states of drive and response systems are globally asymptotically synchronized at the large. Based on the
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
33
given outline for synchronization in Section 2, controller is designed in next section.
systems within unified chaotic system [37]. The dynamics of Bhalekar–Gejji chaotic system is given as:
4. Synchronization of Lu and Bhalekar–Gejji chaotic system
y_ 1 ¼ wy1 y22 þ u1 y_ 2 ¼ lðy3 y2 Þ þ u2 y_ 3 ¼ ay2 by3 þ y1 y2 þ u3
Here, Lu system and Bhalekar–Gejji systems are considered as master and slave respectively. Lu and Chen found in 2002, a critical new chaotic system [33,34] which represents a bridge between the Lorenz and Chen attractors and this new chaotic system is known as Lu system. A unified chaotic system [37] is defined involving Lorenz, Chen and Lu family. Many other chaotic systems like Liu, Pan, Bao, Li are reported. All these systems can be derived from the member of unified chaotic system. Lu system dynamics is given as:
x_ 1 ¼ aðx2 x1 Þ x_ 2 ¼ cx2 x1 x3 x_ 3 ¼ bx3 þ x1 x2
ð10Þ
where x1, x2, x3 are states and a, b, c are the positive parameters of the system (10). The phase plane behaviors for chaotic Lu system are given in the Fig. 1. Chaotic behavior of Lu system is obtained with a = 36, b = 30, c = 20. There are three equilibrium points. The equilibrium points are calculated as E1(0, 0, 0), E2(7.746, 7.746, 20), and E3(7.746, 7.746, 20). Lyapunov exponents are (1.5046, 0, 22.5044) at (0, 0, 0) equilibrium points and thus showing chaotic behavior. The slave system is chosen as a Bhalekar–Gejji chaotic system reported during 2012. This system is neither diffeomorphic nor topologically equivalent with the chaotic
ð11Þ
where, y1, y2, y3 are the states and w < 0, and l, a, b are the positive parameters of the system (11). This system shows two-scroll butterfly-shaped attractor for certain values of parameters. The two-scroll attractor in this system is formed from two one-scroll attractors. Detail forming mechanism of Bhalekar–Gejji using the control parameter is given in [36]. Parameters for chaotic behavior are selected as a = 27.3, b = 1, l = 10, w = 2.667. It has three equilibrium points as E1(0, 0, 0), E2(26.3, 8.3751, 8.3751), and E3(26.3, 8.3751, 8.3751). The Lyapunov exponents are (0.9694, 0.008, 14.6452) at (0, 0, 0) equilibrium point and thus showing chaotic behavior. The phase plane for system (11) with ui = 0, "i are given in Fig. 2. u1, u2, u3 are the active nonlinear controllers to be designed. The synchronization error e 2 R3 is defined as:
ei ¼ yi xi
where i ¼ 1; 2; 3
ð12Þ
The error dynamics equations are obtained as follows:
e_ 1 ¼ we1 þ aðx1 x2 Þ y22 þ wx1 þ u1 e_ 2 ¼ le2 þ lðy3 x2 Þ þ x1 x3 cx2 þ u2 e_ 3 ¼ be3 þ ay2 bx3 þ bx3 þ y1 y2 x1 x2 þ u3
ð13Þ
We need to find the nonlinear active control law for ui, "i in such a manner that the error dynamics of (13) is globally asymptotically stable in the large. Let,
Fig. 1. The phase portraits for chaotic Lu system, (a) on (x1, x2) plane, (b) on (x1, x3) plane, (c) on (x2, x3) plane, (d) on (x1, x2, x3) space.
