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Fractional Order Control and Synchronization of Chaotic Systems pp 497–524Cite as

On New Fractional Inverse Matrix Projective Synchronization Schemes

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Part of the book series: Studies in Computational Intelligence ((SCI,volume 688))

Abstract

In this study, the problem of inverse matrix projective synchronization (IMPS) between different dimensional fractional order chaotic systems is investigated. Based on fractional order Lyapunov approach and stability theory of fractional order linear systems, new complex schemes are proposed to achieve inverse matrix projective synchronization (IMPS) between n-dimension and m-dimension fractional order chaotic systems. To validate the theoretical results and to verify the effectiveness of the proposed schemes, numerical applications and computer simulations are used.

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References

  1. Jumarie, G. (1992). A Fokker-Planck equation of fractional order with respect to time. Journal of Mathematical Physics, 33, 3536–3541.

    Article  MathSciNet  MATH  Google Scholar 

  2. Metzler, R., Glockle, W. G., & Nonnenmacher, T. F. (1994). Fractional model equation for anomalous diffusion. Physica A, 211, 13–24.

    Article  Google Scholar 

  3. Mainardi, F. (1997). Fractional calculus: some basic problems in continuum and statistical mechanics. In A. Carpinteri & F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics. Springer.

    Google Scholar 

  4. Hilfer, R. (2000). Applications of fractional calculus in physics. World Scientific.

    Google Scholar 

  5. Parada, F. J. V., Tapia, J. A. O., & Ramirez, J. A. (2007). Effective medium equations for fractional Fick’s law in porous media. Physica A, 373, 339–353.

    Article  Google Scholar 

  6. Koeller, R. C. (1984). Application of fractional calculus to the theory of viscoelasticity. Journal of Applied Mechanics, 51, 299–307.

    Google Scholar 

  7. Axtell, M., & Bise, E. M. (1990). Fractional calculus applications in control systems. In Proceedings of the IEEE National Aerospace and Electronics Conference New York (pp. 563–566).

    Google Scholar 

  8. Dorčák, L. (1994). Numerical models for the simulation of the fractional-order control systems, UEF-04-94., Institute of Experimental Physics Košice, Slovakia: The Academy of Sciences.

    Google Scholar 

  9. Pires, E. J. S., Machado, J. A. T., & de Moura, P. B. (2003). Fractional order dynamics in a GA planner. Signal Process, 83, 2377–2386.

    Article  MATH  Google Scholar 

  10. Magin, R. L. (2006). Fractional calculus in bioengineering. Begell House Publishers.

    Google Scholar 

  11. Tseng, C. C. (2007). Design of FIR and IIR fractional order Simpson digital integrators. Signal Process, 87, 1045–1057.

    Google Scholar 

  12. da Graca, M. M., Duarte, F. B. M., & Machado, J. A. T. (2008). Fractional dynamics in the trajectory control of redundant manipulators. Communications in Nonlinear Science and Numerical Simulation, 13, 1836–1844.

    Article  Google Scholar 

  13. Hedrih, K. S., & Stanojevic, V. N. (2010). A model of gear transmission: fractional order system dynamics. Mathematical Problems in Engineering, 1–23.

    Google Scholar 

  14. Nakagava, N., & Sorimachi, K. (1992). Basic characteristic of a fractance device. IEICE Transactions on Fundamentals of Electronics, 75, 1814–1818.

    Google Scholar 

  15. Westerlund, S. (2002). Dead Matter Has Memory!. Causal Consulting.

    Google Scholar 

  16. Oldham, K. B., & Spanier, J. (1974). The fractional calculus. Academic.

    Google Scholar 

  17. Miller, K. S., & Ross, B. (1993). An Introduction to the fractional calculus and fractional differential equations. Wiley.

    Google Scholar 

  18. Petráš, I. (2008). A note on the fractional-order Chua’s system. Chaos, Solitons & Fractals, 38, 140–147.

    Google Scholar 

  19. West, B. J., Bologna, M., & Grigolini, P. (2002). Physics of fractal operators. Springer.

    Google Scholar 

  20. Zaslavsky, G.M. (2005). Hamiltonian chaos and fractional dynamics. Oxford University Press.

    Google Scholar 

  21. Azar, A. T., & Vaidyanathan, S. (2015). Chaos modeling and control systems design, studies in computational intelligence (vol. 581). Germany: Springer.

    Google Scholar 

  22. Azar, A. T., & Vaidyanathan, S. (2016). Advances in chaos theory and intelligent control. Studies in Fuzziness and Soft Computing (vol. 337). Germany: Springer. ISBN 978-3-319-30338-3.

