Chaotic Systems 214 0 5 10 15 -1 0 1 eS1 0 5 10 15 -2 -1 0 1 eS2 0 5 10 15 -0.5 0 0.5 1 eS3 Tim e (s ) Fig. 5. Error dynamics of the coupled master-slave Chen system when controller is switched on 4.2 Anti-synchronization of two identical coupled chaotic Lee systems The studied chaotic Lee system is described by the following differential equations (Juhn et al., 2009): 11 212 31 3 10 10 0 40 0 4025 () () () () () () () . () xt xt xt xt xt xt xt xt − ⎡ ⎤⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥⎢ ⎥ =− ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ − ⎣ ⎦⎣ ⎦⎣ ⎦    (62) It exhibits a chaotic attractor, starting at the initial value of the state vector [] 0232() , T x = Fig. 6. -150 -100 -50 0 50 100 150 -20 0 20 0 20 40 60 80 100 120 x1(t) x2(t) x3(t) Fig. 6. Three-dimensional view of the Lee chaotic attractor Coexistence of Synchronization and Anti-Synchronization for Chaotic Systems via Feedback Control 215 Let us consider the master Lee system ( ) m S given by (63): 11 212 31 3 10 10 0 40 0 4025 () () () () () () () . () mm mmm mm m xt xt xt xtxt xt xt xt − ⎡ ⎤⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥⎢ ⎥ =− ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ − ⎣ ⎦⎣ ⎦⎣ ⎦    (63) driving a similar controlled slave Lee system ( ) s S described by (64): 11 212 31 3 10 10 0 0 40 0 1 4025 0 () () () () () () () () . () ss sss ss s xt xt xt xt xt ut xt xt xt − ⎡⎤⎡ ⎤⎡⎤⎡⎤ ⎢⎥⎢ ⎥⎢⎥⎢⎥ =− + ⎢⎥⎢ ⎥⎢⎥⎢⎥ ⎢⎥⎢ ⎥⎢⎥⎢⎥ − ⎣⎦⎣ ⎦⎣⎦⎣⎦    (64) ()ut is the scalar active control. For the following state error vector components, defined relatively to anti-synchronization study: 11 1 22 2 33 3 () () () () () () () () () sm AS sm AS sm AS etxtxt etxtxt etxtxt ⎧ ⎪ ⎨ ⎪ ⎩ =+ =+ =+ (65) the error system can be defined by the following differential equations: () () 112 13 1 3 21 22 11 33 10 10 40 25 4 () () () () () () () () () () () . () () () AS AS AS ss mm AS AS sm AS AS et et et et etxtxtxtxtut et et xtxt ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ =− + =− + + =− + +    (66) The problem of chaos anti-synchronization between two identical Lee chaotic dynamical systems is solved here by the design of a state feedback structure 123(.), , , , i ik ∀= and the choice of nonlinear functions 1 2 3 (.), , , , i fi ∀ = such that: 3 1 123,,() (.) () (.) j ii i jut k x t f = ∀==− − ∑ (67) It comes the following closed-loop dynamical error system: () () (.) () () 1 10 10 0 11 ( ) 40 (.) (.) (.) ( ) ( ) ( ) ( ) ( ) (.) 2123213132 () 2.5 0 0 () 22 33 4() () (.) 113 f et et AS AS e t k k k e t x tx t x tx t f AS AS s s m m et et AS AS xtx t f sm ⎡ ⎤ ⎡⎤ ⎡⎤ ⎢ ⎥ ⎡⎤ − ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢⎥ =− − − − + + ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ − ⎢⎥ ⎢⎥ ⎢ ⎥ ⎣⎦ ⎣⎦ ⎣⎦ ++ ⎢ ⎥ ⎣ ⎦    (68) The nonlinear elements (.) i f and (.) i k have to be chosen to make the instantaneous characteristic matrix of the closed-loop system in the arrow form and the closed-loop error system asymptotically stable. Chaotic Systems 216 From the possible solutions, allowing to put the instantaneous characteristic matrix of (68) under the arrow form, let consider the following: 3 0k = (69) and: () () 3 1 21313 22 311 0 4 () (.) (.) ( ) ( ) ( ) ( ) (.) ( ) ( ) AS ss mm sm t f f x tx t x tx t fe xtxt ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ = =− + =− + (70) For the vector norm [] 123 :(), T AS AS AS AS AS eeeepe= 123 () AS T AS AS AS epeee ⎡ ⎤ ⎣ ⎦ = (71) the overvaluing matrix is in arrow form and has non negative off-diagonal elements and nonlinearities isolated in either one row or one column. By the use of the proposed theorem 2, stability and anti-synchronization properties are satisfied for the both following sufficient conditions (72) and (73): 2 (.) 