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Abstract
In this paper, we have investigated the synchronization behaviour of two identical nonlinear dynamical systems of a rotating ellipsoidal satellite in elliptic orbit under the solar radiation pressure evolving from different initial conditionsusing the active control techniquebased on the Lyapunovstability theory and the Routh-Hurwitz criteria. The designed controller, with our own choice of the coefficient matrix of the error dynamics, are found to be effective in the stabilization of the error states at the origin, thereby, achieving synchronization between the states variables of two dynamical systems under consideration. Numerical simulations are presented to illustrate the effectiveness of the proposed control techniques using mathematica.
Key words
Chaos; Synchronization; Satellite
1.INTRODUCTION
After the pioneering work on chaos control by Ott et al [1] and synchronization of chaotic systems by Pecora and Carroll [2], chaos control and synchronization has received increasing attention [3–7] and has become a very active topic in nonlinear science since last couple of years. Over the last decade various effective methods have been proposed and utilized [8–20] to achieve the control and stabilization of chaotic systems like laser, power electronics etc. The idea of synchronization of two identical chaotic systems that start from different initial conditions consists of linking the trajectory of one system to the same values in the other so that they remain in step with each other, through the transmission of a signal.
The control of physical systems is an important subject in engineering and sciences, thus, in some applications,chaoscanbe usefulwhilein othersit mightbedetrimentalforexamplechaosin powersystems[21–23] and in mechanical systems is objectionable. On the other hand, the idea of chaos synchronization was utilized to build communication systems to ensure the security of information transmitted [24–32]. Several attempts have been made to control and synchronize chaotic systems [2, 16, 18, and 32]. Some of thesemethodsneedseveralcontrollerstorealizesynchronization. TheOGYmethod,forinstance,havebeen successfully applied to many chaotic systems like the periodically driven pendulum [33] and parametric pendulum [34]. Also, the Pyragas time-delayed auto-synchronization method [35, 36] has been shown to be an efficient method that has been realized experimentally in electronic chaos oscillators [37], lasers[38] and chemical systems [39]. In addition, the delayed feedback control, addition of periodic force and adaptive control algorithm has been utilized to control chaos in a symmetric gyro with linear-pluscubic damping [40].
In particular, backstepping design and active control have been recognised as two powerful design methods to control and synchronize chaos. It has been reported [41–43] that backstepping design can guarantee global stability, tracking and transient performance for a broad class of strict-feedback nonlinear systems. In recent time, it has been employed for controlling, tracking and synchronizing many chaotic systems [44–48] as well as hyperchaotic systems [41]. According to ref [45], some of the advantages in the method include applicability to a variety of chaotic systems whether they contain external excitation or not; needs only one controller to realize synchronization between chaotic systems andfinally there are no derivatives in the controller. Zhang [41] states that the controller is singularity free from the nonlinear term of quadratic type, givesflexibility to construct a control law which can be extended to higher dimensional hyperchaotic systems and the closed-loop system is globally stable, while ref [49] adds that it requires less control effort in comparison with the differential geometric method.
The aim of this article is to use the active control technique based on the Lyapunov stability theory and theRouth-Hurwitzcriteriato studythe synchronizationbehaviorofthetwo identicalplanaroscillationof an ellipsoidal satellite in elliptic orbit under solar radiation pressure evolving from different initial conditions.
2.EQUATION OF MOTION OF A SATELLITE IN AN ELLIPTIC ORBIT
Elliptically orbiting planar oscillations of satellites in the solar system make an interesting study, and significant contributions to this end can be found in the works [50-58], all of whom have studied the influence of certain perturbative forces, such as solar radiation pressure, tidal force, and air resistance. In the present work, we consider the planar oscillation of a satellite in elliptic orbit with the spin axisfixed perpendicular to the orbital plane. Let the long axis of the satellite makes an angle xwith a reference axis that isfixed in inertial space, the long axis of the satellite makes an angleφwith satellites planet centre line and the satellite to be a triaxial ellipsoid with principal moments of inertia A < B < C, where C is the moment about the spin axis. The orbit is taken to be afixed ellipse with semi major axis a, eccentricity e, true anomalyν,ω20= 3(B?A)/Cand instantaneous radius r. The equation of motion of satellite planar oscillation in an elliptic orbit around the earth under solar radiation pressure, is
d2x
3.SYNCHRONIZATION VIA ACTIVE CONTROL
For a system of two coupled chaotic oscillators, the master system (˙x = f(x,y)) and the slave system (˙y = g(x,y)),where x(t) and y(t)arethe phase space(state variables),and f(x,y) andg(x,y)are the corresponding nonlinear functions, synchronization in a direct sense implies |x(t)?y(t)|→0 as t→∞. When this occurs the coupled systems are said to be completely synchronized. Chaos synchronizationis related to the observer problem in control theory [59]. The problem may be treated as the design of control laws for full chaotic observer (the slave system) using the known information of the master system so as to ensure that the controlled receiver synchronizes with the master system. Hence, the slave chaotic system completely traces the dynamics of the master in the course of time.
