Impulsive Synchronization of Hyperchaotic Lü Systems with Two Methods
Author(s)
Mingjun Wang*
ABSTRACT
In the paper, impulsive synchronization of two hyperchaotic Lü systems with different initial conditions is studied. The sufficient conditions on feedback strength and impulsive distances are established from two different angles to guarantee the synchronization. The relevant theoretical proofs are presented. Numerical simulations show the effectiveness of the methods.
KEYWORDS
Impulsive Synchronization, Impulsive Distance, Hyperchaotic Lü System, Largest Lyapunov Exponent
Impulsive Synchronization, Impulsive Distance, Hyperchaotic Lü System, Largest Lyapunov Exponent
Cite this paper
Wang, M. (2015) Impulsive Synchronization of Hyperchaotic Lü Systems with Two Methods. International Journal of Modern Nonlinear Theory and Application, 4, 1-9. doi: 10.4236/ijmnta.2015.41001.
Wang, M. (2015) Impulsive Synchronization of Hyperchaotic Lü Systems with Two Methods. International Journal of Modern Nonlinear Theory and Application, 4, 1-9. doi: 10.4236/ijmnta.2015.41001.
References
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http://dx.doi.org/10.1103/PhysRevLett.64.821 [2] Carroll, T. and Pecora, L. (1991) Synchronizing Chaotic Circuits. IEEE Transactions on Circuits Systems, 38, 453-456.
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http://dx.doi.org/10.12693/APhysPolA.123.193 [8] Itoh, M., Yang, T. and Chua, L. (2001) Conditions for Impulsive Synchronization of Chaotic and Hyperchaotic Systems. International Journal of Bifurcation and Chaos, 11, 551-560.
http://dx.doi.org/10.1142/S0218127401002262 [9] Chai, X., Gan, Z. and Shi, C. (2013) Impulsive Synchronization and Adaptive-Impulsive Synchronization of a Novel Financial Hyperchaotic System. Mathematical Problems in Engineering, 2013, 751616.
http://dx.doi.org/10.1155/2013/751616 [10] Senouci, A. and Boukabou, A. (2014) Predictive Control and Synchronization of Chaotic and Hyperchaotic Systems Based on a T?S Fuzzy Model. Mathematics and Computers in Simulation, 105, 62-78.
http://dx.doi.org/10.1016/j.matcom.2014.05.007 [11] Candido, R. and Eisencraft, M. (2014) Channel Equalization for Synchronization of Chaotic Maps. Digital Signal Processing, 33, 42-49. http://dx.doi.org/10.1016/j.dsp.2014.07.001 [12] Xiao, X., Zhou, L. and Zhang, Z. (2014) Synchronization of Chaotic Lur’e Systems with Quantized Sampled-Data Controller. Communications in Nonlinear Science and Numerical Simulation, 19, 2039-2047.
http://dx.doi.org/10.1016/j.cnsns.2013.10.020 [13] Ren, Q. and Zhao, J. (2006) Impulsive Synchronization of Coupled Chaotic Systems via Adaptive-Feedback Approach. Physics Letters A, 355, 342-347. http://dx.doi.org/10.1016/j.physleta.2006.02.053 [14] Chen, Y. and Guo, J. (2014) Synchronization of a Class of Chaotic Systems Using Small Impulsive Signal. International Journal for Light and Electron Optics, 125, 6407-6412.
http://dx.doi.org/10.1016/j.ijleo.2014.06.117 [15] Xi, H., Yu, S., Zhang, R. and Xu, L. (2014) Adaptive Impulsive Synchronization for a Class of Fractional-Order Chaotic and Hyperchaotic Systems. Optik—International Journal for Light and Electron Optics, 125, 2036-2040.
http://dx.doi.org/10.1016/j.ijleo.2013.12.002 [16] Xu, Y., Wang, Y. and Zhao, J. (2014) A New Chaotic System without Linear Term and Its Impulsive Synchronization. Optik—International Journal for Light and Electron Optics, 125, 2526-2530.
http://dx.doi.org/10.1016/j.ijleo.2013.10.123 [17] Chen, A., Lu, J., Lü, J. and Yu, S. (2006) Generating Hyperchaotic Lü Attractor via State Feedback Control. Physica A: Statistical Mechanics and its Applications, 364, 103-110.
http://dx.doi.org/10.1016/j.physa.2005.09.039 [18] Lü, J., Lu, J. and Chen, S. (2002) Analysis and Applications of Chaotic Time Series. Wuhan University Press, Wuhan. [19] Mohammad, H. and Mahsa, D. (2006) Impulsive Synchronization of Chen’s Hyperchaotic System. Physics Letters A, 356, 226-230. http://dx.doi.org/10.1016/j.physleta.2006.03.051
[1] Pecora, L. and Carroll, T. (1990) Synchronization in Chaotic Systems. Physical Review Letters, 64, 821-824.
