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Hybrid Synchronization of Lu and Bhalekar-Gejji Chaotic Systems Using Nonlinear Active Control
Third International Conference on Advances in Control and Optimization of Dynamical Systems March 13-15, 2014. Kanpur, India
Hybrid Synchronization of Lu and Bhalekar-Gejji Chaotic Systems Using Nonlinear Active Control Jay Prakash Singh*. Piyush Pratap Singh,** B K Roy***
*National Institute of Technology Silchar, Silchar, Assam India (e-mail: [email protected]). ** National Institute of Technology Silchar, Silchar, Assam India (e-mail: [email protected]). *** National Institute of Technology Silchar, Silchar, Assam India (e-mail:[email protected]). Abstract: Hybrid synchronization based on the active control theory between two different chaotic systems, i.e. Lu and Bhalekar-Gejji is reported in this paper. Lu chaotic system is taken as master and Bhalekar-Gejji chaotic system as slave. Stabilization of error dynamics is achieved by satisfying the Lyapunov stability conditions. Control is designed by using the relevant variables of master and slave systems. Simulation is presented for verification of proposed scheme. Keywords: Active Control, Asymptotic stability, Chaotic Behavior, Master-Slave System, Nonlinear Control, Synchronization.
and Xu et al., 2007, Jian-Ping et al., 2006), Exponential synchronization (Park, 2007). In similar way, antisynchronization between master-slave systems can be studied. Two chaotic systems (master-slave) will be antisynchronize when their sum of state will be converge to zero asymptotically with time, their amplitude of the state will be same but they will have opposite sign ( Li and Zhou et al., 2007, Liu et al., 2000, Sawalh et al., 2008, Ho et at., 2002b). Hybrid synchronization is the phenomenon when complete synchronization and anti-synchronization occur together between the states of two chaotic systems.
1. INTRODUCTION The important property available in any Chaotic/Dynamical system are sensitivity to initial condition, aperiodic long time behavior, and deterministic in nature (Lorenz, 1963). When the states of any chaotic system drive the state of another system, this phenomenon is called Synchronization. Complete synchronization is the phenomenon in which the differences of the states of synchronized system converge to zero. Synchronization of two identical chaotic systems with different initial condition was first illustrated by (Pecora and Caroll, 1990).
In last decade some paper has absorbed on hybrid synchronization of different chaotic and hyperchaotic system such as; Hybrid synchronization of hyperchaotic Chen system (Sudheer et al., 2009), hybrid synchronization of hyperchaotic Lu system (Sebastian et al., 2009), hybrid synchronization of hyperchaotic Lu system (Sebastian et al., 2009), adaptive hybrid synchronization of two different chaotic system Chen and Lu system (Wei et al., 2010), hybrid synchronization of hyperchaotic Lu system based on Passive control (Xia et al., 2010), hybrid synchronization of Liu and Chen system by active nonlinear control (Sundarpandian, 2011), hybrid synchronization in hyperchaotic Lorenz, Qi system (Yaoyao, 2012), hybrid synchronization of Arneodo and Rossler chaotic system by active nonlinear control (Sundarpandian, 2012), hybrid synchronization of Chen and Lu system (Zheng et al., 2012).
Drive (master) -response (slave) scheme for synchronization of chaotic system is widely used. In this scheme the output of master (drive) system is use to control slave (slave) system. Various schemes have been developed for synchronization of chaotic system in the last two decade such as PC method (Pecora and Carroll, 1990, 1991), OGY method (Ott et al., 1990), active control method (Ho et al., 2002a, Chen, 2005, Bai et al., 1997, Vincet, 2005), adaptive control method (Liao et al., 2000, Sundarpandian, 2013, Yu, 2004), backstepping control method (Park, 2006, Tan et al., 2003), sampled data feedback method (Zhao, et al., 2008), time delay feedback method (Guo et al., 2009), sliding mode control method (Yan et al., 2006, Chen, et al., 2012, Slotine at al., 1983, Utkin, 1993) etc.
Here, we discuss the hybrid synchronization of chaotic Lu system (Lu and Chen, 2002) and chaotic Bhalekar-Gejji systems (Bhalekar et al., 2011, 2012) system, using nonlinear active control.