4.NUMERICAL SIMULATION
For the parameters involvedin system under investigation,e = 0.15, h = 0.1, l = 0.7,μ= 0.02,α= 0.0001, n = 0.1andω0= 0.3andthe initial conditionsformaster andslave systems [x1(0), x2(0), x3(0)] = [0,0.1,0] and#y1(0),y2(0),y3(0)$= [0.1,0.2,0.1]respectively,thesystemhasbeensimulatedusingmathematica. The obtained results show that the system under consideration achieved synchronization. Phase plots of (3.1) and (3.2) (Figure 1), time series analysis of (3.1) and (3.2) (Figure 2) and time series analysis of errors(Figure 3) are the witness of achieving synchronization between master and slave system. Further, it also has been confirmed by the convergenceof the synchronization quality defined by e(t) =G
5.CONCLUSION
In this paper, we have investigated the chaos synchronization behaviour of the two identical planar oscillation of an ellipsoidal satellite in elliptic orbit under solar radiation pressure, evolving from different initial conditions via the active control technique based on the Lyapunov stability theory and the Routh-Hurwitz criteria. The results obtained were validated by numerical simulations using mathematica for the proposed technique.
REFERENCES
[1]E. Ott, C. Grebogi & J. A. Yorke (1990). Controlling Chaos. Phys Rev Lett, 64, 1196-1199.
[2]L. M. Pecora& T. L.Carroll (1990).Synchronizationin ChaoticSystems. Phys Rev.Lett, 64, 821-824.
[3]T. Kapitaniak (1996). Controlling Chaos - Theoretical. Practical Methods in Non-linear Dynamics. London: Academic Press.
[4]Chen G. & Dong X. (1998). From Chaos to Order: Methodologies, Perspectives and Applications. Singapore: World Scientific.
[5]A. S. Pikovsky, M. G. Rosenblum & J. Kurths (2001). Synchronization - A Unified Approach to Nonlinear Science. Cambridge: Cambridge University Press.
[6]M. Lakshmanan & K. Murali (1996). Chaos in Nonlinear Oscillators: Controlling and Synchronization. Singapore: World Scientific.
[7]A. L. Fradkov & A. Yu. Pogromsky(1996). Introductionto Control of Oscillations and Chaos. Singapore: World Scientific.
[8]X. Yu & Song Y. (2001).Chaos Synchronizationvia Controlling Partial State of Chaotic Systems. Int. J. Bifurcation & Chaos, 11, 1737-1741.
[9]C. Wang & S. S. Ge. (2001). Adaptive Synchronization of Uncertain Chaotic Systems via Backstepping Design. Chaos Solitons and Fractals, 212, 1199-1206.
[10] M. C. Ho & Y. C. Hung (2002). Synchronization of Two Different Systems by Using Generalised Active Control. Phys Lett. A., 301, 424-428.
[11] M. T. Yassen (2005). Chaos Synchronization Between Two Different Chaotic Systems Using Active Control. Chaos Solitons and Fractals, 23, 131.
[12] Y. Wang, Z. Guan & H. O. Wang (2003). Feedback and Adaptive Control for the Synchronization of Chen System via a Single Variable. Phys. Lett. A., 312, 34-40.
[13] N.F. Rulkov,M.M.Sushchik,L.S.Tsimring,H.D.I.Abarbanel(1995).GeneralizedSynchronization of Chaos in Directionally Coupled Systems. Phys. Rev. E., 51, 980-994.
[14] A. A. Emadzadeh & M. Haeri (2005). Anti-Synchronization of Two Different Chaotic Systems Via Active Control. Trans on Engg., Comp and Tech., 6, 62-65.
