# American Institute of Mathematical Sciences

## Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters

 1 Department of Mathematics, University of Mentouri Constantine 1, 25000, Algeria 2 Department of Sciences and Technology, Mathematics and their Interactions Laboratory, University of Mila, 43000, Algeria

* Corresponding author: Rabiaa Ouahabi

Received  October 2019 Revised  February 2020 Published  June 2020

This paper proposes a new scheme generalized hybrid projective synchronization for two different chaotic systems using adaptive control, where the master and slave systems do not necessarily have the same number of uncertain parameters. In this method the master system is synchronized by the sum of hybrid state variables for the slave system. Based on Lyapunov stability theory, an adaptive controller for the synchronization of two different chaotic systems is proposed, This method is also applicable if the master and slave systems are identical. As example the generalized hybrid projective synchronization between Vaidyanathan and Zeraoulia chaotic systems are discussed. Numerical simulation are provided to demonstrate the effectiveness of the proposed method.

Citation: Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020182
##### References:
 [1] E. N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar [2] O. E. Rössler, An equation for continuous chaos, Physics Letters A, 57 (1976), 397-398.   Google Scholar [3] J. C. Sprott, Some simple chaotic flows, Physical Review E, 50 (1994), 647-650.  doi: 10.1103/PhysRevE.50.R647.  Google Scholar [4] J. Lü and G. Chen, A new chaotic attractor coined, International Journal of Bifurcation and Chaos, 12 (2002), 659-661.  doi: 10.1142/S0218127402004620.  Google Scholar [5] Z. Elhadj, Analysis of a new three-dimensional quadratic chaotic system, Radioengineering, 17 (2008), 9 pp. Google Scholar [6] M.-S. Abdelouahab and N.-E. Hamri, A new chaotic attractor from hybrid optical bistable system, Nonlinear Dynamics, 67 (2012), 457-463.  doi: 10.1007/s11071-011-9994-5.  Google Scholar [7] S.-J. Prakash and R.-B. Krishna, A more chaotic and easily hardware implementable new 3-D chaotic system in comparison with 50 reported systems, Nonlinear Dynamics, 93 (2018), 1121-1148.   Google Scholar [8] Ü. Çavuşoǧlu, S. Panahi, A. Akgül, S. Jafari and S. Kaçar, A new chaotic system with hidden attractor and its engineering applications: Analog circuit realization and image encryption, Analog Integrated Circuits and Signal Processing, 98 (2019), 85-99.   Google Scholar [9] L. Philippe, S. John and A. N. Jordan, Chaos in continuously monitored quantum systems: An optimal-path approach, Physical Review A, 98 (2018), 012141. Google Scholar [10] P. Sadeghi, S. Panahi, B. Hatef, S. Jafari and J. C. Sprott, A new chaotic model for glucose-insulin regulatory system, Chaos, Solitons & Fractals, 112 (2018), 44-51.  doi: 10.1016/j.chaos.2018.04.029.  Google Scholar [11] K. Uǧur Erkin, S. Çiçek and Y. Uyaroǧlu, Secure communication with chaos and electronic circuit design using passivity-based synchronization, Journal of Circuits, Systems and Computers, 27 (2018), 1850057. doi: 10.1109/81.956024.  Google Scholar [12] D. Ali, U. Yılmaz and A. T. Özcerit, A novel chaotic system for secure communication applications, Information Technology and Control, 44 (2015), 271-278.   Google Scholar [13] T. Yamada and H. Fujisaka, Stability theory of synchronized motion in coupled oscillator systems. II: The mapping approach, Progress of Theoretical Physics, 70 (1983), 1240-1248.  doi: 10.1143/PTP.70.1240.  Google Scholar [14] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Physical Review Letters, 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821.  