0k − < (72) ( ) ( ) 1 10 10 40 (.) (.) 0 12 kk − ⎛⎞ − −− − < ⎜⎟ ⎝⎠ (73) Various choices of the gain vector [ ] 12 (.) (.) 0 ,(.), (.) kkKK = are possible, such as the following linear one: { } [ ] 420 i Kk== (74) By considering the initial condition 0 2 3 2 () , AS T e ⎡ ⎤ ⎣ ⎦ = for the Lee error system (66) when the active controller is deactivated, it is obvious that the error states grow with time chaotically, as shown in Fig. 7., and after activating the controller, Fig. 8. shows three parametrically harmonically excited 3D systems evolve in the opposite direction. The trajectories of error system (68) imply that the asymptotical anti-synchronization has been, successfully, achieved, Fig. 9. 5. Hybrid synchronization by a nonlinear state feedback controller – Application to the Chen–Lee chaotic system (Hammami, 2009) Let consider two coupled chaotic Chen and Lee systems (Juhn et al., 2009). The following nonlinear differential equations, of the form (1), correspond to a master system (Tam & Si Tou, 2008): ( ) () () () mmm xt Axtxt=  Coexistence of Synchronization and Anti-Synchronization for Chaotic Systems via Feedback Control 217 with: () 3 1 2 0 0 0 () () () () m mm m axt Ax t b x t cx t d − ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (75) 12 , mm xx and 3m x are state variables, and , , abc and d four system parameters. For the following parameters ( ) ( ) 5100338,,, , ,., . ,abcd =− − and initial condition 123 000153213() () () . , T T mmm xxx ⎡⎤ ⎡ ⎤ ⎣ ⎦ ⎣⎦ =− the drive system, described by (1) and (75) is a chaotic attractor, as shown in Fig. 10. 0 5 10 15 -20 0 20 eAS1 0 5 10 15 -50 0 50 eAS2 0 5 10 15 0 50 100 eAS3 Tim e (s ) Fig. 7. Error dynamics ( ) 123 ,, AS AS AS eee of the coupled master-slave Lee system when the active controller is deactivated 0 5 10 15 -20 0 20 xm1,xs1 xm1 xs1 0 5 10 15 -50 0 50 xm2,xs2 xm2 xs2 0 5 10 15 -200 0 200 xm3,xs3 Tim e (s ) xm3 xs3 Fig. 8. Partial time series of anti-synchronization for Lee chaotic system when the active controller is switched on Chaotic Systems 218 0 5 10 15 -1 0 1 2 eAS1 0 5 10 15 -2 0 2 4 eAS2 0 5 10 15 -1 0 1 2 eAS3 Tim e (s ) Fig. 9. Error dynamics of the coupled master-slave Lee system when control is activated -40 -20 0 20 40 -50 0 50 0 5 10 15 20 xm1(t) xm2(t) xm3(t) Fig. 10. The 3-dimensional strange attractor of the chaotic Chen–Lee master system For the Chen–Lee slave system, described in the state space by: ( ) () () () () sss xt Axtxt ut=+  (76) with: () 3 1 2 0 0 0 () () () () s ss s axt Ax t b x t cx t d − ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (77) we have selected the anti-synchronization state variables 1m x and 3m x facing to 1s x and 3 , s x and the synchronization state variable 2m x facing to 2 . s x Then, the hybrid synchronization errors between