34
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
Fig. 2. Phase portraits of Bhalekhar–Gejji chaotic system (a) on (y1, y2) plane, (b) on (y1, y3) plane, (c) on (y2, y3) plane, (d) on (y1, y2, y3) space.
u1 ¼ aðx1 x2 Þ þ y22 wx1 u2 ¼ lðy3 x2 Þ x1 x3 þ cx2
ð14Þ
u3 ¼ ay2 þ bx3 bx3 y1 y2 þ x1 x2 Substituting the controller dynamics (14) in error dynamics (13), we have error dynamics as:
e_ 1 ¼ we1 e_ 2 ¼ le2 e_ 3 ¼ be3
ð15Þ
Now, considering a positive definite Lyapunov function as:
anti-synchronization in Section 3, numerical details are given in the next section. 5. Anti-synchronization of Lu and Bhalekar–Gejji chaotic system In this section we discuss anti-synchronization of nonidentical Lu system dynamics (10) and Bhalekar–Gejji system dynamics (11). The anti-synchronization error eas 2 R3 is defined as:
easi ¼ yi þ xi
where i ¼ 1; 2; 3
ð19Þ
The error dynamics for anti-synchronization is given as:
1 VðeÞ ¼ e21 þ e22 þ e23 2
ð16Þ
Assuming first order partial derivatives of (16) exists, we obtained
_ VðeÞ ¼ e1 e_ 1 þ e2 e_ 2 þ e3 e_ 3
_ VðeÞ ¼
e22
l
be23
e_ as3 ¼ y_ 3 þ x_ 3 ¼ beas3 þ bx3 bx3 þ ay2 þ y1 y2 þ x1 x2 þ u3 ð20Þ
ð17Þ
Substituting (15) in (17), we obtained
we21
e_ as1 ¼ y_ 1 þ x_ 1 ¼ weas1 wx1 þ aðx2 x1 Þ y22 þ u1 e_ as2 ¼ y_ 2 þ x_ 2 ¼ leas2 þ lðy3 þ x2 Þ þ cx2 x1 x3 þ u2
Let, the nonlinear control law is defined as
u1 ¼ wx1 aðx2 x1 Þ þ y22 ð18Þ
_ Since w < 0, VðeÞ is a negative definite. Now, as per Lyapunov stability theory [38] error dynamics (15) is globally asymptotically stable in the large for equilibrium state at origin i.e. error dynamics will converge to zero as t ? 1 with the control law in (14). Thus, two chaotic systems Lu (10) and Bhalekar–Gejji (11) are globally asymptotically synchronized for any initial condition. Based on the outline developed for
u2 ¼ lðy3 þ x2 Þ cx2 þ x1 x3
ð21Þ
u3 ¼ bx3 þ bx3 ay2 y1 y2 x1 x2 Then, error dynamics is given as
e_ as1 ¼ weas1 e_ as2 ¼ leas2 e_ as3 ¼ beas3
ð22Þ
Now, consider a positive definite Lyapunov function as:
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
35
Fig. 3. Time response plots of synchronized states of master and slave. (a) First state of master and slave. (b) Second state of master and slave. (c) Third state of master and slave.
Fig. 4. Time response plots of the control input for synchronization between each states of master–slave (a) first control input. (b) Second control input. (c) Third control input.
36
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
Fig. 5. Synchronization error between the states of master and slave systems.
Fig. 6. Time response plots of anti-synchronized states of master and slave. (a) First state of master and slave. (b) Second state of master and slave. (c) Third state of master and slave.
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
37
Fig. 7. Time response plots of control inputs for anti-synchronization between each states of master–slave (a) first control input. (b) Second control input. (c) Third control input.
Fig. 8. Anti-synchronization error between the states of master and slave systems.
