Open Access Open Access  Restricted Access Subscription or Fee Access

Anti-Synchronization of Liu and Li Systems by Active Nonlinear Control

Dr.V. Sundarapandian

Abstract


Since the seminal work by Pecora and Carroll in 1990, the global chaos synchronization problem has been studied extensively in the chaos literature and it has important applications in secure communications and data encryption. This paper investigates the global chaos anti-synchronization of identical Liu systems (2004), identical Li systems (2009) and non-identical Liu and Li chaotic systems. Active nonlinear control is the method adopted to achieve the anti-synchronization of the identical and different Liu and Li systems. Our stability results derived in this paper are established using active control method and Lyapunov stability theory. We use a quadratic Lyapunov function for establishing global asymptotic stability of the error dynamics for the chaos anti-synchronization problems. Since the Lyapunov exponents are not required for these calculations, the nonlinear control method is effective and convenient to synchronize the identical and different Liu and Li systems. Numerical simulations are shown to validate and illustrate the effectiveness of the anti-synchronization results derived in this paper.

Keywords


Chaos Anti-Synchronization, Active Nonlinear Control, Chaos, Liu System, Li System.

Full Text:

PDF

References


K.T. Alligood, T. Sauer and J.A. Yorke, Chaos: An Introduction to Dynamical Systems, New York: Springer-Verlag, 1997.

H. Fujisaka and T. Yamada, “Stability theory of synchronized motion in coupled-oscillator systems,” Progress of Theoretical Physics, vol. 69, pp. 32-47, 1983.

L.M. Pecora and T.L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, pp. 821-824, 1990.

L.M. Pecora and T.L. Carroll, “Synchronizing in chaotic circuits,” IEEE Transactions on Circuits and Systems, vol. 38, pp. 453-456, 1991.

M. Lakshmanan and K. Murali, Chaos in Nonlinear Oscillators: Controlling and Synchronization, World Scientific, Singapore, 1996.

S.K. Han, C. Kerrer and Y. Kuramoto, “D-phasing and bursting in coupled neural oscillators,” Physical Review Letters, vol. 75, pp. 3190-3193, 1995.

B. Blasius, A. Huppert and L. Stone, “Complex dynamics and phase synchronization in spatially extended ecological system,” Nature, Vol. 399, pp. 354-359, 1999.

J. Lu, X. Wu, X. Han and J. Lü, “Adaptive feedback synchronization of a unified chaotic system,” Physics Letters A, vol. 329, pp. 327-333, 2004.

L. Kocarev and U. Partliz, “General approach for chaotic synchronization with applications to communications,” Physical Review Letters, vol. 74, pp. 5028-5030, 1995.

K. Murali and M. Lakshmanan, “Secure communication using a compound signal using sampled-data feedback,” Applied Mathematics and Mechanics, vol. 11, pp. 1309-1315, 2003.

T. Yang and L.O. Chua, “Generalized synchronization of chaos via linear transformation,” International Journal of Bifurcation and Chaos, vol. 9, pp. 215-219, 1999.

E. Ott, C. Grebogi and J.A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, pp. 1196-1199, 1990.

J.H. Park and O.M. Kwon, “A novel criterion for delayed feedback control of time-delay chaotic systems,” Chaos, Solitons and Fractals, vol. 17, pp. 709-716, 2003.

M.C. Ho and Y.C. Hung, “Synchronization of two different systems by using generalized active control,” Phys. Lett. A, vol. 301, pp. 424-428, 2002..

H.K. Chen, “Global chaos synchronization of new chaotic systems via nonlinear control,” Chaos, Solitons and Fractals, vol. 23, pp. 1245-1251, 2005.

V. Sundarapandian and R. Suresh, “Global chaos synchronization of hyperchaotic Qi and Jia systems by nonlinear control,” International Journal of Distributed and Parallel Systems, vol. 2, no. 2, pp. 83-94, 2011.

V. Sundarapandian, “Hybrid chaos synchronization of hyperchaotic Liu and hyperchaotic Chen systems by active nonlinear control,” International Journal of Computer Science, Engineering and Information Technology, vol. 1, no. 2, pp 1-14, 2011.