So many more technique are also available related to synchronization like: Complete synchronization (Fabiny et al., 1991), Lag synchronization (Rosenblum et al., 1997), Phase synchronization (Ge et al., 2004), generalized synchronization (Kocarev et al., 1996), projective synchronization (Li et al., 2001), generalized projective (Li
978-3-902823-60-1 © 2014 IFAC
This paper is organized as follows, In section (2), we discuss about the problem statement for hybrid synchronization of 292
10.3182/20140313-3-IN-3024.00069
2014 ACODS March 13-15, 2014. Kanpur, India
two different chaotic systems. In section (3) we discuss the hybrid synchronization of chaotic Lu system and chaotic Bhalekar-Gejji system. In section (4), simulation results are given for illustration and verification of theory for hybrid synchronization. Finally, conclusion is drawn in section (5).
Now, the asymptotical stabilization of the equation (6) is obtained by designing a controller for stabilizing the error dynamical system (4) at origin. By using the Lyapunov stability theory (Hahn 1976, Khalil 1996), positive definite Lyapunov function
2. PROBLEM STATEMENT FOR HYBRID SYNCHRONIZATION
∑
where, is the order of the system. Assuming that parameters of the drive and response are known and states are measurable. The continuous first partial derivative of (7) is
Consider a chaotic system as drive system: ̇
(1) ̇
where, is state vector, and is the matrix for system parameter, and is the nonlinear part of the system. Consider another chaotic system as a slave system ̇
(7)
∑
(8)
‖ is negative definite, and also as ‖ , then states and are globally asymptotically stable in large, and also the state of drive and response system are globally asymptotically hybrid-synchronized.
(2) 3. HYBRID SYNCHRONIZATION OF LU AND BHALEKAR-GEJJI CHAOTIC SYSTEM
, and
is the matrix for is the non-linear part of is the active non-linear controller to
Here, we apply nonlinear active control technique for the synchronization of Lu system (Lu and Chen, 2002) and Bhalekar-Gejji (Bhalekar et al., 2011, 2012). Master (or drive) system is Lu system dynamics is given:
Our goal is to design an appropriate controller such that state vector of the response system asymptotically approaches a given functional relationship of the state vector of the drive system. State vector and will be state of two identical chaotic system if and . State and will be state of two different chaotic system if and . For our problem functional relationship between master and slave system must include both synchronization and antisynchronization. Therefore relationship can be written as
̇ ̇
(9) ̇
where, , , are the states of system (9) and a, b, c are the parameter of system (9) which are positive. Lu attractors are given in the Fig. 1.
(3) when, , and , the complete synchronization, antisynchronization is obtained respectively, we call this scheme as hybrid synchronization. We, define hybrid synchronization error vector as
(b) 40
20
30
3
2
x state
(a) 40
x state
Having state vector system parameter and the slave system and be designed.
0
20
(4) -20 -20
Now, our aim for hybrid synchronization of chaotic system (1) and (2), we design active control , such that trajectory of system (2) with any initial condition can asymptotically approach the system (1) with initial condition , i.e.
‖
(d)
40
,
20
3
20
0 50 0 0
20
40
x 2 state
Fig. 1. Lu Chaotic attractors (a) on the plane, (c) on the plane, (d) on the
x 2 state
-50 -50
0 x 1 state
50
plane, (b) on the plane.
The slave system is described as Bhalekar-Gejji dynamics
‖ ̇
So, hybrid synchronization error can be written from (1), (3), (5) ̇
20
(c)
30
10 -20
We say that anti-synchronization of two systems (1) and (2) are achieved if the following equation holds: ‖
0 x 1 state
x state
(5)
We say that complete synchronization of two systems (1) and (2) is achieved if the following equation holds: ‖
10 -20
40
40
3
‖
20 x 1 state
40
x state
‖
0
̇
(10) ̇
(6)
Where , , , 293
,
are the states of the system (10) and , >0 are parameters of the system (10), and
2014 ACODS March 13-15, 2014. Kanpur, India
̇
, , are nonlinear active controller element which is to be designed. Bhalekar-Gejji attractors are given in the Fig. 2. (a)
30
y state
10
y state
40
-10 -20 -20
(16)
̇
(17)
is negative definite function on value is .