[15] Y. Lei, W. Xu, H. Zheng (2005). Synchronization of Two Chaotic Nonlinear Gyros Using Active Control. Phys. Lett. A., 343, 153-158.
[16] E. W. Bai, K. E. Lonngren (1997). Synchronization of Two Lorenz Systems Using Active Control. Chaos Solitons and Fractals, 8, 51-58.
[17] U. E. Vincent & J. A. Laoye (2007). Synchronization and Control of Directed Transport in Chaotic Ratchets via Active Control. Phys. Lett. A., 363, 91-95.
[18] E. W. Bai, K. E. Lonngren (2000). Sequential Synchronization of Two Lorenz Systems Using Active Control. Chaos Solitons and Fractals, 11, 1041-1044.
[19] U. E. Vincent (2005). Synchronization of Rikitake Chaotic Attractor Using Active Control. Phys Lett A., 343, 133.
[20] S. Chen & J. Lu (2002).Synchronizationof an UncertainUnified System via AdaptiveControl. Chaos Solitons and Fractals, 14, 643-647.
[21] H. O. Wang & E. H. Abed (1993). Control of Nonlinear Phenomena at the Inception of Voltage Collapse. Proc. 1993 American control conference, San Francisco Jun, 2071 -2075.
[22] E. H. Abed & P. P. Varaiya (1989). Nonlinear Oscillations in Power Systems. Int. J. of Electric Power and Energy System, 6, 37-43.
[23] E. H. Abed & J. H. Fu (1986). Local Feedback Stabilization and Bifurcation Control, I. Hopf Bifurcation. Systems and Control Letters, 7, 11-17.
[24] L. Rosier, G. Millerioux, G. Bloch (2006). Chaos Synchronization for a Class of Discrete Dynamical Systems on the N-dimensional Torus. Systems and Control Letters, 55, 223-231.
[25] T. Yang (2004). A Survey of Chaotic Secure Communication Systems. Int. J. Comp. Cognition, 2, 81-130.
[26] L. Lu, X. Wu & J. L¨Au(2002). Synchronization of a Unified Chaotic System and the Application in Secure Communication. Phys. Lett. A., 305, 365-370.
[27] G. A. Lvarez, F. Montoya, M. Romera, G. Pastor (2004). Crypt Analyzing a Discrete-time Chaos Synchronization Secure Communication System. Chaos Solitons and Fractals, 21, 689-694.
[28] G. A. lvarez, F. Montoya, M. Romera, G. Pastor (1999). Chaotic Cryptosystems. In Larry D. Sanson,(Eds.), 33rdAnnual1999InternationalCarnahanConferenceonSecurityTechnology.IEEE,332-338.
[29] S. Boccaletti, A. Farini, F. T. Arecchi (1997). Adaptive Synchronizationof Chaos for Secure Communication. Phys. Rev. E., 55(5), 4979-4981.
[30] S. Hayes, C. Grebogi, E. Ott, A. Mark (1994). Experimental Control of Chaos for Communication. Phys. Rev. Lett., 73, 1781-1784.
[31] K. M. Cuomo & A. V. Oppenheim (1993). Circuit Implementation of Synchronized Chaos with Applications to Communications. Phys. Rev. Lett., 71, 65-68.
[32] K. M. Cuomo, A. V. Oppenheim, S. H. Strogatz (1993). Synchronization of Lorenz-based Chaotic Circuits with Applications to Communications. IEEE Trans. Circuits Syst., 40, 626-633.
[33] G. L. Baker (1995). Control of the Chaotic Driven Pendulum. Am. J. Phys., 63, 832-838.
[34] J. Starrett, & R. Tagg (1995). Control of a Chaotic Parametrically Driven Pendulum. Phys. Rev. Lett., 74, 1974-1977.
[35] K. Pyragas (1992). Continuous Control of Chaos by Self-controlling Feedback. Phys. Lett. A., 170, 421- 428.
[36] K. Pyragas (2001). Control of Chaos Via an Unstable Delayed Feedback Controller. Phys. Rev. Lett., 86, 2265-2268.
[37] K. Pyragas & A. Tamasiavicius (1993). Experimental Control of Chaos by Delayed Self-controlling Feedback. Phys. Lett A., 180, 99-102.
[38] B. Bielawski, D. Derozier, P. Glorieux (1994). Controlling Unstable Periodic Orbits by a Delayed Continuous Feedback. Phys. Rev. E.,49, 971-974.