http://dx.doi.org/10.1103/PhysRevLett.64.821 [2] Carroll, T. and Pecora, L. (1991) Synchronizing Chaotic Circuits. IEEE Transactions on Circuits Systems, 38, 453-456.
http://dx.doi.org/10.1109/31.75404 [3] Chen, G. and Dong, X. (1998) From Chaos to Order: Methodologies, Perspectives and Applications. World Scientific, Singapore. [4] Wang, G.R., Yu, X.L. and Chen, S.G. (2001) Chaotic Control, Synchronization and Utilizing. National Defence Industry Press, Beijing. [5] Wang, X.Y. (2003) Chaos in the Complex Nonlinearity System. Electronics Industry Press, Beijing. [6] Chen, G.R. and Lü, J.H. (2003) Dynamical Analyses, Control and Synchronization of the Lorenz system family. Science Press, Beijing. [7] Kemih, K., Bouraoui, H., Messadi, M. and Ghanes, M. (2013) Impulsive Control and Synchronization of a New 5D Hyperchaotic System. Acta Physica Polonica A, 123, 193-195.
http://dx.doi.org/10.12693/APhysPolA.123.193 [8] Itoh, M., Yang, T. and Chua, L. (2001) Conditions for Impulsive Synchronization of Chaotic and Hyperchaotic Systems. International Journal of Bifurcation and Chaos, 11, 551-560.
http://dx.doi.org/10.1142/S0218127401002262 [9] Chai, X., Gan, Z. and Shi, C. (2013) Impulsive Synchronization and Adaptive-Impulsive Synchronization of a Novel Financial Hyperchaotic System. Mathematical Problems in Engineering, 2013, 751616.
http://dx.doi.org/10.1155/2013/751616 [10] Senouci, A. and Boukabou, A. (2014) Predictive Control and Synchronization of Chaotic and Hyperchaotic Systems Based on a T?S Fuzzy Model. Mathematics and Computers in Simulation, 105, 62-78.
http://dx.doi.org/10.1016/j.matcom.2014.05.007 [11] Candido, R. and Eisencraft, M. (2014) Channel Equalization for Synchronization of Chaotic Maps. Digital Signal Processing, 33, 42-49. http://dx.doi.org/10.1016/j.dsp.2014.07.001 [12] Xiao, X., Zhou, L. and Zhang, Z. (2014) Synchronization of Chaotic Lur’e Systems with Quantized Sampled-Data Controller. Communications in Nonlinear Science and Numerical Simulation, 19, 2039-2047.
http://dx.doi.org/10.1016/j.cnsns.2013.10.020 [13] Ren, Q. and Zhao, J. (2006) Impulsive Synchronization of Coupled Chaotic Systems via Adaptive-Feedback Approach. Physics Letters A, 355, 342-347. http://dx.doi.org/10.1016/j.physleta.2006.02.053 [14] Chen, Y. and Guo, J. (2014) Synchronization of a Class of Chaotic Systems Using Small Impulsive Signal. International Journal for Light and Electron Optics, 125, 6407-6412.
http://dx.doi.org/10.1016/j.ijleo.2014.06.117 [15] Xi, H., Yu, S., Zhang, R. and Xu, L. (2014) Adaptive Impulsive Synchronization for a Class of Fractional-Order Chaotic and Hyperchaotic Systems. Optik—International Journal for Light and Electron Optics, 125, 2036-2040.
http://dx.doi.org/10.1016/j.ijleo.2013.12.002 [16] Xu, Y., Wang, Y. and Zhao, J. (2014) A New Chaotic System without Linear Term and Its Impulsive Synchronization. Optik—International Journal for Light and Electron Optics, 125, 2526-2530.
http://dx.doi.org/10.1016/j.ijleo.2013.10.123 [17] Chen, A., Lu, J., Lü, J. and Yu, S. (2006) Generating Hyperchaotic Lü Attractor via State Feedback Control. Physica A: Statistical Mechanics and its Applications, 364, 103-110.
http://dx.doi.org/10.1016/j.physa.2005.09.039 [18] Lü, J., Lu, J. and Chen, S. (2002) Analysis and Applications of Chaotic Time Series. Wuhan University Press, Wuhan. [19] Mohammad, H. and Mahsa, D. (2006) Impulsive Synchronization of Chen’s Hyperchaotic System. Physics Letters A, 356, 226-230. http://dx.doi.org/10.1016/j.physleta.2006.03.051