the master and the slave systems 12 3 () () () () , T AS S AS et e t e t e t ⎡⎤ ⎣⎦ = are such as: Coexistence of Synchronization and Anti-Synchronization for Chaotic Systems via Feedback Control 219 11 1 222 33 3 () () () () () () () () () AS s m Ssm AS s m etxtxt et xtx t etxtxt =+ ⎧ ⎪ =− ⎨ ⎪ =+ ⎩ (78) Let compute the following continuous state feedback controller’s structure: ( ) () (), () () ms ut K x t x t et=− (79) to guarantee the asymptotic stability of the error states defined by (78), 12 3 , T AS S AS ee e e ⎡⎤ ⎣⎦ = so that the slave system, characterized by (76) and (77), synchronizes and anti-synchronizes, simultaneously, to the master one, described by (1) and (75), by assuring that the synchronization error 2S e and the anti-synchronization errors 1AS e and 3AS e decay to zero, within a finite time. Thus, for a state feedback controller of the form (79), (.),K { } 123(.) (.) , , , , , ij Kk ij=∀= and by considering the differential systems (1), (75), (76), (77) and (78), we obtain the following state space description of the error resulting system: () (.)()et A et =  (80) with: 11 3 12 13 21 22 1 23 231 32 33 (.) ( ) (.) (.) (.) (.) (.) ( ) (.) ( ) (.) (.) (.) m m m ak x t k k Ak bkxtk cx t k k d k −−− − ⎡ ⎤ ⎢ ⎥ =− − − ⎢ ⎥ ⎢ ⎥ −− − ⎣ ⎦ (81) By respect to the stabilisability conditions announced in the above-mentioned theorem 2, the dynamic error system (80) is first characterized by an instantaneous arrow form matrix (.),A that is to say, the main requirements concerning the choice of the feedback gains 12 (.)k and 21 (.)k are given by: 12 3 21 0 (.) ( ) m kxt k =− ⎧ ⎨ = ⎩ (82) To satisfy that the two first diagonal elements of the characteristic matrix (.) A are strictly negative: 11 22 0 0 (.) (.) ak bk − < ⎧ ⎨ − < ⎩ (83) a possible solution is: 11 22 7 6 k k = ⎧ ⎨ = − ⎩ (84) Besides, to annihilate the nonlinearities in system (80), a solution is: Chaotic Systems 220 23 1 31 2 (.) ( ) (.) ( ) m m kxt kcxt = ⎧ ⎨ = ⎩ (85) Finally, by considering the fixed values of 11 12 21 22 23 , (.), , , (.)kk kkk and 31 (.),k it is relevant to denote that to satisfy the sufficient condition (30) of theorem 2, for any arbitrary chosen parameters of correction 13 (.)k and 32 (.),k it is necessary to tune the remaining design parameter 33 (.),k guaranteeing the hybrid synchronization of the coupled chaotic studied system such that: 33 0(.)dk − < (86) Then, for the following instantaneous gain matrix (.),K easily obtained: 3 1 2 70 06 01 () (.) ( ) () m m m xt Kxt cx t − ⎡ ⎤ ⎢ ⎥ =− ⎢ ⎥ ⎢ ⎥ − ⎣ ⎦ (87) the studied dynamic error system (80) is asymptotically stable. For the following initial master and slave systems conditions, 0155213,() . T m x ⎡ ⎤ ⎣ ⎦ =− 021015() , T s x ⎡⎤ ⎣⎦ =− − and without activation of the designed controller, the numerical simulation results of the above master-slave system are shown in Fig. 11. It is obvious, from Fig. 12., that the error states grow with time chaotically. Therefore, by designing an adequate nonlinear controlled slave system and under mild conditions, the hybrid synchronization is achieved within a shorter time, as it is shown in Fig. 13., with an exponentially decaying error, Fig. 14. The obtained phase trajectories of the Fig. 15., show that the Chen–Lee slave chaotic attractor