    Google Scholar 

  23. Azar, A. T., & Vaidyanathan, S. (2015). Computational intelligence applications in modeling and control. Studies in Computational Intelligence (vol. 575). Germany: Springer. ISBN 978-3-319-11016-5.

    Google Scholar 

  24. Azar, A. T., & Vaidyanathan, S. (2015). Handbook of research on advanced intelligent control engineering and automation. Advances in Computational Intelligence and Robotics (ACIR) Book Series, IGI Global, USA. ISBN 9781466672482.

    Google Scholar 

  25. Zhu, Q., & Azar, A. T. (2015). Complex system modelling and control through intelligent soft computations. Studies in Fuzziness and Soft Computing (vol. 319). Germany: Springer. ISBN 978-3-319-12882-5.

    Google Scholar 

  26. Azar, A. T., & Zhu, Q. (2015). Advances and applications in sliding mode control systems. Studies in Computational Intelligence (vol. 576). Germany: Springer. ISBN 978-3-319-11172-8.

    Google Scholar 

  27. Filali, R. L., Benrejeb, M., & Borne, P. (2014). On observer-based secure communication design using discrete-time hyperchaotic systems. Communications in Nonlinear Science and Numerical Simulation, 19, 1424–1432.

    Article  MathSciNet  Google Scholar 

  28. Sheikhan, M., Shahnazi, M., & Garoucy, S. (2013). Hyperchaos synchronization using PSO-optimized RBF-based controllers to improve security of communication systems. Neural Computing & Applications, 22(5), 835–846.

    Google Scholar 

  29. Fernando, J. (2011). Applying the theory of chaos and a complex model of health to establish relations among financial indicators. Procedia Computer Science, 3, 982–986.

    Article  Google Scholar 

  30. Zsolt, B. (1997). Chaos theory and power spectrum analysis in computerized cardiotocography. European Journal of Obstetrics & Gynecology and Reproductive Biology, 71(2), 163–168.

    Google Scholar 

  31. Chen, G., & Dong, X. (1989). From chaos to order. World Scientific.

    Google Scholar 

  32. Ott, E., Grebogi, C., & Yorke, J. A. (1990). Controling chaos. Physical Review Letters, 64, 1196–1199.

    Article  MathSciNet  MATH  Google Scholar 

  33. Boccalettis, C., Grebogi, Y. C., LAI, M. H., & Maza, D. (2000). The control of chaos: theory and application. Physics Reports, 329, 103–197.

    Google Scholar 

  34. Yamada, T., & Fujisaka, H. (1983). Stability theory of synchroized motion in coupled-oscillator systems. Progress of Theoretical Physics, 70, 1240–1248.

    Article  MathSciNet  MATH  Google Scholar 

  35. Pecora, L. M., & Carroll, T. L. (1990). Synchronization in chaotic systems. Physical Review Letters, 64, 821–827.

    Article  MathSciNet  MATH  Google Scholar 

  36. Carroll, T. L., & Pecora, L. M. (1991). Synchronizing a chaotic systems. IEEE Transactions on Circuits and Systems, 38, 453–456.

    Article  Google Scholar 

  37. Pikovsky, A., Rosenblum, M., & Kurths, J. (2001). Synchronization an universal concept in nonlinear sciences. Cambridge university press.

    Google Scholar 

  38. Boccaletti, S., Kurths, J., Osipov, G., Valladares, D. L., & Zhou, C. S. (2002). The synchronization of chaotic systems. Physics Reports, 366, 1–101.

    Article  MathSciNet  MATH  Google Scholar 

  39. Aziz-Alaoui,  M. A. (2006). Synchronization of chaos. Encyclopedia of Mathematical Physics, 5, 213–226.

    Google Scholar 

  40. Luo, A. (2009). A theory for synchronization of dynamical systems. Communications in Nonlinear Science and Numerical Simulation, 14, 1901–1951.

    Article  MathSciNet  MATH  Google Scholar 

  41. Vaidyanathan, S., Sampath, S., & Azar, A. T. (2015). Global chaos synchronisation of identical chaotic systems via novel sliding mode control method and its application to Zhu system. International Journal of Modelling, Identification and Control (IJMIC), 23(1), 92–100.

    Google Scholar 

  42. Vaidyanathan, S., Azar, A. T., Rajagopal, K., & Alexander, P. (2015). Design and SPICE implementation of a 12-term novel hyperchaotic system and its synchronization via active control, (2015). International Journal of Modelling. Identification and Control (IJMIC), 23(3), 267–277.