Google Scholar [15] M. S. Abd-Elouahab, N. Hamri and J. Wang, Chaos control of afractional-order financial system, Mathematical Problems in Engineering, 2010 (2010). doi: 10.1155/2010/270646.  Google Scholar [16] D. Chen, R. Zhang, M. Xiaoyi and S. Liu, Chaotic synchronization and anti-synchronization for a novel class of multiple chaotic systems via a sliding mode control scheme, Nonlinear Dynamics, 69 (2012), 35-55.  doi: 10.1007/s11071-011-0244-7.  Google Scholar [17] A. Senouci and A. Boukabou, Predictive control and synchronization of chaotic and hyperchaotic systems based on a T–S fuzzy model, Mathematics and Computers in Simulation, 105 (2014), 62-78.  doi: 10.1016/j.matcom.2014.05.007.  Google Scholar [18] K. Ayub, B. Mridula and I. Aysha, Multi-switching compound synchronization of four different chaotic systems via active backstepping method, International Journal of Dynamics and Control, 6 (2018), 1126-1135.  doi: 10.1007/s40435-017-0365-z.  Google Scholar [19] G. Li and S. Chunxiang, Adaptive neural network backstepping control of fractional-order Chua–Hartley chaotic system, Advances in Difference Equations, 2019 (2019), 148. doi: 10.1186/s13662-019-2099-z.  Google Scholar [20] L. Jianquan and C. Jinde, Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters, Chaos: An Interdisciplinary Journal of Nonlinear Science, 15 (2005), 043901. doi: 10.1063/1.2089207.  Google Scholar [21] G. Zheng-Ming and C. Chien-Cheng, Phase synchronization of coupled chaotic multiple time scales systems, Chaos, Solitons & Fractals, 20 (2004), 639-647.  doi: 10.1016/j.chaos.2004.11.032.  Google Scholar [22] S. Wen, Z. Zeng, T. Huang and Q. Meng, Lag synchronization of switched neural networks via neural activation function and applications in image encryption, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 1493-1502.  doi: 10.1109/TNNLS.2014.2387355.  Google Scholar [23] Z. Xuebing and Z. Honglan, Anti-synchronization of two different hyperchaotic systems via active and adaptive control, International Journal of Nonlinear Science, 6 (2008), 216-223.   Google Scholar [24] L. Chengren, L. Ling, Z. Guannan, L. Gang, T. Jing, G. Jiajia and W. Zhouyang, Projective synchronization of uncertain scale-free network based on modified sliding mode control technique, Physica A: Statistical Mechanics and its Applications, 473 (2017), 511-521.  doi: 10.1016/j.physa.2017.01.040.  Google Scholar [25] R. Mainieri and J. Rehacek, Projective synchronization in three-dimensional chaotic systems, Physical Review Letters, 82 (1999), 3042. Google Scholar [26] G.-H. Li, Modified projective synchronization of chaotic system, Chaos, Solitons Fractals, 32 (2007), 1786-1790.  doi: 10.1016/j.chaos.2005.12.009.  Google Scholar [27] J. Sun, J. Guo, C. Yang, A. Zheng and X. Zhang, Adaptive generalized hybrid function projective dislocated synchronization of new four-dimentional uncertain chaotic systems, Applied Mathematics and Computation, 252 (2015), 304-314.  doi: 10.1016/j.amc.2014.12.004.  Google Scholar [28] J.Chen, J. Sun, M. Chi and C. Xin-Ming, A novel scheme adaptive hybrid dislocated synchronization for two identical and different memristor chaotic oscillator systems with uncertain parameters, Abstract and Applied Analysis, 2014 (2014). doi: 10.1155/2014/675840.  Google Scholar [29] M. Krsti, K. Ioannis and V. Petar, Nonlinear and Adaptive Control Design, {Wiley New York}, (1995), 576. Google Scholar [30] S. Vaidyanathan, A new eight-term 3-D polynomial chaotic system with three quadratic nonlinearities, Far East J. Math. Sci, 84 (2014), 219-226.   Google Scholar [31] W. Hahn, Stability of Motion, Die Grundlehren der mathematischen Wissenschaften, 138, Springer-Verlag New York, Inc., New York, 1967.  Google Scholar