20
for condition as parameter
Now, we can say that according to Lyapunov stability theory (Hahn 1976, Khalil 1996) error dynamics (12) globally asymptotically stable for equilibrium state at origin i.e. error dynamics will converge to zero as t . We obtained the following result.
10 -10
0 y 1 state
10
0 -20
20
-10
0 y 1 state
10
20
(d)
(c) 40 40
y state
30
0 20
10 0 -20
Remarks. The chaotic Lu (9) and chaotic Bhalekar-Gejji (10) system are exponentially and globally hybrid synchronized for any value of initial conditions with the nonlinear controller defined by (13).
20
3
20
3
y state
̇
3
2
0
̇ +
Substituting the error dynamics (14) into (16) we obtained
(b)
20
̇ +
0 -10
0 y 2 state
10
20
y 2 state
0 y 1 state
20
4. SIMULATION RESULTS
Fig. 2. Bhalekhar-Gejji Chaotic attractors (a) on the plane, (c) on the plane, (d) on the
The hybrid-synchronization error
-20 -20
plane, (b) on the plane.
We are using fourth-order Runga-Kutta method for solving the dynamics (9), and (10) and active nonlinear controller (13) and simulating the result with time step h= .
which is defined as: (11)
where,
For chaotic behavior of Lu system the parameter is selected as follows (Lu and Chen, 2002), and the parameters of the Bhalekar-Gejji system for chaotic behavior are selected as following (Bhalekar, 2012).
̇ , and for synchronization and for anti-synchronization. Hence, (11) can be written
as:
The initial condition for plotting the synchronization of state, controller, error dynamics of the master (9) and slave system (10) are as and .
States are synchronized and anti-synchronized alternatively. The error dynamics equation is obtained as follows:
4.1 Time Response plots of Master and Slave states ̇ ̇
(12) x &y
1
20 x1
0
y1
1
̇
-20
(13)
200
300
400 500 600 t ( time in seconds ) (a)
700
800
900
20
y2
-20
3 3
x &y
̇ (14) ̇
100
200
300
400 500 600 t ( time in seconds ) (b)
700
800
900
40 20
1000
x3
0 -20
1000
x2
0 0
Substituting the controller dynamics (13) into error dynamics (12), we obtain the error dynamics as: ̇
100
2
x &y
2
We consider the active nonlinear controller for above error dynamics (12) are as follows:
0
y3 0
100
200
300
400 500 600 t ( time in seconds ) (c)
700
800
900
1000
Fig. 3. Time Response plots of hybrid synchronized states of Master and Slave. (a) Time response of first state of Master and Slave. (b) Time response of second state of Master and Slave. (c) Time response of third state of Master and Slave.
Now, considering positive definite Lyapunov function on defined by (15) Taking the continuous first partial derivative of (15), we obtained the following 294
2014 ACODS March 13-15, 2014. Kanpur, India
4.2 Control inputs for hybrid synchronization between Master-Slave
REFERENCES
control input u
1
Bai, E.W. and Lonngran, E.E. (1997). Synchronization of two Lorenz systems using active control. Chaos Solitons and Fractals, 8, 1, 51–58. Bhalekar, S., Gejji, V.D. (2011). A new chaotic dynamical system and its synchronization, Proceedings of the international conference on mathematical sciences. Kerla, India. Bhalekar, S. B. (2012). Forming Mechanism of Bhalekar Gejji Chaotic Dynamical System, American Journal of Computational and Applied Mathematics. 2, 6, 257-259. Chen, H. K. (2005). Global chaos synchronization of new chaotic systems via nonlinear control. Chaos, Solitons and Fractals, 23, 1245-1251. Chen, D., Zhang, R., Ma, X., Liu, S. (2012). Chaotic synchronization and anti-synchronization for a novel class of multiple chaotic systems via a sliding mode control scheme, Nonlinear Dynamics. 