[39] P. Parmanada, R. Madrigal, M. Rivera (1999). Stabilization of Unstable Steady States and Periodic Orbits in an Electrochemical System Using Delayed-feedback Control. Phys. Rev. E., 59, 5266.
[40] H. K. Chen (2002). Chaos and Chaos Synchronization of a Symmetric Gyro with Linear-plus-cubic Damping. J. Sound Vib., 255(4), 719-740.
[41] H. Zhang, X. Ma, M. Li, J. Zou (2005). Controlling and Tracking Hyperchaotic Rossler System Via Active Backstepping Design. Chaos Solitons and Fractals, 26, 353-361.
[42] P. V. Kokotovic (1992). The Joy of Feedback: Nonlinear and Adaptive. IEEE Control Syst. Mag., 6, 7-17.
[43] M. Krstic, I. Kanellakopoulus, P. Kokotovic (1995). Nonlinear and Adaptive Control Design. New York: John Wiley.
[44] A. M. Harb (2004). Nonlinear Chaos Control in a Permanent Magnet Reluctance Machine. Chaos Solitons and Fractals, 19, 1217-1224.
[45] X. Tan, J. Zhang, Y. Yang (2003).SynchronizingChaotic Systems Using BacksteppingDesign. Chaos Solitons and Fractals, 16, 37-45.
[46] A. M. Harb, B. A. Harb (2004). Chaos Control of Third-order Phase-locked Loops Using Backstepping Nonlinear Controller. Chaos Solitons and Fractals, 20(4), 719-723.
[47] J. A. Laoye, U. E. Vincent, S. O. Kareem (2009). Chaos Control of 4-D Chaotic System Using Recursive Backstepping Nonlinear Controller. Chaos, Solitons and Fractals, 39(1), 356-362.
[48] U. E. Vincent,A. N. Njah, J. A. Laoye(2007).ControllingChaoticMotionand DeterministicDirected Transport in Chaotic Ratchets Using Backstepping Nonlinear Controller. Physica D, 231, 130.
[49] S. Mascolo ( ). Backstepping Design for Controlling Lorenz Chaos, Proceedings of the 36th IEEE CDC San Diego. CA 1500-1501.
[50] V. V. Beletskii (1966). Motion of an Artificial Satellite about Its Center of Mass (Jerusalem: Israel Program Sci. Transl.).
[51] V. V. Beletskii, M. L. Pivovarov, E. L. Starostin (1996). Regular and Chaotic Motions in Applied Dynamics of a Rigid Body Chaos, 6, 155-166.
[52] R. B.Singh,V.G.Demin(1972).AbouttheMotionofaHeavyFlexibleStringAttachedtotheSatellite in the Central Field of Attraction Celest. Mech. & Dyn. Astron., 6(3), 268-277
[53] C. Soto-Trevino& T. J. Kaper (1996).Higher-orderMelnikovTheory for Adiabatic Systems. J. Math. Phys., 37, 6220-6249.
[54] L. S. Wang, P. S. Krishnaprasad, J. H. Maddocks (1991). Hamiltonian Dynamics of a Rigid Body in a Central Gravitational Field. Celest. Mech. & Dyn. Astron., 50(4), 349-386.
[55] J. Wisdom (1987). Rotational Dynamics of Irregularly Shaped Natural Satellites. A .J., 94, 1350-60.
[56] J. Wisdom, S. J. Peale, F. Mignard (1984). The Chaotic Rotation of Hyperion Icarus, 58, 137-152
[57] P. Goldreich & S. Peale (1996). Spin-orbit Coupling in the Solar System. A. J., 71, 425-438
[58] A. Khan, R. Sharma, L. M. Saha (1998). Chaotic Motion of an Ellipsoidal Satellite I. Astron. J., 116, 2058-66.
[59] H. Nimeijer, M. Y. Mareels Ivan(1997). An Observer Looks at Synchronization. Circ. Syst. (IEEE Trans.), 144, 882-890.
[60] L. Zengrong ( ). Several Academic Problems about Synchronization. Science Forum Ziran Zazhi, 26(5).
[61] L. Youming, X. Wei, X. Wenxian (2007). Synchronizationof Two Chaotic Four-dimensionalSystems Using Active Control. Chaos Solitons and Fractals, 32, 1823-1829.