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##### References:
 [1] E. N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar [2] O. E. Rössler, An equation for continuous chaos, Physics Letters A, 57 (1976), 397-398.   Google Scholar [3] J. C. Sprott, Some simple chaotic flows, Physical Review E, 50 (1994), 647-650.  doi: 10.1103/PhysRevE.50.R647.  Google Scholar [4] J. Lü and G. Chen, A new chaotic attractor coined, International Journal of Bifurcation and Chaos, 12 (2002), 659-661.  doi: 10.1142/S0218127402004620.  Google Scholar [5] Z. Elhadj, Analysis of a new three-dimensional quadratic chaotic system, Radioengineering, 17 (2008), 9 pp. Google Scholar [6] M.-S. Abdelouahab and N.-E. Hamri, A new chaotic attractor from hybrid optical bistable system, Nonlinear Dynamics, 67 (2012), 457-463.  doi: 10.1007/s11071-011-9994-5.  Google Scholar [7] S.-J. Prakash and R.-B. Krishna, A more chaotic and easily hardware implementable new 3-D chaotic system in comparison with 50 reported systems, Nonlinear Dynamics, 93 (2018), 1121-1148.   Google Scholar [8] Ü. Çavuşoǧlu, S. Panahi, A. Akgül, S. Jafari and S. Kaçar, A new chaotic system with hidden attractor and its engineering applications: Analog circuit realization and image encryption, Analog Integrated Circuits and Signal Processing, 98 (2019), 85-99.   Google Scholar [9] L. Philippe, S. John and A. N. Jordan, Chaos in continuously monitored quantum systems: An optimal-path approach, Physical Review A, 98 (2018), 012141. Google Scholar [10] P. Sadeghi, S. Panahi, B. Hatef, S. Jafari and J. C. Sprott, A new chaotic model for glucose-insulin regulatory system, Chaos, Solitons & Fractals, 112 (2018), 44-51.  doi: 10.1016/j.chaos.2018.04.029.  Google Scholar [11] K. Uǧur Erkin, S. Çiçek and Y. Uyaroǧlu, Secure communication with chaos and electronic circuit design using passivity-based synchronization, Journal of Circuits, Systems and Computers, 27 (2018), 1850057. doi: 10.1109/81.956024.  Google Scholar [12] D. Ali, U. Yılmaz and A. T. Özcerit, A novel chaotic system for secure communication applications, Information Technology and Control, 44 (2015), 271-278.   Google Scholar [13] T. Yamada and H. Fujisaka, Stability theory of synchronized motion in coupled oscillator systems. II: The mapping approach, Progress of Theoretical Physics, 70 (1983), 1240-1248.  doi: 10.1143/PTP.70.1240.  Google Scholar [14] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Physical Review Letters, 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821.  Google Scholar [15] M. S. Abd-Elouahab, N. Hamri and J. Wang, Chaos control of afractional-order financial system, Mathematical Problems in Engineering, 2010 (2010). doi: 10.1155/2010/270646.  Google Scholar [16] D. Chen, R. Zhang, M. Xiaoyi and S. Liu, Chaotic synchronization and anti-synchronization for a novel class of multiple chaotic systems via a sliding mode control scheme, Nonlinear Dynamics, 69 (2012), 35-55.  doi: 10.1007/s11071-011-0244-7.  Google Scholar [17] A. Senouci and A. Boukabou, Predictive control and synchronization of chaotic and hyperchaotic systems based on a T–S fuzzy model, Mathematics and Computers in Simulation, 105 (2014), 62-78.  doi: 10.1016/j.matcom.2014.05.007.  Google Scholar [18] K. Ayub, B. Mridula and I. Aysha, Multi-switching compound synchronization of four different chaotic systems via active backstepping method, International Journal of Dynamics and Control, 6 (2018), 1126-1135.  doi: 10.1007/s40435-017-0365-z.  Google Scholar [19] G. Li and S. Chunxiang, Adaptive neural network backstepping control of fractional-order Chua–Hartley chaotic system, Advances in Difference Equations, 2019 (2019), 148. doi: 10.1186/s13662-019-2099-z.  