is synchronized in a hybrid manner with the master one. 0 5 10 15 -50 0 50 xm1,xs1 xm1 xs1 0 5 10 15 -50 0 50 xm2,xs2 xm2 xs2 0 5 10 15 -40 -20 0 20 xm3,xs3 Time (s) xm3 xs3 Fig. 11. Error dynamics between the master Chen–Lee system and its corresponding slave system before their hybrid synchronization Coexistence of Synchronization and Anti-Synchronization for Chaotic Systems via Feedback Control 221 0 5 10 15 -100 0 100 eAS1 0 5 10 15 -100 0 100 eS2 0 5 10 15 -60 -40 -20 0 20 eAS3 Tim e (s ) Fig. 12. Evolutions of the hybrid synchronization errors versus time when the proposed controller is turned off 0 5 10 15 -50 0 50 xm1,xs1 xm1 xs1 0 5 10 15 -50 0 50 xm2 0 5 10 15 -50 0 50 xs2 0 5 10 15 -20 0 20 xm3,xs3 Tim e (s ) xm3 xs3 Fig. 13. Hybrid synchronization of the master-slave Chen–lee chaotic system Chaotic Systems 222 0 5 10 15 -1 0 1 eAS1 0 5 10 15 -100 -50 0 50 eS2 0 5 10 15 -2 -1 0 1 eAS3 Tim e (s) Fig. 14. Exponential convergence of the error dynamics -30 -20 -10 0 10 20 30 0 5 10 15 20 xm2(t) xm3(t) xm2 vs xm3 -30 -20 -10 0 10 20 30 -20 -15 -10 -5 0 xs2(t) xs3(t) xs2 vs xs3 Fig. 15. 2-D projection of the hybrid synchronized attractors associated to the Chen–Lee chaotic system 6. Conclusion Appropriate feedback controllers are designed, in this chapter, for the chosen slave system states to be synchronized, anti-synchronized as well as synchronized in a hybrid manner with the target master system states. It is shown that by applying a proposed control scheme, the variance of both synchronization and anti-synchronization errors can converge to zero. The synchronisation of two identical Chen chaotic systems, the anti-synchronization of two identical Lee chaotic systems and, finally, the coexistence of both synchronization and anti-synchronization for two identical Chen–Lee chaotic systems, considered as a coupled master-slave systems, are guaranteed by using the practical stability criterion of Borne and Gentina, associated to the specific matrix description, namely the arrow form matrix. Coexistence of Synchronization and Anti-Synchronization for Chaotic Systems via Feedback Control 223 7. References Bai, E.W. & lonngsen, K.E. (1997). Synchronization of two Lorenz systems using active control. Chaos, Solitons and Fractals, 8, 51–58, 0960-0779 Benrejeb, M. (1980). Sur l’analyse et la synthèse de processus complexes hiérarchisés. Application aux systèmes singulièrement perturbés. Thèse de Doctorat ès Sciences Physiques, Université des Sciences et Techniques de Lille, France Benrejeb, M. (2010). Stability study of two level hierarchical nonlinear systems. Plenary lecture, 12 th International Federation of Automatic Control Large Scale Systems Symposium: Theory and Applications, IFAC – LSS 2010, Lille, July 2010, France Benrejeb, M. & Hammami, S. (2008). New approach of stabilization of nonlinear continuous monovariable processes characterized by an arrow form matrix. 