69, 35-55. Fabiny, L. and Wiesenfield, K. (1991). Clustering behavior of oscillator arrays, Physical Review A, 43, 2640-2648. Guo, H. and Zhong, S. (2009). Synchronization criteria of time-delay feedback control system with sector bounded nonlinearity, Applied Mathematics and Computation, 191, 2, 550-557. Ge, Z.M. and Chen, C.C. (2004). Phase synchronization of coupled chaotic multiple time scales systems, Chaos, Solitons and Fractals, 20, 639-647. Hahn, W. (1967). The Stability of Motion, Springer, Berlin Ho, M. C., and Hung Y. C. (2002a).Synchronization of two different chaotic systems by using generalized active control. Physics Letters A, 301, 424-428. Ho, M.C., Hung Y.C., Chou ,C.H. (2002b). Phase and anti phase synchronization of two chaotic systems by using active control, Phys. Lett. A., 296, 43–48. Jian-Ping, Y., Chang-Pin, L. (2006). Generalized projective synchronization for the chaotic Lorenz systems and the chaotic Chen system, Journal of Shanghai University, 10, 299–304. Khalil, H. K. (1996). Nonlinear Systems, 2nd edition, Prentice Hall, NJ. Kocarev, L., Parlitz, U. (1996). Generalized synchronization, predictability and equivalence of unidirectionally coupled systems. Phys. Rev. Lett. 76, 11, 1816–1819. Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130-141. Liao T. L. and Tsai S. H. (2000). Adaptive synchronization of chaotic systems and its applications to secure communications. Chaos, Solitons and Fractals, 11, 1387-1396. Li, Guo-Hui, Zhou, Shi-Ping. (2007). Anti-synchronization in different chaotic systems, Chaos, Solitons and Fractals, 32, 516–520. Li, R. H., Xu, W. and Li, S. (2007). Adaptive generalized projective synchronization in different chaotic systems based on parameter identifiction, Phys. Lett. A, 367, 199–206. Liu, J. B., Ye, C. F., Zhang, S. J., Song, W. T. (2000). Antiphase synchronization in coupled map lattices, Phys Lett. A., 274, 27–29.
1000 0 0
100
200
300
400 500 600 t ( time in seconds ) (a)
700
800
900
1000
0
100
200
300
400 500 600 t ( time in seconds ) (b)
700
800
900
1000
0
100
200
300
400 500 600 t ( time in seconds ) (c)
700
800
900
1000
control input u
2
-1000
1000 0
control input u
3
-1000
1000 0 -1000
Fig. 4. Time Response plots of the Control Input states for the hybrid synchronization between each states of Master-Slave (a) First Control input. (b) Second Control input. (c) Third Control input.
4.3 Error Response between the States of Master and Slave (a)
error e
1
10 0 -10
0
1
2
3
4 5 6 t ( time in seconds ) (b)
7
8
9
10
0
1
2
3
4 5 6 t ( time in seconds ) (c)
7
8
9
10
0
1
2
3
4 5 6 t ( time in seconds )
7
8
9
10
error e
2
10 0 -10
error e
3
10 0 -10
Fig. 5. Convergence of error between the states of Master and Slave Systems. (a) between first states. (b) between second states. (c) between third states.
5. CONCLUSIONS In this paper, hybrid synchronization between two different chaotic systems, i.e. Lu and Bhalekar-Gejji system is addressed. Active control technique is used to realize the hybrid synchronization since the parameters of the systems are known. Two states of master and slave system are synchronized and one state is anti-synchronized. MATLAB simulation is done. Simulation results verify the effectiveness of the proposed scheme. Literature review suggests that hybrid synchronization involving Bhalekar-Gejji chaotic system is not reported by others. Hardware implementation is being done and will be reported soon.