38
Vðeas Þ ¼
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39
1 2 eas1 þ e2as2 þ e2as3 2
ð23Þ
Assuming first order partial derivatives exist, we obtained
_ as Þ ¼ eas e_ as þ eas e_ as þ eas e_ as Vðe 1 1 2 2 3 3
ð24Þ
Hence, substituting error dynamics (22) in (24), we obtained
_ as Þ ¼ we2 le2 be2 < 0 Vðe as1 as2 as3
ð25Þ
Hence, error dynamics (22) is globally asymptotically stable at the large for equilibrium state at origin i.e. error dynamics will converge to zero as t ? 1 with the proposed control law in (21). Thus, two chaotic systems Lu (10) and Bhalekar–Gejji (11) are globally asymptotically anti-synchronized for any initial condition. Based on the theoretical explanation developed in Sections 4 and 5, corresponding simulation results are given and discussed in next section. 6. Results and discussion Forth-order Runge–Kutta method is used for solving the dynamics (10) and (11) with nonlinear active control law (14) for synchronization, and nonlinear active control law (21) for anti-synchronization. Simulation is done with time step h = 0.005. For chaotic behavior of Lu system (10) and Bhalekar–Gejji system (11), parameters are selected as a = 36, b = 3, c = 20 and a = 27.3, b = 1, l = 10, w = 2.667 respectively for chaotic behavior. 6.1. Synchronization of Lu and Bhalekar–Gejji chaotic dynamics The initial condition for simulation is assumed arbitrarily as xð0Þ ¼ ½ 10 20 30 T and yð0Þ ¼ ½ 17 22 9 T for synchronization. Time response plots of master and slave synchronization of states are given in Fig. 3. Time responses of the control input for synchronization between each states of master–slave are given in Fig. 4. Synchronization errors between the states of master and slave systems are given in Fig. 5. It is apparent from Figs. 3 and 5 that the expected synchronization between master and slave systems is achieved. 6.2. Anti-synchronization of Lu and Bhalekar–Gejji chaotic dynamics The initial condition for simulation is considered as xð0Þ ¼ ½ 10 20 17 T and yð0Þ ¼ ½ 7 7 9 T , for anti-synchronization. Time responses between master and slave anti-synchronized states are given in Fig. 6. Time responses of the control input for anti-synchronization between each states of master–slave are given in Fig. 7. Anti-synchronization errors between the states of master and slave systems are given in Fig. 8. Anti-synchronization of states between master and slave is successfully achieved as seen in the Figs. 6 and 8.
7. Conclusions In this paper global synchronization and anti-synchronization schemes are presented to investigate synchronization and anti-synchronization of chaotic Lu from unified chaotic system and newly developed Bhalekar– Gejji systems. In most of the reported paper of synchronization and anti-synchronization between chaotic systems involving Lu and other system is chosen from the members of unified chaotic system. But Bhalekar–Gejji does not fit in the unified chaotic systems. This paper deals with Lu and Bhalekar–Gejji, a combination which is not reported earlier. Nonlinear active controllers have been proposed to guarantee global asymptotically stability at the large. It has been shown that the master and slave systems are synchronized and anti-synchronized by proper choice of control inputs. Finally, simulations are presented to demonstrate effectiveness of the proposed synchronization and anti-synchronization schemes. The simulation results establish feasibility and effectiveness of proposed theoretical design. Synchronization and anti-synchronization of chaotic system can be used for secure communication and complex dynamical networks. Circuit implementation is in progress and we will be reported in future. Apart from these issues, other control and stabilization scheme can be used for synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems. References [1] Lorenz EN. Deterministic non-periodic flow. J Atmos Sci 1963;20:130–41. [2] Andrievskii BR, Fradkov AL. Control of chaos: methods and applications. II. Applications. Autom Remote Control 2004;65:505–33. [3] Fujisaka H, Yamada T. Stability theory of synchronized motion in coupled-oscillator systems. Prog Theor Phys 1983;69:32–47. [4] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. [5] Pecora LM, Carroll TL. Synchronizing chaotic circuits. IEEE Trans Circuits Syst 1991;38:453–6. [6] Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. [7] Chen HK. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos Solitons Fractals 2005;23:1245–51. [8] Chen Hsien-Keng. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos Solitons Fractals 2005;23: 1245–50. [9] Sundarapandian V. Global chaos synchronization of Lorenz and Pehlivan chaotic systems by nonlinear control. Int J Adv Sci Technol 2011;2:19–28. [10] Liao TL, Tsai SH. Adaptive synchronization of chaotic systems and its applications to secure communications. Chaos Solitons Fractals 2000;11:1387–96. [11] Sundarapandian V. Adaptive synchronization of hyperchaotic Lorenz and hyperchaotic Liu systems. Int J Instrum Control Syst 2011;1:1–18. [12] Yu YG, Zhang SC. Adaptive backstepping synchronization of uncertain chaotic systems. Chaos Solitons Fractals 2004;21:643–9. [13] Tan X, Zhang J, Yang Y. Synchronizing chaotic systems using backstepping design. Chaos Solitons Fractals 2003;16:37–45. [14] P.P. Singh, B.K. Roy, H. Handa, Observer based synchronization of 4D modified Lorenz-Stenflo chaotic system. In: 2013 Annual IEEE india conference (INDICON). (2013). p. 1–6. [15] Guo H, Zhong S. Synchronization criteria of time-delay feedback control system with sector bounded nonlinearity. Appl Math Comput 2009;191:550–7. [16] Yan J, Hung M, Chiang T, Yang Y. Robust synchronization of chaotic systems via adaptive sliding mode control. Phys Lett A 2006;356:220–5.