    Google Scholar 

  43. Vaidyanathan, S., & Azar, A. T. (2016). Takagi-sugeno fuzzy logic controller for liu-chen four-scroll chaotic system. International Journal of Intelligent Engineering Informatics, 4(2), 135–150.

    Article  Google Scholar 

  44. Ouannas, A., Azar, A. T., & Abu-Saris, R. (2016). A new type of hybrid synchronization between arbitrary hyperchaotic maps. International Journal of Machine Learning and Cybernetics. doi:10.1007/s13042-016-0566-3.

  45. Vaidyanathan, S., Azar, A. T. (2015). Anti-Synchronization of identical chaotic systems using sliding mode control and an application to vaidyanathan-madhavan chaotic systems. In A. T. Azar & Q. Zhu (Eds.), Advances and applications in sliding mode control systems. Studies in Computational Intelligence book Series (vol. 576, pp. 527–547), GmbH Berlin/Heidelberg: Springer. doi:10.1007/978-3-319-11173-5_19.

  46. Vaidyanathan, S., & Azar, A. T. (2015). Hybrid synchronization of identical chaotic systems using sliding mode control and an application to vaidyanathan chaotic systems. In A. T. Azar & Q. Zhu, (Eds.), Advances and applications in sliding mode control systems. Studies in Computational Intelligence book Series, (vol. 576, pp. 549–569), GmbH Berlin/Heidelberg: Springer. doi:10.1007/978-3-319-11173-5_20.

  47. Vaidyanathan, S., & Azar, A. T. (2015). analysis, control and synchronization of a Nine-Term 3-D novel chaotic system. In A. T. Azar & S. Vaidyanathan (Eds.), Chaos Modeling and Control Systems Design, Studies in Computational Intelligence (vol. 581, pp. 3–17), GmbH Berlin/Heidelberg: Springer. doi:10.1007/978-3-319-13132-0_1.

  48. Vaidyanathan, S., & Azar, A. T. (2015). Analysis and control of a 4-D novel hyperchaotic system. In A. T. Azar & S. Vaidyanathan (Eds.), Chaos modeling and control systems design, Studies in Computational Intelligence (vol. 581, pp. 19–38), GmbH Berlin/Heidelberg: Springer. dpoi:10.1007/978-3-319-13132-0_2.

  49. Vaidyanathan, S., Idowu, B. A., & Azar, A. T. (2015). Backstepping controller design for the global chaos synchronization of sprott’s jerk systems. In A. T. Azar & S. Vaidyanathan (Eds.), chaos modeling and control systems design, Studies in Computational Intelligence, (vol. 581, pp. 39–58), GmbH Berlin/Heidelberg: Springer. doi:10.1007/978-3-319-13132-0_3.

  50. Boulkroune, A., Bouzeriba, A., Bouden, T., & Azar, A. T. (2016). Fuzzy adaptive synchronization of uncertain fractional-order chaotic systems. In A. T Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in Fuzziness and Soft Computing (vol. 337). Germany: Springer.

    Google Scholar 

  51. Boulkroune, A., Hamel, S., & Azar, A. T. (2016). Fuzzy control-based function synchronization of unknown chaotic systems with dead-zone input. Advances in chaos theory and intelligent control. Studies in Fuzziness and Soft Computing (vol. 337), Germany: Springer.

    Google Scholar 

  52. Vaidyanathan, S., & Azar, A. T. (2016). Dynamic analysis, adaptive feedback control and synchronization of an Eight-Term 3-D novel chaotic system with three quadratic nonlinearities. Advances in chaos theory and intelligent control. Studies in Fuzziness and Soft Computing (vol. 337). Germany: Springer.

    Google Scholar 

  53. Vaidyanathan, S., & Azar, A. T. (2016). Qualitative study and adaptive control of a novel 4-D hyperchaotic system with three quadratic nonlinearities. Advances in chaos theory and intelligent control. Studies in Fuzziness and Soft Computing (vol. 337). Germany: Springer.

    Google Scholar 

  54. Vaidyanathan, S., & Azar, A. T. (2016). A novel 4-D four-wing chaotic system with four quadratic nonlinearities and its synchronization via adaptive control method. Advances in chaos theory and intelligent control. Studies in Fuzziness and Soft Computing (vol. 337). Germany: Springer.

    Google Scholar 

  55. Vaidyanathan, S., & Azar, A. T. (2016). Adaptive control and synchronization of halvorsen circulant chaotic systems. Advances in chaos theory and intelligent control. Studies in Fuzziness and Soft Computing (vol. 337). Germany: Springer.