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2014 ACODS March 13-15, 2014. Kanpur, India
Lu, J., Chen, G. (2002). A new chaotic attractor coined, Int. J. Bifurc. Chaos,12, 3, 659–61. Li, Z. and Xu, D. (2001). Stability criterion for projective synchronization in three-dimensional chaotic systems, Physics Letters A, 282, 175-179. Ott, E., Grebogi, C. and Yorke, J. A. (1990). Controlling chaos. Physical Review Letters, 64, 11, 1196- 1199. Pecora, L.M. and Carroll, T.L. (1990). Synchronization in chaotic systems. Phys. Rev. Lett., 64, 821-824. Pecora, L.M and Carroll, T.L. (1991). Synchronizing chaotic circuits. IEEE Trans. Circ. Sys., 38, 453-456. Park, J. H. (2006). Synchronization of Genesio chaotic system via backstepping approach. Chaos, Solitons and Fractals, 27, 5, 1369-1375. Park, Ju. H. (2007). Exponential synchronization of the Genesio- Tesi chaotic system via a novel feedback control, Phys. Scr, 76, 6, 617–622. Rosenblum, M. G., Pikovsky, A. S., Kurths, J. (1997). From phase to lag synchronization in coupled chaotic oscillators, Phys.Rev. Lett. 78, 22, 4193–4196. Sundarapandian, V. (2013). Adaptive design of controller and synchronizer for Lu-Xiao chaotic system with unknown parameters. International Journal of Computer Science and Information Technology, 5, 1, 197-210. Slotine, J. E., and Sastry, S. S. (1983). Tracking control of nonlinear systems using sliding surface with application to robotic manipulators, Internat. J. Control, 38, 465492. Sawalh, Al., Noorani, M. M., M. S. M. (2008). Chaos antisynchronization between two novel different hyperchaotic systems, Chin Phys Lett, 25, 8, 2743–7. Sudheer, S. K. and Sabir, M. (2009). Hybrid Synchronization of Hyperchaotic Chen System. National Conference on Nonlinear Systems & Dynamics, Kolkata, India. Sebastian, S. K., and Sabir, M. (2009). Hybrid synchronization of hyperchaotic Lu system, Pramana J. Phys., 73, 4, 781-786. Sundarapandian, V. (2011). Hybrid Synchronization of Liu and Chen Systems by Active Nonlinear Control, Int. J. of Mathematical Sciences and Applications, 1, 3, 11391146. Sundarapandian, V. and Suresh, R. (2012). Hybrid Synchronization of Arneodo and Rossler Chaotic Systems by Active Nonlinear Control, CCSIT, 84, 73– 82. Tan, X., Zhang, J. & Yang, Y. (2003). Synchronizing chaotic systems using backstepping design, Chaos, Solitons & Fractals, 16, 37-45. Utkin, V.I. (1993). Sliding mode control design principles and applications to electric drives, IEEE Trans. Industrial Electronics, 40, 23-36, 1993. Vincent, U.E. (2005). Synchronization of Rikitake chaotic attractor using active control. Phys. Lett. A, 343, 133– 138. Wei, H., Li, L. (2010). Adaptive Hybrid Synchronization of Two Different Chaotic Systems with Unknown Parameters, 2nd International Conference on Signal Processing Systems (ICSPS), 3, 609-613. Xia Huang, Zhen Wang, Yuxia Li, (2010). Hybrid Synchronization of Hyperchaotic Lu System Based on
Passive Control, International Workshop on ChaosFractal Theory and its Applications, 34-38. Yan, Z. Y. (2005). Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic system-A symbolic-numeric computation approach, Chaos, Solitons, and Fractals, 15, 023902029. Yu, Y. G., and Zhang, S. C. (2004). Adaptive backstepping synchronization of uncertain chaotic systems, Chaos, Solitons and Fractals, 21, 3, 643-649. Yan, J., Hung, M., Chiang, T., and Yang, Y. (2006). Robust synchronization of chaotic systems via adaptive sliding mode control, Physics Letters A, 356, 3, 220-225. Yaoyao, L. A., Qingguo, L. I. (2012).Hybrid Synchronization in Hyperchaotic Systems, Journal of Computational Information Systems, 18, 7699-7707. Zhao, J., and Lu, J. (2008). Using sampled-data feedback control and linear feedback synchronization in a new hyperchaotic system, Chaos, Solitons and Fractals, 35, 2, 376-382. Zhang, Y., Sun, J. (2004). Chaotic synchronization and anti synchronization based on suitable separation, Phys. Lett. A, 330, 442-7. Zheng, Z., Tan, M., Wang, Q. (2012).Hybrid Synchronization of Two Delayed Systems with Uncertain Parameters. 285-292.