1 st International Conference Systems ENgineering Design and Applications, SENDA 2008, Monastir, October 2008, Tunisia Borne, P. & Benrejeb, M. (2008). On the representation and the stability study of large scale systems. International Journal of Computers Communications and Control, 3, 55–66, 1841-9836 Borne, P.; Vanheeghe, P. & Dufols, E. (2007). Automatisation des processus dans l’espace d’état, Editions Technip, 9782710808794, Paris Fallahi, K.; Raoufi, R. & Khoshbin, H. (2008). An application of Chen system for secure chaotic communication based on extended Kalman filter and multi-shift cipher algorithm. Communications in Nonlinear Science and Numerical Simulation, 763–81, 1007-5704 Hammami, S. (2009). Sur la stabilisation de systèmes dynamiques continus non linéaires exploitant les matrices de formes en flèche. Application à la synchronisation de systèmes chaotiques. Thèse de Doctorat, École Nationale d’Ingénieurs de Tunis, Tunisia Hammami, S.; Benrejeb, M. & Borne, P. (2010a). On nonlinear continuous systems stabilization using arrow form matrices. International Review of Automatic Control, 3, 106–114, 1974-6067 Hammami, S.; Ben saad, K. & Benrejeb, M. (2009). On the synchronization of identical and non-dentical 4-D chaotic systems using arrow form matrix. Chaos, Solitons and Fractals, 42, 101–112, 0960-0779 Hammami, S.; Benrejeb, M.; Feki, M. & Borne, P. (2010b). Feedback control design for Rössler and Chen chaotic systems anti-synchronization. Phys. Lett. A, 374, 2835– 2840, 0375-9601 Juhn, H.C.; Hsien, K.C. & Yu, K.L. (2009). Synchronization and anti-synchronization coexist in Chen–Lee chaotic systems. Chaos, Solitons and Fractals, 39, 707–716, 0960-0779 Kapitanialc, T. (2000). Chaos for Engineers: Theory, Applications and Control, Second revised Springer edition, 3540665749, Berlin Li, G.H. (2005). Synchronization and anti-synchronization of Colpitts oscillators using active control. Chaos, Solitons and Fractals, 26, 87–93, 0960-0779 Michael, G.R.; Arkady, S.P. & Jürgen, K. (1996). Phase synchronization of chaotic oscillators. Phys. Rev. Lett., 76, 1804–1807, 0031-9007 Pecora, L.M. & Carroll, T.L. (1990). Synchronization in chaotic systems. Phys. Rev. Lett., 64, 821-824, 0031-9007 [...]... communication systems International Journal of Bifurcation and Chaos, 3, 1619–1627, 1793-6551 Yang, S.S & Duan, K (1998) Generalized synchronization in chaotic systems Chaos, Solitons and Fractals, 10, 1703–1707, 0960-0779 Yassen, M.T (2005) Chaotic synchronization between two different chaotic systems using active control Chaos, Solitons and Fractals, 131–140, 0960-0779 Zhang, Y & Sun, J (2004) Chaotic. .. and 3-D saturated n × m × l-grid scroll chaotic attractors, where n, m and l are integers and can have the same values Since this multi-scroll system can be designed by using PWL functions, it is 234 Chaotic Systems Fig 5 Hyperchaos with: (a) R7=10kΩ, (b) R7=7kΩ, (c) R7=5kΩ, and (d) R7=500Ω with persistence a good chaos system suitable for the development of a systematic design automation process by... CCII+s 5LM and 6LM) 232 Chaotic Systems RM=3032 Y X CCII+ 3LM Y C3M=4.7nF R5M =2360 C 2M =47nF C 1M =4.7nF X X Y CCII+ 4 LM R 3M =470 R 4M =10K Z CCII+ 1 DM Y Z X CCII+ 2DM Z Z R 1M =6 810 R2M =3979 Y Z CCII+ 5L M X R6M =2360 X Z CCII+ 6LM Y Y X CCII+ 2C1 Z R7=10k X CCII+ 2C2 Y Z R8=1k Y X RS=3116 CCII+ 3LS Y C3S=4.7nF R5S =2360 X Y C2S =47nF C 1S =4.7nF X CCII+ 4LS R 3S =470 R 4S =10K Z CCII+ 1 DS Y Z... generators The design of multi-scroll chaotic attractors can be performed via PWL functions (Elhadj & Sprott, 2 010; Lin & Wang, 2 010; Lü et al., 