    Google Scholar 

  56. Vaidyanathan, S., & Azar, A. T. (2016). Adaptive backstepping control and synchronization of a novel 3-D jerk system with an exponential nonlinearity. Advances in chaos theory and intelligent control. Studies in Fuzziness and Soft Computing (vol. 337). Germany: Springer.

    Google Scholar 

  57. Vaidyanathan, S., & Azar, A. T. (2016). Generalized projective synchronization of a novel hyperchaotic Four-Wing system via adaptive control method. Advances in chaos theory and intelligent control. Studies in Fuzziness and Soft Computing. (vol. 337). Germany: Springer.

    Google Scholar 

  58. Hartley, T., Lorenzo, C., & Qammer, H. (1995). Chaos in a fractional order Chua’s system. IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications, 42, 485–490.

    Google Scholar 

  59. Grigorenko, I., & Grigorenko, E. (2003). Chaotic dynamics of the fractional Lorenz system. Physical Review Letters, 91, 034101.

    Article  Google Scholar 

  60. Li, C., & Chen, G. (2004). Chaos in the fractional order Chen system and its control. Chaos Solitons Fractals, 22, 549–554.

    Article  MATH  Google Scholar 

  61. Lu, J. G., & Chen, G. (2006). A note on the fractional-order Chen system. Chaos Solitons Fractals, 27, 685–688.

    Article  MATH  Google Scholar 

  62. Li, C., & Chen, G. (2004). Chaos and hyperchaos in fractional order R össler equations. Physica A, 341, 55–61.

    Article  MathSciNet  Google Scholar 

  63. Lu, J. G. (2005). Chaotic dynamics and synchronization of fractional-order Arneodo’s systems. Chaos Solitons Fractals, 26, 1125–1133.

    Article  MATH  Google Scholar 

  64. Deng, W. H., & Li, C. P. (2005). Chaos synchronization of the fractional Lü system. Physica A, 353, 61–72.

    Article  Google Scholar 

  65. Guo, L. J. (2005). Chaotic dynamics and synchronization of fractional-order Genesio-Tesi systems. Chinese Physics, 14, 1517–1521.

    Article  Google Scholar 

  66. Ge, Z. M., & Ou, C. Y. (2007). Chaos in a fractional order modified Duffing system. Chaos Solitons Fractals, 34, 262–291.

    Article  MATH  Google Scholar 

  67. Chen, W. C. (2008). Nonlinear dynamic and chaos in a fractional-order financial system. Chaos Solitons Fractals, 36, 1305–1314.

    Article  Google Scholar 

  68. Sheu, L. J., Chen, H. K., Chen, J. H., Tam, L. M., Chen, W. C., Lin, K. T., et al. (2008). Chaos in the Newton-Leipnik system with fractional order. Chaos Solitons Fractals, 36, 98–103.

    Article  MathSciNet  MATH  Google Scholar 

  69. Ahmed, E., El-Sayed, A. M. A., & El-Saka, H. A. A. (2007). Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. Journal of Mathematical Analysis and Applications, 325, 542–553.

    Article  MathSciNet  MATH  Google Scholar 

  70. Gejji, V. D., & Bhalekar, S. (2010). Chaos in fractional order Liu system. Computers & Mathematics with Applications, 59, 1117–1127.

    Google Scholar 

  71. Li, C., & Zhou, T. (2005). Synchronization in fractional-order differential systems. Physica D, 212, 111–125.

    Article  MathSciNet  MATH  Google Scholar 

  72. Wang, J., Xiong, X., & Zhang, Y. (2006). Extending synchronization scheme to chaotic fractional-order Chen systems. Physica A, 370, 279–285.

    Article  Google Scholar 

  73. Li, C. P., Deng, W. H., & Xu, D. (2006). Chaos synchronization of the Chua system with a fractional order. Physica A, 360, 171–185.

    Article  MathSciNet  Google Scholar 

  74. Peng, G. (2007). Synchronization of fractional order chaotic systems. Physics Letters A, 363, 426–432.

    Article  MathSciNet  MATH  Google Scholar 

  75. Yan, J., & Li, C. (2007). On chaos synchronization of fractional differential equations. Chaos Solitons Fractals, 32, 725–735.

    Article  MathSciNet  MATH  Google Scholar 

  76. Li, C., & Yan, J. (2007). The synchronization of three fractional differential systems. Chaos Solitons Fractals, 32, 751–757.

    Article  Google Scholar 

  77. Zhou, S., Li, H., Zhu, Z., & Li, C. (2008). Chaos control and synchronization in a fractional neuron network system. Chaos Solitons Fractals, 36, 973–984.

    Article  MathSciNet  MATH  Google Scholar 

  78. Zhu, H., Zhou, S., & Zhang, J. (2009). Chaos and synchronization of the fractional-order Chua’s system. Chaos Solitons Fractals, 39, 1595–1603.

