Generalized Projective Synchronization of Fractional-order Hyperchaotic Lu and Hyperchaotic Cai System via Active Control
Keywords:
Generalized projective synchronization, Fractional-order hyperchaotic systems, Hyperchaotic Lu system, Hyperchaotic Cai system, Active control
Abstract
This paper studies the design of feedback controllers to achieve generalized projective synchronization (GPS) between two fractional derivative order hyperchaotic Lu systems (similar),and between hyperchaotic Cai system and fractional-order hyperchaotic Lu system (dissimilar). First, new sliding surface is defined for GPS of fractional-order Lu-Lu and Lu-Cai systems. Then the results of the GPS achieved in this article is proved using Lyapunov stability theorem. Without using Lyapunov exponents, the active feedback control method to achieve GPS of fractional-order hyperchaotic Lu-Lu and Cai-Lu is calculated. Performance of the proposed method is shown, using numerical simulations.
References
[1] K. T. Alligood, T. Sauer, J. A. Yorke, “An Introduction to Dynamical Systems”, Springer, NewYork, 1997.
[2] S. Pakiriswamy, V. Sundarapandian, “Generalized Projective Synchronization of Hyperchaotic Lu and Hyperchaotic Cai systems via Active Control” CCSIT 2012, Part I, LNICST 84, pp. 53–62, 2012
[3] S. Banerjee, S. Mukhopadhyay, L. Rondoni, “Multi-image encryption based on synchronization of chaotic lasers and iris authentication”, Optics and Lasers in Engineering, vol 50, pp. 950–957, 2012.
[4] X. Zhao, Z. Li, S. Li, “Synchronization of a chaotic finance system”, Applied Mathematics and Computation, vol. 217, pp. 6031–6039, 2011.
[5] Y.N. Li, L. Chen, Z.S. Cai, X.Z. Zhao, “Study on chaos synchronization in the Belousov–Zhabotinsky chemical system”, Chaos, Solitons and Fractals, vol. 17, pp. 699–707, 2003.
[6] L. Basnarkov, G. S. Duane, L. Kocarev, “Generalized synchronization and coherent structures in spatially extended systems”, Chaos, Solitons & Fractals, vol. 59, pp. 35–41, 2014
[7] S. Z. Ali, M.K. Islam, M. Zafrullah, “Effect of transmission fiber on dense wavelength division multiplexed (DWDM) chaos synchronization”, Optik, vol.124, pp. 1108– 1112, 2013.
[8] F.T. Arecchi, R. Meucci, A. Di Garbo, E. Allaria, “Homoclinic chaos in a laser: synchronization and its implications in biological systems”, Optics and Lasers in Engineering, vol. 39, pp. 293–304, 2003.
[9] Y.Y. Hou, B.Y. Liau, H.C. Chen, “Synchronization of Unified Chaotic Systems Using Sliding Mode Controller”, Mathematical Problems in Engineering, vol. 2012, Article ID 632712, 10 pages, 2012.
[10] S.K.Agrawal, M. Srivastava, S. Das, “Synchronization of fractional order chaotic systems using active control method”, Chaos, Solitons & Fractals, vol. 45, pp. 737–752, 2012.
[11] S. Vaidyanathan, S. Sampath, “Complete Synchronization of Hyperchaotic Systems via Novel Sliding Mode Control”, Advances in Chaos Theory and Intelligent Control, Springer, Switzerland pp 327-347 (2016), DOI 10.1007/978-3-319-30340-6_12.
[12] S. Vaidyanathan, S. Sampath, “Anti-synchronization of Hyperchaotic Systems via Novel Sliding Mode Control and Its Application to Vaidyanathan Hyperjerk System”, Advances and Applications in Nonlinear Control Systems, pp 143-158 Springer (2016), DOI 10.1007/978-3-319-30169-3_8.
[13] H-L Li · Y.L Jiang · Z.L Wang. “Anti-synchronization and intermittent anti-synchronization of two identical hyperchaotic Chua systems via impulsive control”, Nonlinear Dyn, vol. 79, pp. 919–925, 2015.
[14] M. Srivastava, S. P Ansari, Agrawal. S. K, Das. S, Leung. A. Y. T. “Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method”, Nonlinear Dyn, vol. 76, pp.905–914, 2014.
[15] C. Jiang, S. Liu, “Generalized combination complex synchronization of new hyperchaotic complex Lü-like systems, Advances in Difference Equations”, Advances in Difference Equations, 2015:214, 2015.
[16] X. Shi, Z.Wang, “Hybrid synchronization phenomenon in two coupled delay hyperchaotic Lorenz systems with unknown parameters via a single linear controller, Journal of Vibration and Control, 19: 1051, 2013.
[17] S. Vaidyanathan, A. Taher Azar, “Generalized Projective Synchronization of a Novel Hyperchaotic Four-Wing System via Adaptive Control Method”, Advances in Chaos Theory and Intelligent Control, Springer, pp 275-296 (2014), DOI 10.1007/978-3-319-30340-6_12
[18] C.Li, J. Xiongb, W. Li, “A new hyperchaotic system and its generalized synchronization”, Optik 125, vol. 1, pp. 575– 579, 2014.