V. Sundarapandian and R. Karthikeyan, “Active controller design for global chaos anti-synchronization of Li and Tigan chaotic systems,” International Journal of Computer Science and Information Technology, vol. 3, no. 4, pp 255-268, 2011.

V. Sundarapandian, “Global chaos synchronization of Li and Pan chaotic systems by active nonlinear control,” CIIT International Journal of Digital Signal Processing, vol. 3, no. 7, pp. 308-312, 2011.

V. Sundarapandian, “Anti-synchronization of Cai and Tigan systems by active nonlinear control,” CIIT International Journal of Programmable Device Circuits and Systems, vol. 3, no. 9, pp. 456-460, 2011.

S.S. Ge, C. Wang and T.H. Lee, “Adaptive backstepping control of a class of chaotic systems,” International Journal of Bifurcation and Chaos, vol. 10, pp. 1149-1156, 2000.

X. Wu and J. Lü, “Parameter identification and backstepping control of uncertain Lü sysetm,” Chaos, Solitons and Fractals, vol. 18, pp. 721-729, 2003.

J.H. Park, “Synchronization of Genesio chaotic system via backstepping approach,” Chaos, Solitons and Fractals, vol. 27, pp. 1369-1375, 2006.

D. Vincent, “Chaos synchronization using active control and backstepping control: A comparative analysis,” Nonlinear Analysis, vol. 13, no. 2, pp. 253-261, 2008.

T.L. Liao and S.H. Tsai, “Adaptive synchronization of chaotic systems and its applications to secure communications”, Chaos, Solitons and Fractals, vol. 11, pp. 1387-1396, 2000.

J.H. Park, S.M. Lee and O.M. Kwon, “Adaptive synchronization of Genesio-Tesi chaotic system via a novel feedback control,” Physics Letters A, vol. 371, pp. 263-270, 2007.

J. Lu, X. Wu, X. Han and J. Lü, “Adaptive feedback synchronization of a unified chaotic system,” Physics Letters A, vol. 329, pp 327-333, 2004.

V. Sundarapandian, “Adaptive control and synchronization of hyperchaotic Liu system,” International Journal of Computer Science, Engineering and Information Technology, vol. 1, no. 2, pp. 29-40, 2011.

V. Sundarapandian, “Adaptive control and synchronization of hyperchaotic Newton-Leipnik system,” International Journal of Advanced Information Technology, vol. 1, no. 3, pp. 22-33, 2011.

V. Sundarapandian, “Adaptive control and synchronization of a highly chaotic attractor,” International Journal of Information Sciences and Techniques, vol. 1, no. 2, pp 1-11, 2011.

H.T. Yau, “Design of adaptive sliding mode controller for chaos synchronization with uncertaintities,” Chaos, Solitons and Fractals, vol. 22, pp. 341-347, 2004.

V.I. Utkin, “Sliding mode control design principles and applications to electric drives,” IEEE Trans. Industrial Electronics, vol. 40, pp 23-36, 1993.

V. Sundarapandian, “Adaptive control and synchronization of hyperchaotic Newton-Leipnik system,” International Journal of Advanced Information Technology, vol. 1, no. 3, pp 22-33, 2011.

V. Sundarapandian, “Global chaos synchronization of hyperchaotic Newton-Leipnik systems by sliding mode control,” International Journal of Computer Science, Engineering and Applications, vol. 1, no. 4, pp 127-138, 2011.

V. Sundarapandian, “Hybrid chaos synchronization of hyperchaotic Newton-Leipnik systems by sliding mode control,” International Journal of Control Theory and Computer Modeling, vol. 1, no. 2, pp 1-10, 2011.

C. Liu, T. Liu, L. Liu and K. Liu, “A new chaotic attractor,” Chaos, Solitons and Fractals, vol. 22, pp. 1031-1038, 2004.

X.F. Li, K.E. Chlouveakis and D.L. Xu, “Nonlinear dynamics and circuit realization of a new chaotic flow: A variant of Lorenz, Chen and Lü,” Nonlinear Analysis, vol. 10, pp. 2357-2368, 2009.

W. Hahn, The Stability of Motion, Springer, Berlin, 1967.


Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.