We realize the function projective synchronization (FPS) between two discrete-time hyperchaotic systems, that is, the drive state vectors and the response state vectors can evolve in a proportional scaling function matrix. In this paper, a systematic scheme is explored to investigate the function projective synchronization of two identical discrete-time hyperchaotic systems using the backstepping method. Additionally, FPS of two different hyperchaotic systems is also realized. Numeric simulations are given to verify the effectiveness of our scheme.
References
[1]
Ott, E., Grebogi, C. and Yorke, J.A. (1990) Controlling Chaos. Physical Review Letters, 64, 1196-1199. https://doi.org/10.1103/PhysRevLett.64.1196
[2]
Wang, H., Ye, J.M., Miao, Z.H. and Jonckheere, E.A. (2018) Robust Finite-Time Chaos Synchronization of Time-Delay Chaotic Systems and Its Application in Secure Communication. Transactions of the Institute of Measurement and Control, 40, 1177-1187. https://doi.org/10.1177/0142331216678311
[3]
Hamel, S. and Boulkroune, A. (2016) A Generalized Function Projective Synchronization Scheme for Uncertain Chaotic Systems Subject to Input Nonlinearities. International Journal of General Systems, 45, 689-710. https://doi.org/10.1080/03081079.2015.1118094
[4]
Boccaletti, S., Kurth, J., Osipov, G., Valladares, D.L. and Zhou, C.S. (2002) The Synchronization of Chaotic Systems. Physics Reports, 366, 1-101. https://doi.org/10.1016/S0370-1573(02)00137-0
[5]
Al-Azzawi, S.F. and Aziz, M.M. (2018) Chaos Synchronization of Nonlinear Dynamical Systems via a Novel Analytical Approach. Alexandria Engineering Journal, 57, 3493-3500. https://doi.org/10.1016/j.aej.2017.11.017
[6]
Chen, G.R. and Dong, X.N. (1998) From Chaos to Order: Methodologies, Perspectives and Applications. World Scientific, Singapore. https://doi.org/10.1142/3033
[7]
Yan, Z.Y. (2005) Q-S (Lag or Anticipated) Synchronization Backstepping Scheme in a Class of Discrete-Time Chaotic (Hyperchaotic) Systems: A Symbolic-Numeric Computation Approach. Chaos, 15, Article ID: 023902. https://doi.org/10.1063/1.1876612
[8]
Yang, S., Hu, C., Yu, J. and Jiang, H.J. (2021) Exponential Synchronization of Fractional-Order Reaction-Diffusion Coupled Neural Networks with Hybrid Delay-Dependent Impulses. Journal of the Franklin Institute, 358, 3167-3192.
[9]
Mainieri, R. and Rehacek, J. (1999) Projective Synchronization in Three-Dimensional Chaotic Systems. Physical Review Letters, 82, 3042-3045. https://doi.org/10.1103/PhysRevLett.82.3042
[10]
Li, G.H. (2007) Generalized Projective Synchronization between Lorenz System and Chen’s System. Chaos, Solitons & Fractals, 32, 1454-1458. https://doi.org/10.1016/j.chaos.2005.11.073
[11]
Chen, Y. and Li, X. (2007) Function Projective Synchronization between Two Different Chaotic Systems. Zeitschrift für Naturforschung A, 62, 29-33. https://doi.org/10.1515/zna-2007-1-205
[12]
Chen, Y. and Li, X. (2007) Function Projective Synchronization between Two Identical Chaotic Systems. International Journal of Modern Physics C, 18, 883-888. https://doi.org/10.1142/S0129183107010607
[13]
Li, X. and Chen, Y. (2007) Generalized Projective Synchronization between Rössler System and New Unified Chaotic System. Communications in Theoretical Physics, 48, 132-136. https://doi.org/10.1088/0253-6102/48/1/027
[14]
Qiu, X.L., Bin, H.H. and Chu, L.C. (2017) Modified Function Projective Synchronization of Complex Networks with Multiple Proportional Delays. Applied Mathematics, 8, 537-549. https://doi.org/10.4236/am.2017.84043
[15]
Li, X. and Chen, Y. (2009) Stabilizing of Two-Dimensional Discrete Lorenz Chaotic System and Three-Dimensional Discrete Rössler Hyperchaotic System. Chinese Physics Letters, 26, Article ID: 090503. https://doi.org/10.1088/0256-307X/26/9/090503
[16]
Li, C.D., Liao, X.F. and Wong, K.W. (2005) Lag Synchronization of Hyperchaos with Application to Secure Communications. Chaos, Solitons & Fractals, 23, 183-193. https://doi.org/10.1016/j.chaos.2004.04.025
[17]
Vicente, R., Dauden, J., Colet, P. and Toral, R. (2005) Analysis and Characterization of the Hyperchaos Generated by a Semiconductor Laser Subject to a Delayed Feedback Loop. IEEE Journal of Quantum Electronics, 41, 541-548. https://doi.org/10.1109/JQE.2005.843606
[18]
Baleanu, D., Jajarmi, A., Sajjadi, S.S. and Mozyrska, D. (2019) A New Fractional Model and Optimal Control of a Tumor-Immune Surveillance with Non-Singular Derivative Operator. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29, Article ID: 083127. https://doi.org/10.1063/1.5096159
[19]
Itoh, M. and Chua, L.O. (2003) Equivalent CNN Cell Models and Patterns. International Journal of Bifurcation and Chaos, 13, 1055-1161. https://doi.org/10.1142/S0218127403007151
[20]
Wang, X. Y. (2003) Chaos in Complex Nonlinear Systems. Publishing House of Electronics Industry, Beijing.
[21]
Yan, Z.Y. (2006) Q-S (Complete or Anticipated) Synchronization Backstepping Scheme in a Class of Discrete-Time Chaotic (Hyperchaotic) Systems: A Symbolic-Numeric Computation Approach. Chaos, 16, Article ID: 013119. https://doi.org/10.1063/1.1930727
[22]
Li, X., Chen, Y. and Li, Z.B. (2008) Function Projective Synchronization of Discrete-Time Chaotic Systems. Zeitschrift für Naturforschung A, 63, 7-14. https://doi.org/10.1515/zna-2008-1-202
[23]
Krstic, M., Kaneuakopoulos, I. and Kokotovic, P.V. (1995) Nonlinear Adaptive Control Design. Wiley, New York.
[24]
He, P., Ma, S.H. and Fan, T. (2012) Finite-Time Mixed Outer Synchronization of Complex Networks with Coupling Time-Varying Delay. Chaos, 22, Article ID: 043151. https://doi.org/10.1063/1.4773005
[25]
Rössler, O.E. (1979) An Equation for Hyperchaos. Physics Letters A, 71, 155-157. https://doi.org/10.1016/0375-9601(79)90150-6
[26]
Yan, Z.Y. (2005) Q-S Synchronization in 3D Hénon-Like Map and Generalized Hénon Map via a Scalar Controller. Physics Letters A, 342, 309-317. https://doi.org/10.1016/j.physleta.2005.04.049
[27]
Wu, G.C., Baleanu, D., Xie, H.P. and Chen, F.L. (2016) Chaos Synchronization of Fractional Chaotic Maps Based on the Stability Condition. Physica A: Statistical Mechanics and its Applications, 460, 374-383. https://doi.org/10.1016/j.physa.2016.05.045