    Article  MATH  Google Scholar 

  79. Wu, X., Wang, H., & Lu, H. (2012). Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication. Nonlinear Analysis: Real World Applications, 13(2012), 1441–1450.

    Article  MathSciNet  MATH  Google Scholar 

  80. Liang, H., Wang, Z., Yue, Z., & Lu, R. (2012). Generalized synchronization and control for incommensurate fractional unified chaotic system and applications in secure communication. Kybernetika, 48, 190–205.

    MathSciNet  MATH  Google Scholar 

  81. Muthukumar, P., & Balasubramaniam, P. (2013). Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography. Nonlinear Dynamics, 1169–1181.

    Google Scholar 

  82. Chen, L., Wu, R., He, Y., & Chai, Y. (2015). Adaptive sliding-mode control for fractional-order uncertain linear systems with nonlinear disturbances. Nonlinear Dynamics, 80, 51–58.

    Article  MathSciNet  MATH  Google Scholar 

  83. Liu, L., Ding, W., Liu, C., Ji, H., & Cao, C. (2014). Hyperchaos synchronization of fractional-order arbitrary dimensional dynamical systems via modified sliding mode control. Nonlinear Dynamics, 76, 2059–2071.

    Article  MathSciNet  MATH  Google Scholar 

  84. Zhang, L., & Yan, Y. (2014). Robust synchronization of two different uncertain fractional-order chaotic systems via adaptive sliding mode control. Nonlinear Dynamics, 76, 1761–1767.

    Google Scholar 

  85. Srivastava, M., Ansari, S. P., Agrawal, S. K., Das, S., & Leung, A. Y. T. (2014). Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method. Nonlinear Dynamics, 76, 905–914.

    Article  MathSciNet  Google Scholar 

  86. Agrawal, S. K., & Das, S. (2013). A modified adaptive control method for synchronization of some fractional chaotic systems with unknown parameters. Nonlinear Dynamics, 73, 907–919.

    Google Scholar 

  87. Zhou, P., & Bai, R. (2015). The adaptive synchronization of fractional-order chaotic system with fractional-order 1 \(<\) q \(<\) 2 via linear parameter update law. Nonlinear Dynamics, 80, 753–765.

    Google Scholar 

  88. Odibat, Z. (2010). Adaptive feedback control and synchronization of non-identical chaotic fractional order systems. Nonlinear Dynamics, 60, 479–487.

    Article  MathSciNet  MATH  Google Scholar 

  89. Yuan, W. X., & Mei, S. J. (2009). Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control. Communications in Nonlinear Science and Numerical Simulation, 14, 3351–3357.

    Article  MATH  Google Scholar 

  90. Odibat, Z. M., Corson, N., Aziz-Alaoui, M. A., & Bertelle, C. (2010). Synchronization of chaotic fractional-order systems via linear control. International Journal of Bifurcation and Chaos, 20, 81–97.

    Article  MathSciNet  MATH  Google Scholar 

  91. Chen, X. R., & Liu, C. X. (2012). Chaos synchronization of fractional order unified chaotic system via nonlinear control. International Journal of Modern Physics B, 25, 407–415.

    Article  MATH  Google Scholar 

  92. Cafagna, D., & Grassi, G. (2012). Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic Rössler systems. Nonlinear Dynamics, 68, 117–128.

    Article  MathSciNet  MATH  Google Scholar 

  93. Peng, G., & Jiang, Y. (2008). Generalized projective synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal. Physics Letters A, 372, 3963–3970.

    Article  MathSciNet  MATH  Google Scholar 

  94. Odibat, Z. M. (2012). A note on phase synchronization in coupled chaotic fractional order systems. Nonlinear Analysis: Real World Applications, 13, 779–789.

    Article  MathSciNet  MATH  Google Scholar 

  95. Chen, F., Xia, L., & Li, C. G. (2012). Wavelet phase synchronization of fractional-order chaotic systems. Chinese Physics Letters, 29, 6–070501.

    Google Scholar 

  96. Razminiaa, A., & Baleanu, D. (2013). Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics, 23, 873–879.

    Article  Google Scholar 

  97. Pan, L., Zhou, W., Fang, J., & Li, D. (2010). Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems. Communications in Nonlinear Science and Numerical Simulation, 15, 3754–3762.

    Article  MathSciNet  MATH  Google Scholar 

  98. Liu, F. C., Li, J. Y., & Zang, X. F. (2011). Anti-synchronization of different hyperchaotic systems based on adaptive active control and fractional sliding mode control. Acta Physica Sinica, 60, 030504.