[19] K.S. Ojo, S.T. Ogunjo, A. N.Njah, I.A. Fuwape, “Increased-order generalized synchronization of chaotic and hyperchaotic systems”, Pramana journal of physics, pp. 33–45, January 2015.
[20] N. Yang, C. Liu, “A novel fractional-order hyperchaotic system stabilization via fractional sliding-mode control”, Nonlinear Dyn, vol. 74, pp.721–732, 2013.
[21] J. Vargas, E. Grzeida, E. Hemerly, “Robust adaptive synchronization of a hyperchaotic finance”, Nonlinear Dyn, vol. 80, pp. 239–248, 2015.
[22] A. Nourian, S. Balochian, “The adaptive synchronization of fractional-order Liu chaotic system with unknown parameters”, Pramana journal of physics, vol. 84, no. 1, pp. 33–45, 2015.
[23] X. Wang, P. Sun, “Multi-switching synchronization of chaotic system with adaptive controllers and unknown parameters”, Nonlinear Dyn, vol. 63, pp. 599–609, 2011.
[24] G. Alamdar, S. Balochian, “Chaos control of permanent magnet synchronous generator via sliding mode controller, Majlesi journal of Electrical Engineering, accepted 2018, In press March 2019.
[25] M. Feng, X. Wang, Q. Wei, “Adaptive robust synchronization of fractional-order chaotic system with disturbance”, Journal of Vibration and Control, vol. 21, no. 1, 2015.
[27] C. Li, K. Su, Y. Tong, H. Li, “Robust synchronization for a class of fractional-order chaotic and hyperchaotic Systems”, Optik, vol. 124, 3242– 3245, 2013.
[28] V. A. Slipko, Y. V. Pershin, “Switching synchronization in one-dimensional memristive networks: An exact solution”, Physical Review E, vol. 96, no. 6, 2017.
[29] C. Yin, Y. Cheng, Y. Chen , B. Stark, S. Zhong, “Adaptive fractional-order switching-type control method design for 3D fractional-order nonlinear systems”, Nonlinear Dyn,vol. 82, pp.39–52, 2015.
[30] S. Bhalekar, “Synchronization of Fractional Chaotic and Hyperchaotic Systems Using an Extended Active Control”, Advances in Chaos Theory and Intelligent Control, Springer, Switzerland, pp 53-73, 2016
[31] G. Zhang, Z. Liu, Z. Ma, “Generalized synchronization of different dimensional chaotic dynamical systems”, Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 773–779, 2007.
[32] H. Du, Q. Zeng, C. Wang, M. Ling, “Function projective synchronization in coupled chaotic systems”, Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp.705–712, 2010.
[2] S. Pakiriswamy, V. Sundarapandian, “Generalized Projective Synchronization of Hyperchaotic Lu and Hyperchaotic Cai systems via Active Control” CCSIT 2012, Part I, LNICST 84, pp. 53–62, 2012
[3] S. Banerjee, S. Mukhopadhyay, L. Rondoni, “Multi-image encryption based on synchronization of chaotic lasers and iris authentication”, Optics and Lasers in Engineering, vol 50, pp. 950–957, 2012.
[4] X. Zhao, Z. Li, S. Li, “Synchronization of a chaotic finance system”, Applied Mathematics and Computation, vol. 217, pp. 6031–6039, 2011.
[5] Y.N. Li, L. Chen, Z.S. Cai, X.Z. Zhao, “Study on chaos synchronization in the Belousov–Zhabotinsky chemical system”, Chaos, Solitons and Fractals, vol. 17, pp. 699–707, 2003.
[6] L. Basnarkov, G. S. Duane, L. Kocarev, “Generalized synchronization and coherent structures in spatially extended systems”, Chaos, Solitons & Fractals, vol. 59, pp. 35–41, 2014
[7] S. Z. Ali, M.K. Islam, M. Zafrullah, “Effect of transmission fiber on dense wavelength division multiplexed (DWDM) chaos synchronization”, Optik, vol.124, pp. 1108– 1112, 2013.
[8] F.T. Arecchi, R. Meucci, A. Di Garbo, E. Allaria, “Homoclinic chaos in a laser: synchronization and its implications in biological systems”, Optics and Lasers in Engineering, vol. 39, pp. 293–304, 2003.
[9] Y.Y. Hou, B.Y. Liau, H.C. Chen, “Synchronization of Unified Chaotic Systems Using Sliding Mode Controller”, Mathematical Problems in Engineering, vol. 2012, Article ID 632712, 10 pages, 2012.
[10] S.K.Agrawal, M. Srivastava, S. Das, “Synchronization of fractional order chaotic systems using active control method”, Chaos, Solitons & Fractals, vol. 45, pp. 737–752, 2012.