2004; Muñoz-Pacheco & Tlelo-Cuautle, 2009; 2 010; Design and Applications of Continuous-Time Chaos Generators 233 Fig 4 Hyperchaos with: (a) R7=10kΩ, (b) R7=7kΩ, (c) R7=5kΩ, and (d) R7=500Ω without persistence Sánchez-López et al., 2 010; Suykens et al., 1997; Trejo-Guerra,... family of systems depends on the number of nonlinear functions, for example: three nonlinear functions are needed to generate 3D-scrolls In (Lü et al., 2004) a saturated multi-scroll chaotic system based on saturated function series, is introduced That system can produce three different types of attractors, as follows: 1-D saturated n-scroll chaotic attractors, 2-D saturated n × m-grid scroll chaotic. ..224 Chaotic Systems Taherion, I.S & Lai, Y.C (1999) Observability of lag synchronization of coupled chaotic oscillators Phys Rev E, 59, 6247–6250, 1539-3755 Tam, L.M & Si Tou, W.M (2008) Parametric study of the fractional order Chen–Lee System Chaos Solitons and Fractals, 37, 817–826, 0960-0779 Wu, C.W & Chua, L.O (1993) A simple way to synchronize chaotic systems with applications... function frequently used in system modeling is the piecewise-linear (PWL) approximation, which consists of a set of linear relations valid in different regions (Elhadj & Sprott, 2 010; Lin & Wang, 2 010; Lü et al., 2004; Muñoz-Pacheco & Tlelo-Cuautle, 2009; Sánchez-López et al., 2 010; Suykens et al., 1997; Trejo-Guerra, Sánchez-López, Tlelo-Cuautle, Cruz-Hernández & Muñoz-Pacheco, 2 010; Yalçin et al., 2002)... (Cook, 1994) 230 Chaotic Systems 2 Chua’s circuit and Hyperchaos Over the last two decades, theoretical design and circuit implementation of various chaos generators have been a focal subject of increasing interest due to their promising applications in various real-world chaos-based technologies and information systems (Cook, 1994; Cruz-Hernández et al., 2005; Ergun & Ozoguz, 2 010; Gámez-Guzmán et... 1994; Cruz-Hernández et al., 2005; Ergun & Ozoguz, 2 010; Gámez-Guzmán et al., 2008; Lin & Wang, 2 010; Ott, 1994; Strogatz, 2001; Trejo-Guerra et al., 2009) In electronics, among the currently available chaotic oscillators, Chua’s circuit has been the most used one (Chakraborty & Dana, 2 010; Elhadj & Sprott, 2 010; Sánchez-López et al., 2008; Senani & Gupta, 1998; Suykens et al., 1997; Trejo-Guerra et al.,... complementary metal-oxide-semiconductor (CMOS) technology (Trejo-Guerra, Tlelo-Cuautle, Muñoz-Pacheco, Cruz-Hernández & Sánchez-López, 2 010) 228 Chaotic Systems The usefulness of the chaos generators is highlighted through the physical realization of a secure communication system by applying Hamiltonian forms and observer approach (Cruz-Hernández et al., 2005) This chapter finishes by listing several trends . system Chaotic Systems 222 0 5 10 15 -1 0 1 eAS1 0 5 10 15 -100 -50 0 50 eS2 0 5 10 15 -2 -1 0 1 eAS3 Tim e (s) Fig. 14. Exponential convergence of the error dynamics -30 -20 -10 0 10. Anti-Synchronization for Chaotic Systems via Feedback Control 221 0 5 10 15 -100 0 100 eAS1 0 5 10 15 -100 0 100 eS2 0 5 10 15 -60 -40 -20 0 20 eAS3 Tim e (s ) Fig. 12. Evolutions of the. anti-synchronization for Lee chaotic system when the active controller is switched on Chaotic Systems 218 0 5 10 15 -1 0 1 2 eAS1 0 5 10 15 -2 0 2 4 eAS2 0 5 10 15 -1 0 1 2 eAS3 Tim e (s