    Google Scholar 

  99. Al-sawalha, M. M., Alomari, A. K., Goh, S. M., & Nooran, M. S. M. (2011). Active anti-synchronization of two identical and different fractional-order chaotic systems. International Journal of Nonlinear Science, 11, 267–274.

    MathSciNet  MATH  Google Scholar 

  100. Li, C. G. (2006). Projective synchronization in fractional order chaotic systems and its control. Progress of Theoretical Physics, 115, 661–666.

    Article  Google Scholar 

  101. Shao, S. Q., Gao, X., & Liu, X. W. (2007). Projective synchronization in coupled fractional order chaotic Rössler system and its control. Chinese Physics, 16, 2612–2615.

    Article  Google Scholar 

  102. Wang, X. Y., & He, Y. J. (2008). Projective synchronization of fractional order chaotic system based on linear separation. Physics Letters A, 372, 435–441.

    Article  MATH  Google Scholar 

  103. Si, G., Sun, Z., Zhang, Y., & Chen, W. (2012). Projective synchronization of different fractional-order chaotic systems with non-identical orders. Nonlinear Analysis: Real World Applications, 13, 1761–1771.

    Article  MathSciNet  MATH  Google Scholar 

  104. Agrawal, S. K., & Das, S. (2014). Projective synchronization between different fractional-order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique. Mathematical Methods in the Applied Sciences, 37, 2164–2176.

    Google Scholar 

  105. Chang, C. M., & Chen, H. K. (2010). Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen-Lee systems. Nonlinear Dynamics, 62, 851–858.

    Article  MathSciNet  MATH  Google Scholar 

  106. Wang, S., Yu, Y. G., & Diao, M. (2010). Hybrid projective synchronization of chaotic fractional order systems with different dimensions. Physica A, 389, 4981–4988.

    Article  Google Scholar 

  107. Zhou, P., & Zhu, W. (2011). Function projective synchronization for fractional-order chaotic systems. Nonlinear Analysis: Real World Applications, 12, 811–816.

    Article  MathSciNet  MATH  Google Scholar 

  108. Zhou, P., & Cao, Y. X. (2010). Function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems. Chinese Physics B, 19, 100507.

    Article  Google Scholar 

  109. Xi, H., Li, Y., & Huang, X. (2015). Adaptive function projective combination synchronization of three different fractional-order chaotic systems. Optik, 126, 5346–5349.

    Article  Google Scholar 

  110. Peng, G. J., Jiang, Y. L., & Chen, F. (2008). Generalized projective synchronization of fractional order chaotic systems. Physica A, 387, 3738–3746.

    Article  Google Scholar 

  111. Shao, S. Q. (2009). Controlling general projective synchronization of fractional order Rössler systems. Chaos Solitons Fractals, 39, 1572–1577.

    Article  MATH  Google Scholar 

  112. Wu, X. J., & Lu, Y. (2009). Generalized projective synchronization of the fractional-order Chen hyperchaotic system. Nonlinear Dynamics, 57, 25–35.

    Article  MATH  Google Scholar 

  113. Zhou, P., Kuang, F., & Cheng, Y. M. (2010). Generalized projective synchronization for fractional order chaotic systems. The Chinese Journal of Physics, 48, 49–56.

    MathSciNet  Google Scholar 

  114. Deng, W. H. (2007). Generalized synchronization in fractional order systems. Physical Review E, 75, 056201.

    Article  Google Scholar 

  115. Zhou, P., Cheng, X. F., & Zhang, N. Y. (2008). Generalized synchronization between different fractional-order chaotic systems. Communications in Theoretical Physics, 50, 931–934.

    Article  Google Scholar 

  116. Zhang, X. D., Zhao, P. D., & Li, A. H. (2010). Construction of a new fractional chaotic system and generalized synchronization. Communications in Theoretical Physics, 53, 1105–1110.

    Article  MATH  Google Scholar 

  117. Jun, W. M., & Yuan, W. X. (2011). Generalized synchronization of fractional order chaotic systems. International Journal of Modern Physics B, 25, 1283–1292.

    Article  MATH  Google Scholar 

  118. Wu, X. J., Lai, D. R., & Lu, H. T. (2012). Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes. Nonlinear Dynamics, 69, 667–683.

    Article  MathSciNet  MATH  Google Scholar 

  119. Xiao, W., Fu, J., Liu, Z., & Wan, W. (2012). Generalized synchronization of typical fractional order chaos system. Journal of Computer, 7, 1519–1526.