In this paper, we have investigated the synchronization behaviour of two identical nonlinear dynamical systems of a rotating ellipsoidal satellite in elliptic orbit under the solar radiation pressure evolving from different initial conditionsusing the active control techniquebased on the Lyapunovstability theory and the Routh-Hurwitz criteria. The designed controller, with our own choice of the coefficient matrix of the error dynamics, are found to be effective in the stabilization of the error states at the origin, thereby, achieving synchronization between the states variables of two dynamical systems under consideration. Numerical simulations are presented to illustrate the effectiveness of the proposed control techniques using mathematica.
Key words
Chaos; Synchronization; Satellite
1.INTRODUCTION
After the pioneering work on chaos control by Ott et al [1] and synchronization of chaotic systems by Pecora and Carroll [2], chaos control and synchronization has received increasing attention [3–7] and has become a very active topic in nonlinear science since last couple of years. Over the last decade various effective methods have been proposed and utilized [8–20] to achieve the control and stabilization of chaotic systems like laser, power electronics etc. The idea of synchronization of two identical chaotic systems that start from different initial conditions consists of linking the trajectory of one system to the same values in the other so that they remain in step with each other, through the transmission of a signal.
The control of physical systems is an important subject in engineering and sciences, thus, in some applications,chaoscanbe usefulwhilein othersit mightbedetrimentalforexamplechaosin powersystems[21–23] and in mechanical systems is objectionable. On the other hand, the idea of chaos synchronization was utilized to build communication systems to ensure the security of information transmitted [24–32]. Several attempts have been made to control and synchronize chaotic systems [2, 16, 18, and 32]. Some of thesemethodsneedseveralcontrollerstorealizesynchronization. TheOGYmethod,forinstance,havebeen successfully applied to many chaotic systems like the periodically driven pendulum [33] and parametric pendulum [34]. Also, the Pyragas time-delayed auto-synchronization method [35, 36] has been shown to be an efficient method that has been realized experimentally in electronic chaos oscillators [37], lasers[38] and chemical systems [39]. In addition, the delayed feedback control, addition of periodic force and adaptive control algorithm has been utilized to control chaos in a symmetric gyro with linear-pluscubic damping [40].
In particular, backstepping design and active control have been recognised as two powerful design methods to control and synchronize chaos. It has been reported [41–43] that backstepping design can guarantee global stability, tracking and transient performance for a broad class of strict-feedback nonlinear systems. In recent time, it has been employed for controlling, tracking and synchronizing many chaotic systems [44–48] as well as hyperchaotic systems [41]. According to ref [45], some of the advantages in the method include applicability to a variety of chaotic systems whether they contain external excitation or not; needs only one controller to realize synchronization between chaotic systems andfinally there are no derivatives in the controller. Zhang [41] states that the controller is singularity free from the nonlinear term of quadratic type, givesflexibility to construct a control law which can be extended to higher dimensional hyperchaotic systems and the closed-loop system is globally stable, while ref [49] adds that it requires less control effort in comparison with the differential geometric method.
The aim of this article is to use the active control technique based on the Lyapunov stability theory and theRouth-Hurwitzcriteriato studythe synchronizationbehaviorofthetwo identicalplanaroscillationof an ellipsoidal satellite in elliptic orbit under solar radiation pressure evolving from different initial conditions.
2.EQUATION OF MOTION OF A SATELLITE IN AN ELLIPTIC ORBIT
Elliptically orbiting planar oscillations of satellites in the solar system make an interesting study, and significant contributions to this end can be found in the works [50-58], all of whom have studied the influence of certain perturbative forces, such as solar radiation pressure, tidal force, and air resistance. In the present work, we consider the planar oscillation of a satellite in elliptic orbit with the spin axisfixed perpendicular to the orbital plane. Let the long axis of the satellite makes an angle xwith a reference axis that isfixed in inertial space, the long axis of the satellite makes an angleφwith satellites planet centre line and the satellite to be a triaxial ellipsoid with principal moments of inertia A < B < C, where C is the moment about the spin axis. The orbit is taken to be afixed ellipse with semi major axis a, eccentricity e, true anomalyν,ω20= 3(B?A)/Cand instantaneous radius r. The equation of motion of satellite planar oscillation in an elliptic orbit around the earth under solar radiation pressure, is
d2x
3.SYNCHRONIZATION VIA ACTIVE CONTROL
For a system of two coupled chaotic oscillators, the master system (˙x = f(x,y)) and the slave system (˙y = g(x,y)),where x(t) and y(t)arethe phase space(state variables),and f(x,y) andg(x,y)are the corresponding nonlinear functions, synchronization in a direct sense implies |x(t)?y(t)|→0 as t→∞. When this occurs the coupled systems are said to be completely synchronized. Chaos synchronizationis related to the observer problem in control theory [59]. The problem may be treated as the design of control laws for full chaotic observer (the slave system) using the known information of the master system so as to ensure that the controlled receiver synchronizes with the master system. Hence, the slave chaotic system completely traces the dynamics of the master in the course of time.