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Hybrid Synchronization of Lu and Bhalekar-Gejji Chaotic Systems Using Nonlinear Active Control Jay Prakash Singh*. Piyush Pratap Singh,** B K Roy***
*National Institute of Technology Silchar, Silchar, Assam India (e-mail: [email protected]). ** National Institute of Technology Silchar, Silchar, Assam India (e-mail: [email protected]). *** National Institute of Technology Silchar, Silchar, Assam India (e-mail:[email protected]). Abstract: Hybrid synchronization based on the active control theory between two different chaotic systems, i.e. Lu and Bhalekar-Gejji is reported in this paper. Lu chaotic system is taken as master and Bhalekar-Gejji chaotic system as slave. Stabilization of error dynamics is achieved by satisfying the Lyapunov stability conditions. Control is designed by using the relevant variables of master and slave systems. Simulation is presented for verification of proposed scheme. Keywords: Active Control, Asymptotic stability, Chaotic Behavior, Master-Slave System, Nonlinear Control, Synchronization.
and Xu et al., 2007, Jian-Ping et al., 2006), Exponential synchronization (Park, 2007). In similar way, antisynchronization between master-slave systems can be studied. Two chaotic systems (master-slave) will be antisynchronize when their sum of state will be converge to zero asymptotically with time, their amplitude of the state will be same but they will have opposite sign ( Li and Zhou et al., 2007, Liu et al., 2000, Sawalh et al., 2008, Ho et at., 2002b). Hybrid synchronization is the phenomenon when complete synchronization and anti-synchronization occur together between the states of two chaotic systems.
1. INTRODUCTION The important property available in any Chaotic/Dynamical system are sensitivity to initial condition, aperiodic long time behavior, and deterministic in nature (Lorenz, 1963). When the states of any chaotic system drive the state of another system, this phenomenon is called Synchronization. Complete synchronization is the phenomenon in which the differences of the states of synchronized system converge to zero. Synchronization of two identical chaotic systems with different initial condition was first illustrated by (Pecora and Caroll, 1990).
In last decade some paper has absorbed on hybrid synchronization of different chaotic and hyperchaotic system such as; Hybrid synchronization of hyperchaotic Chen system (Sudheer et al., 2009), hybrid synchronization of hyperchaotic Lu system (Sebastian et al., 2009), hybrid synchronization of hyperchaotic Lu system (Sebastian et al., 2009), adaptive hybrid synchronization of two different chaotic system Chen and Lu system (Wei et al., 2010), hybrid synchronization of hyperchaotic Lu system based on Passive control (Xia et al., 2010), hybrid synchronization of Liu and Chen system by active nonlinear control (Sundarpandian, 2011), hybrid synchronization in hyperchaotic Lorenz, Qi system (Yaoyao, 2012), hybrid synchronization of Arneodo and Rossler chaotic system by active nonlinear control (Sundarpandian, 2012), hybrid synchronization of Chen and Lu system (Zheng et al., 2012).
Drive (master) -response (slave) scheme for synchronization of chaotic system is widely used. In this scheme the output of master (drive) system is use to control slave (slave) system. Various schemes have been developed for synchronization of chaotic system in the last two decade such as PC method (Pecora and Carroll, 1990, 1991), OGY method (Ott et al., 1990), active control method (Ho et al., 2002a, Chen, 2005, Bai et al., 1997, Vincet, 2005), adaptive control method (Liao et al., 2000, Sundarpandian, 2013, Yu, 2004), backstepping control method (Park, 2006, Tan et al., 2003), sampled data feedback method (Zhao, et al., 2008), time delay feedback method (Guo et al., 2009), sliding mode control method (Yan et al., 2006, Chen, et al., 2012, Slotine at al., 1983, Utkin, 1993) etc.
Here, we discuss the hybrid synchronization of chaotic Lu system (Lu and Chen, 2002) and chaotic Bhalekar-Gejji systems (Bhalekar et al., 2011, 2012) system, using nonlinear active control.