    Google Scholar 

  120. Martínez-Guerra, R., & Mata-Machuca, J. L. (2014). Fractional generalized synchronization in a class of nonlinear fractional order systems. Nonlinear Dynamics, 77, 1237–1244.

    Article  MathSciNet  MATH  Google Scholar 

  121. Yi, C., Liping, C., Ranchao, W., & Juan, D. (2013). Q-S synchronization of the fractional-order unified system. Pramana, 80, 449–461.

    Article  Google Scholar 

  122. Mathiyalagan, K., Park, J. H., & Sakthivel, R. (2015). Exponential synchronization for fractional-order chaotic systems with mixed uncertainties. Complexity, 21, 114–125.

    Article  MathSciNet  Google Scholar 

  123. Aghababa, M. P. (2012). Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique. Nonlinear Dynamics, 69, 247–261.

    Article  MathSciNet  MATH  Google Scholar 

  124. Li, D., Zhang, X. P., Hu, Y. T., & Yang, Y. Y. (2015). Adaptive impulsive synchronization of fractional order chaotic system with uncertain and unknown parameters. Neurocomputing, 167, 165–171.

    Article  Google Scholar 

  125. Xi, H., Yu, S., Zhang, R., & Xu, L. (2014). Adaptive impulsive synchronization for a class of fractional-order chaotic and hyperchaotic systems. Optik, 125, 2036–2040.

    Article  Google Scholar 

  126. Ouannas, A., Al-sawalha, M. M., & Ziar, T. (2016). Fractional chaos synchronization schemes for different dimensional systems with non-Identical fractional-orders via two scaling matrices. Optik, 127, 8410–8418.

    Article  Google Scholar 

  127. Ouannas, A., Azar, A. T., & Vaidyanathan, S. (2016). A robust method for new fractional hybrid chaos synchronization. Mathematical Methods in the Applied Sciences, 1–9.

    Google Scholar 

  128. Hu, M., Xu, Z., Zhang, R., & Hu, A. (2007). Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order. Physics Letters A, 365, 315–327.

    Article  Google Scholar 

  129. Zhang, Q., & Lu, J. (2008). Full state hybrid lag projective synchronization in chaotic (hyperchaotic) systems. Physics Letters A, 372, 1416–1421.

    Article  MathSciNet  MATH  Google Scholar 

  130. Hu, M., Xu, Z., Zhang, R., & Hu, A. (2007). Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyperchaotic) systems. Physics Letters A, 361, 231–237.

    Article  MathSciNet  MATH  Google Scholar 

  131. Tang, Y., Fang, J. A., & Chen, L. (2010). Lag full state hybrid projective synchronization in different fractional-order chaotic systems. International Journal of Modern Physics B, 24, 6129–61411.

    Article  MathSciNet  MATH  Google Scholar 

  132. Feng, H., Yang, Y., & Yang, S. P. (2013). A new method for full state hybrid projective synchronization of different fractional order chaotic systems. Applied Mechanics and Materials, 385–38, 919–922.

    Article  Google Scholar 

  133. Razminia, A. (2013). Full state hybrid projective synchronization of a novel incommensurate fractional order hyperchaotic system using adaptive mechanism. Indian Journal of Physics, 87, 161–167.

    Article  Google Scholar 

  134. Zhang, L., & Liu, T. (2016). Full state hybrid projective synchronization of variable-order fractional chaotic/hyperchaotic systems with nonlinear external disturbances and unknown parameters. The Journal of Nonlinear Science and Applications, 9, 1064–1076.

    MathSciNet  MATH  Google Scholar 

  135. Manieri, R., & Rehacek, J. (1999). Projective synchronization in three-dimensional chaotic systems. Physical Review Letters, 82, 3042–3045.

    Article  Google Scholar 

  136. Hu, M., Xu, Z., Zhang, R., & Hu, A. (2008). Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems. Communications in Nonlinear Science and Numerical Simulation, 13, 456–464.

    Article  MathSciNet  MATH  Google Scholar 

  137. Hu, M., Xu, Z., Zhang, R., & Hu, A. (2008). Full state hybrid projective synchronization of a general class of chaotic maps. Communications in Nonlinear Science and Numerical Simulation, 13, 782–789.

    Article  MathSciNet  MATH  Google Scholar 

  138. Grassi, G. (2012). Arbitrary full-state hybrid projective synchronization for chaotic discrete-time systems via a scalar signal. Chinese Physics B, 21, 060504.

    Article  Google Scholar 

  139. Ouannas, A., & Abu-Saris, R. (2016). On matrix projective synchronization and inverse matrix projective synchronization for different and identical dimensional discrete-time chaotic systems. Journal of Control Science and Engineering, 1–7.

