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.
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
Wang, X. Y. (2003) Chaos in Complex Nonlinear Systems. Publishing House of Electronics Industry, Beijing.
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
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
Krstic, M., Kaneuakopoulos, I. and Kokotovic, P.V. (1995) Nonlinear Adaptive Control Design. Wiley, New York.
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
Rössler, O.E. (1979) An Equation for Hyperchaos. Physics Letters A, 71, 155-157. https://doi.org/10.1016/0375-9601(79)90150-6
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
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