4.NUMERICAL SIMULATION
For the parameters involvedin system under investigation,e = 0.15, h = 0.1, l = 0.7,μ= 0.02,α= 0.0001, n = 0.1andω0= 0.3andthe initial conditionsformaster andslave systems [x1(0), x2(0), x3(0)] = [0,0.1,0] and#y1(0),y2(0),y3(0)$= [0.1,0.2,0.1]respectively,thesystemhasbeensimulatedusingmathematica. The obtained results show that the system under consideration achieved synchronization. Phase plots of (3.1) and (3.2) (Figure 1), time series analysis of (3.1) and (3.2) (Figure 2) and time series analysis of errors(Figure 3) are the witness of achieving synchronization between master and slave system. Further, it also has been confirmed by the convergenceof the synchronization quality defined by e(t) =G
5.CONCLUSION
In this paper, we have investigated the chaos synchronization behaviour of the two identical planar oscillation of an ellipsoidal satellite in elliptic orbit under solar radiation pressure, evolving from different initial conditions via the active control technique based on the Lyapunov stability theory and the Routh-Hurwitz criteria. The results obtained were validated by numerical simulations using mathematica for the proposed technique.
REFERENCES
[1]E. Ott, C. Grebogi & J. A. Yorke (1990). Controlling Chaos. Phys Rev Lett, 64, 1196-1199.
[2]L. M. Pecora& T. L.Carroll (1990).Synchronizationin ChaoticSystems. Phys Rev.Lett, 64, 821-824.
[3]T. Kapitaniak (1996). Controlling Chaos - Theoretical. Practical Methods in Non-linear Dynamics. London: Academic Press.
[4]Chen G. & Dong X. (1998). From Chaos to Order: Methodologies, Perspectives and Applications. Singapore: World Scientific.
[5]A. S. Pikovsky, M. G. Rosenblum & J. Kurths (2001). Synchronization - A Unified Approach to Nonlinear Science. Cambridge: Cambridge University Press.
[6]M. Lakshmanan & K. Murali (1996). Chaos in Nonlinear Oscillators: Controlling and Synchronization. Singapore: World Scientific.
[7]A. L. Fradkov & A. Yu. Pogromsky(1996). Introductionto Control of Oscillations and Chaos. Singapore: World Scientific.
[8]X. Yu & Song Y. (2001).Chaos Synchronizationvia Controlling Partial State of Chaotic Systems. Int. J. Bifurcation & Chaos, 11, 1737-1741.
[9]C. Wang & S. S. Ge. (2001). Adaptive Synchronization of Uncertain Chaotic Systems via Backstepping Design. Chaos Solitons and Fractals, 212, 1199-1206.
[10] M. C. Ho & Y. C. Hung (2002). Synchronization of Two Different Systems by Using Generalised Active Control. Phys Lett. A., 301, 424-428.
[11] M. T. Yassen (2005). Chaos Synchronization Between Two Different Chaotic Systems Using Active Control. Chaos Solitons and Fractals, 23, 131.
[12] Y. Wang, Z. Guan & H. O. Wang (2003). Feedback and Adaptive Control for the Synchronization of Chen System via a Single Variable. Phys. Lett. A., 312, 34-40.
[13] N.F. Rulkov,M.M.Sushchik,L.S.Tsimring,H.D.I.Abarbanel(1995).GeneralizedSynchronization of Chaos in Directionally Coupled Systems. Phys. Rev. E., 51, 980-994.
[14] A. A. Emadzadeh & M. Haeri (2005). Anti-Synchronization of Two Different Chaotic Systems Via Active Control. Trans on Engg., Comp and Tech., 6, 62-65.