    Google Scholar 

  140. Gorenflo, R., & Mainardi, F. (1997). Fractional calculus: Integral and differential equations of fractional order. In A. Carpinteri & F. Mainardi (Eds.), The book Fractals and fractional calculus, New York.

    Google Scholar 

  141. Caputo, M. (1967). Linear models of dissipation whose \(Q\) is almost frequency independent-II. Geophysical Journal of the Royal Astronomical Society, 13, 529–539.

    Article  Google Scholar 

  142. Samko, S. G., Klibas, A. A., & Marichev, O. I. (1993). Fractional integrals and derivatives: theory and applications. Gordan and Breach.

    Google Scholar 

  143. Podlubny, I. (1999). Fractional differential equations. Academic Press.

    Google Scholar 

  144. Heymans, N., & Podlubny, I. (2006). Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheologica Acta, 45, 765–772.

    Article  Google Scholar 

  145. Matignon, D. (1996). Stability results of fractional differential equations with applications to control processing. In: IMACS, IEEE-SMC, Lille, France.

    Google Scholar 

  146. Li, Y., Chen, Y., & Podlubny, I. (2010). Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag Leffler stability. Computers & Mathematics with Applications, 59, 21–1810.

    Google Scholar 

  147. Chen, D., Zhang, R., Liu, X., & Ma, X. (2014). Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks. Communications in Nonlinear Science and Numerical Simulation, 19, 4105–4121.

    Article  MathSciNet  Google Scholar 

  148. Aguila-Camacho, N., Duarte-Mermoud, M. A., & Gallegos, J. A. (2014). Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation, 19, 2951–2957.

    Article  MathSciNet  Google Scholar 

  149. Ouannas, A., & Mahmoud, E. (2014). Inverse matrix projective synchro-nization for discrete chaotic systems with different dimensions. Intelligence and Electronic Systems, 3, 188–192.

    Article  Google Scholar 

  150. Wang, X.-Y., & Zhang, H. (2013). Bivariate module-phase synchronization of a fractional-order lorenz system in diFFerent dimensions. Journal of Computational and Nonlinear Dynamics, 8, 031017.

    Article  Google Scholar 

  151. Zhou, P., Wei, L. J., & Cheng, X. F. (2009). A novel fractional-order hyperchaotic system and its synchronization. Chinese Physics B, 18, 2674.

    Article  Google Scholar 

  152. Liu, C., Liu, T., Liu, L., & Liu, K. (2004). A new chaotic attractor. Chaos Solitons Fractals, 22, 1031–1038.

    Article  MathSciNet  MATH  Google Scholar 

  153. Han, Q., Liu, C. X., Sun, L., & Zhu, D. R. (2013). A fractional order hyperchaotic system derived from a Liu system and its circuit realization. Chinese Physics B, 22, 6–020502.

    Google Scholar 

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Authors and Affiliations

  1. Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Larbi Tebessi, 12002, Tebessa, Algeria

    Adel Ouannas

  2. Faculty of Computers and Information, Benha University, Banha, Egypt

    Ahmad Taher Azar

  3. Nanoelectronics Integrated Systems Center (NISC), Nile University, Cairo, Egypt

    Ahmad Taher Azar

  4. Department of Material Sciences, University of Tebessa, 12002, Tébessa, Algeria

    Toufik Ziar

  5. Research and Development Centre, Vel Tech University, Avadi, Chennai, 600062, Tamil Nadu, India

    Sundarapandian Vaidyanathan

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  1. Adel Ouannas
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  2. Ahmad Taher Azar
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  3. Toufik Ziar
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  4. Sundarapandian Vaidyanathan
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Correspondence to Ahmad Taher Azar .

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  1. Faculty of Computers and Information, Benha University Faculty of Computers and Information, AND Nanoelectronics Integrated Systems Center (NISC), Nile University, Egypt. , Benha, Egypt

    Ahmad Taher Azar

  2. Research and Development Centre, Vel Tech University Research and Development Centre, Chennai, Tamil Nadu, India

    Sundarapandian Vaidyanathan

  3. Department of Mathematics and CS, University of Tebessa Department of Mathematics and CS, Tebessa, Algeria

    Adel Ouannas

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Ouannas, A., Azar, A.T., Ziar, T., Vaidyanathan, S. (2017). On New Fractional Inverse Matrix Projective Synchronization Schemes. In: Azar, A., Vaidyanathan, S., Ouannas, A. (eds) Fractional Order Control and Synchronization of Chaotic Systems. Studies in Computational Intelligence, vol 688. Springer, Cham. https://doi.org/10.1007/978-3-319-50249-6_17

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