So many more technique are also available related to synchronization like: Complete synchronization (Fabiny et al., 1991), Lag synchronization (Rosenblum et al., 1997), Phase synchronization (Ge et al., 2004), generalized synchronization (Kocarev et al., 1996), projective synchronization (Li et al., 2001), generalized projective (Li
978-3-902823-60-1 © 2014 IFAC
This paper is organized as follows, In section (2), we discuss about the problem statement for hybrid synchronization of 292
10.3182/20140313-3-IN-3024.00069
2014 ACODS March 13-15, 2014. Kanpur, India
two different chaotic systems. In section (3) we discuss the hybrid synchronization of chaotic Lu system and chaotic Bhalekar-Gejji system. In section (4), simulation results are given for illustration and verification of theory for hybrid synchronization. Finally, conclusion is drawn in section (5).
Now, the asymptotical stabilization of the equation (6) is obtained by designing a controller for stabilizing the error dynamical system (4) at origin. By using the Lyapunov stability theory (Hahn 1976, Khalil 1996), positive definite Lyapunov function
2. PROBLEM STATEMENT FOR HYBRID SYNCHRONIZATION
∑
where, is the order of the system. Assuming that parameters of the drive and response are known and states are measurable. The continuous first partial derivative of (7) is
Consider a chaotic system as drive system: ̇
(1) ̇
where, is state vector, and is the matrix for system parameter, and is the nonlinear part of the system. Consider another chaotic system as a slave system ̇
(7)
∑
(8)
‖ is negative definite, and also as ‖ , then states and are globally asymptotically stable in large, and also the state of drive and response system are globally asymptotically hybrid-synchronized.
(2) 3. HYBRID SYNCHRONIZATION OF LU AND BHALEKAR-GEJJI CHAOTIC SYSTEM
, and
is the matrix for is the non-linear part of is the active non-linear controller to
Here, we apply nonlinear active control technique for the synchronization of Lu system (Lu and Chen, 2002) and Bhalekar-Gejji (Bhalekar et al., 2011, 2012). Master (or drive) system is Lu system dynamics is given:
Our goal is to design an appropriate controller such that state vector of the response system asymptotically approaches a given functional relationship of the state vector of the drive system. State vector and will be state of two identical chaotic system if and . State and will be state of two different chaotic system if and . For our problem functional relationship between master and slave system must include both synchronization and antisynchronization. Therefore relationship can be written as
̇ ̇
(9) ̇
where, , , are the states of system (9) and a, b, c are the parameter of system (9) which are positive. Lu attractors are given in the Fig. 1.
(3) when, , and , the complete synchronization, antisynchronization is obtained respectively, we call this scheme as hybrid synchronization. We, define hybrid synchronization error vector as
(b) 40
20
30
3
2
x state
(a) 40
x state
Having state vector system parameter and the slave system and be designed.
0
20
(4) -20 -20
Now, our aim for hybrid synchronization of chaotic system (1) and (2), we design active control , such that trajectory of system (2) with any initial condition can asymptotically approach the system (1) with initial condition , i.e.
‖
(d)
40
,
20
3
20
0 50 0 0
20
40
x 2 state
Fig. 1. Lu Chaotic attractors (a) on the plane, (c) on the plane, (d) on the
x 2 state
-50 -50
0 x 1 state
50
plane, (b) on the plane.
The slave system is described as Bhalekar-Gejji dynamics
‖ ̇
So, hybrid synchronization error can be written from (1), (3), (5) ̇
20
(c)
30
10 -20
We say that anti-synchronization of two systems (1) and (2) are achieved if the following equation holds: ‖
0 x 1 state
x state
(5)
We say that complete synchronization of two systems (1) and (2) is achieved if the following equation holds: ‖
10 -20
40
40
3
‖
20 x 1 state
40
x state
‖
0
̇
(10) ̇
(6)
Where , , , 293
,
are the states of the system (10) and , >0 are parameters of the system (10), and
2014 ACODS March 13-15, 2014. Kanpur, India
̇
, , are nonlinear active controller element which is to be designed. Bhalekar-Gejji attractors are given in the Fig. 2. (a)
30
y state
10
y state
40
-10 -20 -20
(16)
̇
(17)
is negative definite function on value is .
20
for condition as parameter
Now, we can say that according to Lyapunov stability theory (Hahn 1976, Khalil 1996) error dynamics (12) globally asymptotically stable for equilibrium state at origin i.e. error dynamics will converge to zero as t . We obtained the following result.