[11] S. Vaidyanathan, S. Sampath, “Complete Synchronization of Hyperchaotic Systems via Novel Sliding Mode Control”, Advances in Chaos Theory and Intelligent Control, Springer, Switzerland pp 327-347 (2016), DOI 10.1007/978-3-319-30340-6_12.
[12] S. Vaidyanathan, S. Sampath, “Anti-synchronization of Hyperchaotic Systems via Novel Sliding Mode Control and Its Application to Vaidyanathan Hyperjerk System”, Advances and Applications in Nonlinear Control Systems, pp 143-158 Springer (2016), DOI 10.1007/978-3-319-30169-3_8.
[13] H-L Li · Y.L Jiang · Z.L Wang. “Anti-synchronization and intermittent anti-synchronization of two identical hyperchaotic Chua systems via impulsive control”, Nonlinear Dyn, vol. 79, pp. 919–925, 2015.
[14] M. Srivastava, S. P Ansari, Agrawal. S. K, Das. S, Leung. A. Y. T. “Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method”, Nonlinear Dyn, vol. 76, pp.905–914, 2014.
[15] C. Jiang, S. Liu, “Generalized combination complex synchronization of new hyperchaotic complex Lü-like systems, Advances in Difference Equations”, Advances in Difference Equations, 2015:214, 2015.
[16] X. Shi, Z.Wang, “Hybrid synchronization phenomenon in two coupled delay hyperchaotic Lorenz systems with unknown parameters via a single linear controller, Journal of Vibration and Control, 19: 1051, 2013.
[17] S. Vaidyanathan, A. Taher Azar, “Generalized Projective Synchronization of a Novel Hyperchaotic Four-Wing System via Adaptive Control Method”, Advances in Chaos Theory and Intelligent Control, Springer, pp 275-296 (2014), DOI 10.1007/978-3-319-30340-6_12
[18] C.Li, J. Xiongb, W. Li, “A new hyperchaotic system and its generalized synchronization”, Optik 125, vol. 1, pp. 575– 579, 2014.
[19] K.S. Ojo, S.T. Ogunjo, A. N.Njah, I.A. Fuwape, “Increased-order generalized synchronization of chaotic and hyperchaotic systems”, Pramana journal of physics, pp. 33–45, January 2015.
[20] N. Yang, C. Liu, “A novel fractional-order hyperchaotic system stabilization via fractional sliding-mode control”, Nonlinear Dyn, vol. 74, pp.721–732, 2013.
[21] J. Vargas, E. Grzeida, E. Hemerly, “Robust adaptive synchronization of a hyperchaotic finance”, Nonlinear Dyn, vol. 80, pp. 239–248, 2015.
[22] A. Nourian, S. Balochian, “The adaptive synchronization of fractional-order Liu chaotic system with unknown parameters”, Pramana journal of physics, vol. 84, no. 1, pp. 33–45, 2015.
[23] X. Wang, P. Sun, “Multi-switching synchronization of chaotic system with adaptive controllers and unknown parameters”, Nonlinear Dyn, vol. 63, pp. 599–609, 2011.
[24] G. Alamdar, S. Balochian, “Chaos control of permanent magnet synchronous generator via sliding mode controller, Majlesi journal of Electrical Engineering, accepted 2018, In press March 2019.
[25] M. Feng, X. Wang, Q. Wei, “Adaptive robust synchronization of fractional-order chaotic system with disturbance”, Journal of Vibration and Control, vol. 21, no. 1, 2015.
[27] C. Li, K. Su, Y. Tong, H. Li, “Robust synchronization for a class of fractional-order chaotic and hyperchaotic Systems”, Optik, vol. 124, 3242– 3245, 2013.
[28] V. A. Slipko, Y. V. Pershin, “Switching synchronization in one-dimensional memristive networks: An exact solution”, Physical Review E, vol. 96, no. 6, 2017.
[29] C. Yin, Y. Cheng, Y. Chen , B. Stark, S. Zhong, “Adaptive fractional-order switching-type control method design for 3D fractional-order nonlinear systems”, Nonlinear Dyn,vol. 82, pp.39–52, 2015.
[30] S. Bhalekar, “Synchronization of Fractional Chaotic and Hyperchaotic Systems Using an Extended Active Control”, Advances in Chaos Theory and Intelligent Control, Springer, Switzerland, pp 53-73, 2016
[31] G. Zhang, Z. Liu, Z. Ma, “Generalized synchronization of different dimensional chaotic dynamical systems”, Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 773–779, 2007.
[32] H. Du, Q. Zeng, C. Wang, M. Ling, “Function projective synchronization in coupled chaotic systems”, Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp.705–712, 2010.
Published
2019-06-01
How to Cite
Balochian, S. (2019). Generalized Projective Synchronization of Fractional-order Hyperchaotic Lu and Hyperchaotic Cai System via Active Control. Majlesi Journal of Mechatronic Systems, 8(2), 15-21. Retrieved from https://ms.majlesi.info/index.php/ms/article/view/398
Issue
Section
Articles