[15] Y. Lei, W. Xu, H. Zheng (2005). Synchronization of Two Chaotic Nonlinear Gyros Using Active Control. Phys. Lett. A., 343, 153-158.
[16] E. W. Bai, K. E. Lonngren (1997). Synchronization of Two Lorenz Systems Using Active Control. Chaos Solitons and Fractals, 8, 51-58.
[17] U. E. Vincent & J. A. Laoye (2007). Synchronization and Control of Directed Transport in Chaotic Ratchets via Active Control. Phys. Lett. A., 363, 91-95.
[18] E. W. Bai, K. E. Lonngren (2000). Sequential Synchronization of Two Lorenz Systems Using Active Control. Chaos Solitons and Fractals, 11, 1041-1044.
[19] U. E. Vincent (2005). Synchronization of Rikitake Chaotic Attractor Using Active Control. Phys Lett A., 343, 133.
[20] S. Chen & J. Lu (2002).Synchronizationof an UncertainUnified System via AdaptiveControl. Chaos Solitons and Fractals, 14, 643-647.
[21] H. O. Wang & E. H. Abed (1993). Control of Nonlinear Phenomena at the Inception of Voltage Collapse. Proc. 1993 American control conference, San Francisco Jun, 2071 -2075.
[22] E. H. Abed & P. P. Varaiya (1989). Nonlinear Oscillations in Power Systems. Int. J. of Electric Power and Energy System, 6, 37-43.
[23] E. H. Abed & J. H. Fu (1986). Local Feedback Stabilization and Bifurcation Control, I. Hopf Bifurcation. Systems and Control Letters, 7, 11-17.
[24] L. Rosier, G. Millerioux, G. Bloch (2006). Chaos Synchronization for a Class of Discrete Dynamical Systems on the N-dimensional Torus. Systems and Control Letters, 55, 223-231.
[25] T. Yang (2004). A Survey of Chaotic Secure Communication Systems. Int. J. Comp. Cognition, 2, 81-130.
[26] L. Lu, X. Wu & J. L¨Au(2002). Synchronization of a Unified Chaotic System and the Application in Secure Communication. Phys. Lett. A., 305, 365-370.
[27] G. A. Lvarez, F. Montoya, M. Romera, G. Pastor (2004). Crypt Analyzing a Discrete-time Chaos Synchronization Secure Communication System. Chaos Solitons and Fractals, 21, 689-694.
[28] G. A. lvarez, F. Montoya, M. Romera, G. Pastor (1999). Chaotic Cryptosystems. In Larry D. Sanson,(Eds.), 33rdAnnual1999InternationalCarnahanConferenceonSecurityTechnology.IEEE,332-338.
[29] S. Boccaletti, A. Farini, F. T. Arecchi (1997). Adaptive Synchronizationof Chaos for Secure Communication. Phys. Rev. E., 55(5), 4979-4981.
[30] S. Hayes, C. Grebogi, E. Ott, A. Mark (1994). Experimental Control of Chaos for Communication. Phys. Rev. Lett., 73, 1781-1784.
[31] K. M. Cuomo & A. V. Oppenheim (1993). Circuit Implementation of Synchronized Chaos with Applications to Communications. Phys. Rev. Lett., 71, 65-68.
[32] K. M. Cuomo, A. V. Oppenheim, S. H. Strogatz (1993). Synchronization of Lorenz-based Chaotic Circuits with Applications to Communications. IEEE Trans. Circuits Syst., 40, 626-633.
[33] G. L. Baker (1995). Control of the Chaotic Driven Pendulum. Am. J. Phys., 63, 832-838.
[34] J. Starrett, & R. Tagg (1995). Control of a Chaotic Parametrically Driven Pendulum. Phys. Rev. Lett., 74, 1974-1977.
[35] K. Pyragas (1992). Continuous Control of Chaos by Self-controlling Feedback. Phys. Lett. A., 170, 421- 428.
[36] K. Pyragas (2001). Control of Chaos Via an Unstable Delayed Feedback Controller. Phys. Rev. Lett., 86, 2265-2268.
[37] K. Pyragas & A. Tamasiavicius (1993). Experimental Control of Chaos by Delayed Self-controlling Feedback. Phys. Lett A., 180, 99-102.
[38] B. Bielawski, D. Derozier, P. Glorieux (1994). Controlling Unstable Periodic Orbits by a Delayed Continuous Feedback. Phys. Rev. E.,49, 971-974.