10 -10
0 y 1 state
10
0 -20
20
-10
0 y 1 state
10
20
(d)
(c) 40 40
y state
30
0 20
10 0 -20
Remarks. The chaotic Lu (9) and chaotic Bhalekar-Gejji (10) system are exponentially and globally hybrid synchronized for any value of initial conditions with the nonlinear controller defined by (13).
20
3
20
3
y state
̇
3
2
0
̇ +
Substituting the error dynamics (14) into (16) we obtained
(b)
20
̇ +
0 -10
0 y 2 state
10
20
y 2 state
0 y 1 state
20
4. SIMULATION RESULTS
Fig. 2. Bhalekhar-Gejji Chaotic attractors (a) on the plane, (c) on the plane, (d) on the
The hybrid-synchronization error
-20 -20
plane, (b) on the plane.
We are using fourth-order Runga-Kutta method for solving the dynamics (9), and (10) and active nonlinear controller (13) and simulating the result with time step h= .
which is defined as: (11)
where,
For chaotic behavior of Lu system the parameter is selected as follows (Lu and Chen, 2002), and the parameters of the Bhalekar-Gejji system for chaotic behavior are selected as following (Bhalekar, 2012).
̇ , and for synchronization and for anti-synchronization. Hence, (11) can be written
as:
The initial condition for plotting the synchronization of state, controller, error dynamics of the master (9) and slave system (10) are as and .
States are synchronized and anti-synchronized alternatively. The error dynamics equation is obtained as follows:
4.1 Time Response plots of Master and Slave states ̇ ̇
(12) x &y
1
20 x1
0
y1
1
̇
-20
(13)
200
300
400 500 600 t ( time in seconds ) (a)
700
800
900
20
y2
-20
3 3
x &y
̇ (14) ̇
100
200
300
400 500 600 t ( time in seconds ) (b)
700
800
900
40 20
1000
x3
0 -20
1000
x2
0 0
Substituting the controller dynamics (13) into error dynamics (12), we obtain the error dynamics as: ̇
100
2
x &y
2
We consider the active nonlinear controller for above error dynamics (12) are as follows:
0
y3 0
100
200
300
400 500 600 t ( time in seconds ) (c)
700
800
900
1000
Fig. 3. Time Response plots of hybrid synchronized states of Master and Slave. (a) Time response of first state of Master and Slave. (b) Time response of second state of Master and Slave. (c) Time response of third state of Master and Slave.
Now, considering positive definite Lyapunov function on defined by (15) Taking the continuous first partial derivative of (15), we obtained the following 294
2014 ACODS March 13-15, 2014. Kanpur, India
4.2 Control inputs for hybrid synchronization between Master-Slave
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control input u
1
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1000 0 0
100
200
300
400 500 600 t ( time in seconds ) (a)
700
800
900
1000
0
100
200
300
400 500 600 t ( time in seconds ) (b)
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800
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400 500 600 t ( time in seconds ) (c)
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-1000
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control input u
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-1000
1000 0 -1000
Fig. 4. Time Response plots of the Control Input states for the hybrid synchronization between each states of Master-Slave (a) First Control input. (b) Second Control input. (c) Third Control input.
4.3 Error Response between the States of Master and Slave (a)
error e
1
10 0 -10
0
1
2
3
4 5 6 t ( time in seconds ) (b)
7
8
9
10
0
1
2
3
4 5 6 t ( time in seconds ) (c)
7
8
9
10
0
1
2
3
4 5 6 t ( time in seconds )
7
8
9
10
error e
2
10 0 -10
error e
3
10 0 -10
Fig. 5. Convergence of error between the states of Master and Slave Systems. (a) between first states. (b) between second states. (c) between third states.
5. CONCLUSIONS In this paper, hybrid synchronization between two different chaotic systems, i.e. Lu and Bhalekar-Gejji system is addressed. Active control technique is used to realize the hybrid synchronization since the parameters of the systems are known. Two states of master and slave system are synchronized and one state is anti-synchronized. MATLAB simulation is done. Simulation results verify the effectiveness of the proposed scheme. Literature review suggests that hybrid synchronization involving Bhalekar-Gejji chaotic system is not reported by others. Hardware implementation is being done and will be reported soon.
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