Google Scholar [20] L. Jianquan and C. Jinde, Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters, Chaos: An Interdisciplinary Journal of Nonlinear Science, 15 (2005), 043901. doi: 10.1063/1.2089207.  Google Scholar [21] G. Zheng-Ming and C. Chien-Cheng, Phase synchronization of coupled chaotic multiple time scales systems, Chaos, Solitons & Fractals, 20 (2004), 639-647.  doi: 10.1016/j.chaos.2004.11.032.  Google Scholar [22] S. Wen, Z. Zeng, T. Huang and Q. Meng, Lag synchronization of switched neural networks via neural activation function and applications in image encryption, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 1493-1502.  doi: 10.1109/TNNLS.2014.2387355.  Google Scholar [23] Z. Xuebing and Z. Honglan, Anti-synchronization of two different hyperchaotic systems via active and adaptive control, International Journal of Nonlinear Science, 6 (2008), 216-223.   Google Scholar [24] L. Chengren, L. Ling, Z. Guannan, L. Gang, T. Jing, G. Jiajia and W. Zhouyang, Projective synchronization of uncertain scale-free network based on modified sliding mode control technique, Physica A: Statistical Mechanics and its Applications, 473 (2017), 511-521.  doi: 10.1016/j.physa.2017.01.040.  Google Scholar [25] R. Mainieri and J. Rehacek, Projective synchronization in three-dimensional chaotic systems, Physical Review Letters, 82 (1999), 3042. Google Scholar [26] G.-H. Li, Modified projective synchronization of chaotic system, Chaos, Solitons Fractals, 32 (2007), 1786-1790.  doi: 10.1016/j.chaos.2005.12.009.  Google Scholar [27] J. Sun, J. Guo, C. Yang, A. Zheng and X. Zhang, Adaptive generalized hybrid function projective dislocated synchronization of new four-dimentional uncertain chaotic systems, Applied Mathematics and Computation, 252 (2015), 304-314.  doi: 10.1016/j.amc.2014.12.004.  Google Scholar [28] J.Chen, J. Sun, M. Chi and C. Xin-Ming, A novel scheme adaptive hybrid dislocated synchronization for two identical and different memristor chaotic oscillator systems with uncertain parameters, Abstract and Applied Analysis, 2014 (2014). doi: 10.1155/2014/675840.  Google Scholar [29] M. Krsti, K. Ioannis and V. Petar, Nonlinear and Adaptive Control Design, {Wiley New York}, (1995), 576. Google Scholar [30] S. Vaidyanathan, A new eight-term 3-D polynomial chaotic system with three quadratic nonlinearities, Far East J. Math. Sci, 84 (2014), 219-226.   Google Scholar [31] W. Hahn, Stability of Motion, Die Grundlehren der mathematischen Wissenschaften, 138, Springer-Verlag New York, Inc., New York, 1967.  Google Scholar
Estimated unknown parameters $\overset{\sim }{\alpha }_{1}\left( t\right) ,$ $\overset{\sim }{\alpha }_{2}\left( t\right) ,$ $\overset{\sim }{\alpha }_{3}\left( t\right)$ and $\overset{\sim }{\alpha }_{4}\left( t\right)$ of the master Vaidyanathan system (19), we observe that the estimation values of unknown parameters converge to their real values $\alpha _{1} = 25, \alpha _{2} = 33, \alpha _{3} = 11, \alpha _{4} = 6$
Estimated unknown parameters $\overset{\sim }{\beta } _{1}\left( t\right) ,$ $\overset{\sim }{\beta }_{2}\left( t\right) ,$ $\overset{\sim }{\beta }_{3}\left( t\right)$ of the slave Zeraoulia system (20), we observe that the estimation values of unknown parameters converge to their real values $\beta _{1} = 36,$ $\beta _{2} = 25,$ $\beta _{3} = 3$
Synchronization errors $e_1, e_2, e_3$ between Vaidynathan and Zeraoulia systems (19) and (20), we observe that the errors converge to zero when the time increases
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