P. P. Singh et al. / Chaos, Solitons & Fractals 69 (2014) 31–39 [17] Chen D, Zhang R, Ma X, Liu S. Chaotic synchronization and antisynchronization for a novel class of multiple chaotic systems via a sliding mode control scheme. Nonlinear Dyn 2012;69:35–55. http:// dx.doi.org/10.1007/s11071- 011-0244-7. [18] Fabiny L, Wiesenfield K. Clustering behaviour of oscillator arrays. Phys Rev A 1991;43:2640–8. [19] Rosenblum MG, Pikovsky AS, Kurths J. From phase to lag synchronization in coupled chaotic oscillators. Phys Rev Lett 1997;78:4193–6. [20] Ge ZM, Chen CC. Phase synchronization of coupled chaotic multiple time scales systems. Chaos Solitons Fractals 2004;20:639–47. [21] Wang YW, Guan ZH. Generalized synchronization of continuous chaotic systems. Chaos Solitons Fractals 2006;27:97–101. [22] Qiang J. Projective synchronization of a new hyper chaotic Lorenz system. Phys Lett A 2007;370:40–5. [23] Jian-Ping Y, Chang-Pin L. Generalized projective synchronization for the chaotic Lorenz systems and the chaotic Chen system. J Shanghai Univ 2006;10:299–304. [24] Yan ZY. Q–S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic system-A symbolic–numeric computation approach. Chaos 2005;15:20–9. [25] Hramov AE, Koronovski AA. An approach to chaotic synchronization, chaos: an interdisciplinary. J Nonlinear Sci 2004;14:603–10. [26] Ju Park H. Exponential synchronization of the Genesio–Tesi chaotic system via a novel feedback control. Phys Scr 2007;76:617–22. [27] Zhang X, Zhu H. Anti-synchronization of two different hyperchaotic systems via active and adaptive control. Int J Nonlinear Sci 2008;6:216–23.
39
[28] Li GH. Synchronization and anti-synchronization of Colpitts oscillators using active control. Chaos Solitons Fractals 2005;26: 87–93. [29] Khan A, Prasad RP. Anti-synchronization of Pan and Lorenz–Lu–Liu– Cai chaotic systems by active nonlinear control. Int J Artif Life Res 2012;3:15–25. [30] Li W, Chen X, Zhiping S. Anti-synchronization of two different chaotic systems. Physica A 2008;387:3747–50. [31] Li GH, Zhou SP. Anti-synchronization in different chaotic systems. Chaos Solitons Fractals 2007;32:516–20. [32] Bai EW, Lonngren KE. Synchronization of two lorenz systems using active control. Chaos Solitons Fractals 1997;8(1):51–8. [33] Lu J, Chen G. A new chaotic attractor coined. Int J Bifurcation Chaos 2002;12:659–61. [34] Lu J, Chen G, Zhang S. Dynamical analysis of a new chaotic attractor. Int J Bifurcation Chaos 2002;12:1001–15. [35] S.B. Bhalekar, V. Daftardar-Gejji, A new chaotic dynamical system and its synchronization. In: Proceedings of the international conference on mathematical sciences in honor of Prof. A. M. Mathai, Palai, Kerla, India, 2011. [36] Bhalekar SB. Forming mechanizm of Bhalekar–Gejji chaotic dynamical system. Am J Comput Appl Math 2012;2:257–9. [37] Lu J, Chen G, Cheng D, Celikovsky S. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurcation Chaos 2002;12:2917–26. [38] Hahn W. The stability of motion. Berlin: Springer; 1967.