[39] P. Parmanada, R. Madrigal, M. Rivera (1999). Stabilization of Unstable Steady States and Periodic Orbits in an Electrochemical System Using Delayed-feedback Control. Phys. Rev. E., 59, 5266.
[40] H. K. Chen (2002). Chaos and Chaos Synchronization of a Symmetric Gyro with Linear-plus-cubic Damping. J. Sound Vib., 255(4), 719-740.
[41] H. Zhang, X. Ma, M. Li, J. Zou (2005). Controlling and Tracking Hyperchaotic Rossler System Via Active Backstepping Design. Chaos Solitons and Fractals, 26, 353-361.
[42] P. V. Kokotovic (1992). The Joy of Feedback: Nonlinear and Adaptive. IEEE Control Syst. Mag., 6, 7-17.
[43] M. Krstic, I. Kanellakopoulus, P. Kokotovic (1995). Nonlinear and Adaptive Control Design. New York: John Wiley.
[44] A. M. Harb (2004). Nonlinear Chaos Control in a Permanent Magnet Reluctance Machine. Chaos Solitons and Fractals, 19, 1217-1224.
[45] X. Tan, J. Zhang, Y. Yang (2003).SynchronizingChaotic Systems Using BacksteppingDesign. Chaos Solitons and Fractals, 16, 37-45.
[46] A. M. Harb, B. A. Harb (2004). Chaos Control of Third-order Phase-locked Loops Using Backstepping Nonlinear Controller. Chaos Solitons and Fractals, 20(4), 719-723.
[47] J. A. Laoye, U. E. Vincent, S. O. Kareem (2009). Chaos Control of 4-D Chaotic System Using Recursive Backstepping Nonlinear Controller. Chaos, Solitons and Fractals, 39(1), 356-362.
[48] U. E. Vincent,A. N. Njah, J. A. Laoye(2007).ControllingChaoticMotionand DeterministicDirected Transport in Chaotic Ratchets Using Backstepping Nonlinear Controller. Physica D, 231, 130.
[49] S. Mascolo ( ). Backstepping Design for Controlling Lorenz Chaos, Proceedings of the 36th IEEE CDC San Diego. CA 1500-1501.
[50] V. V. Beletskii (1966). Motion of an Artificial Satellite about Its Center of Mass (Jerusalem: Israel Program Sci. Transl.).
[51] V. V. Beletskii, M. L. Pivovarov, E. L. Starostin (1996). Regular and Chaotic Motions in Applied Dynamics of a Rigid Body Chaos, 6, 155-166.
[52] R. B.Singh,V.G.Demin(1972).AbouttheMotionofaHeavyFlexibleStringAttachedtotheSatellite in the Central Field of Attraction Celest. Mech. & Dyn. Astron., 6(3), 268-277
[53] C. Soto-Trevino& T. J. Kaper (1996).Higher-orderMelnikovTheory for Adiabatic Systems. J. Math. Phys., 37, 6220-6249.
[54] L. S. Wang, P. S. Krishnaprasad, J. H. Maddocks (1991). Hamiltonian Dynamics of a Rigid Body in a Central Gravitational Field. Celest. Mech. & Dyn. Astron., 50(4), 349-386.
[55] J. Wisdom (1987). Rotational Dynamics of Irregularly Shaped Natural Satellites. A .J., 94, 1350-60.
[56] J. Wisdom, S. J. Peale, F. Mignard (1984). The Chaotic Rotation of Hyperion Icarus, 58, 137-152
[57] P. Goldreich & S. Peale (1996). Spin-orbit Coupling in the Solar System. A. J., 71, 425-438
[58] A. Khan, R. Sharma, L. M. Saha (1998). Chaotic Motion of an Ellipsoidal Satellite I. Astron. J., 116, 2058-66.
[59] H. Nimeijer, M. Y. Mareels Ivan(1997). An Observer Looks at Synchronization. Circ. Syst. (IEEE Trans.), 144, 882-890.
[60] L. Zengrong ( ). Several Academic Problems about Synchronization. Science Forum Ziran Zazhi, 26(5).
[61] L. Youming, X. Wei, X. Wenxian (2007). Synchronizationof Two Chaotic Four-dimensionalSystems Using Active Control. Chaos Solitons and Fractals, 32, 1823-1829.