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Dynamics of Fractional Order Complex Uçar System

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Fractional Order Control and Synchronization of Chaotic Systems

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

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Abstract

The fractional order delay differential equations are models with rich dynamical properties. Both fractional order and delay are useful in modelling memory and hereditary properties in the physical system. In this chapter, we have proposed a complex version of fractional order Uçar system with delay. The stability of the numerical methods for solving such equations is discussed. It is observed that a slight modification in the proposed system generates chaotic trajectories. The bifurcation and chaos is studied in the modified system. The delayed feedback method is used to control chaos in the system. Finally, the system is synchronized by using the method of projective synchronization.

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References

  1. Adomian, G. (1994). Solving frontier problems of physics: The decomposition method. Dordrecht: Kluwer Academic.

    Book  MATH  Google Scholar 

  2. Agladze, K. I., Krinsky, V. I., & Pertsov, A. M. (1984). Chaos in the non-stirred Belousov-Zhabotinsky reaction is induced by interaction of waves and stationary dissipative structures. Nature, 308(5962), 834–835.

    Article  Google Scholar 

  3. AL-Mutib, A. N. (1984). Stability properties of numerical methods for solving delay differential equations. Journal of Computational and Applied Mathematics, 10(1), 71–79.

    Google Scholar 

  4. Alligood, K. T., Sauer, T. D., & Yorke, J. A. (2008). Chaos: An introduction to dynamical systems. New York: Springer.

    MATH  Google Scholar 

  5. Al-Bassam, M. A. (1965). Some existence theorems on differential equations of generalized order. Journal für die reine und angewandte Mathematik, 218, 70–78.

    MathSciNet  MATH  Google Scholar 

  6. Anastasio, T. J. (1994). The fractional-order dynamics of Brainstem Vestibulo-Oculomotor neurons. Biological Cybernetics, 72(1), 69–79.

    Article  Google Scholar 

  7. Asea, P., & Zak, P. (1999). Time-to-build and cycles. Journal of Economic Dynamics and Control, 23(8), 1155–1175.

    Article  MathSciNet  MATH  Google Scholar 

  8. Azar, A. T., & Serrano, F. E. (2014). Robust IMC-PID tuning for cascade control systems with gain and phase margin specifications. Neural Computing and Applications, 25(5), 983–995. Springer. doi:10.1007/s00521-014-1560-x.

  9. Azar, A. T., & Serrano, F. E. (2015). Adaptive sliding mode control of the Furuta pendulum. In A. T. Azar & Q. Zhu (Eds.), Advances and applications in sliding mode control systems (Vol. 576, pp. 1–42). Studies in computational intelligence. Berlin: Springer.

    Google Scholar 

  10. 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. USA: IGI Global.

    Google Scholar 

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

    Google Scholar 

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

    MATH  Google Scholar 

  13. Azar, A. T., & Zhu, Q. (2015). Advances and applications in sliding mode control systems (Vol. 576). Studies in computational intelligence. Germany: Springer. ISBN: 978-3-319-11172-8.

    Google Scholar 

  14. Azar, A. T., & Serrano, F. E. (2015). Deadbeat control for multivariable systems with time varying delays. In A. T. Azar & S. Vaidyanathan (Eds.), Chaos modeling and control systems design (Vol. 581, pp. 97–132). Studies in computational intelligence. Berlin: Springer.

    Google Scholar 

  15. Azar, A. T., & Zhu, Q. (2015). Advances and applications in sliding mode control systems (Vol. 576). Studies in computational intelligence. Germany: Springer.

    Google Scholar 

  16. Azar, A. T., & Serrano, F. E. (2015). Stabilization and control of mechanical systems with backlash. In A. T. Azar & S. Vaidyanathan (Eds.), Advanced intelligent control engineering and automation. Advances in computational intelligence and robotics (ACIR) book series. USA: IGI-Global.

    Google Scholar 

  17. Azar, A. T., & Serrano, F. E. (2015). Design and modeling of anti wind up PID controllers. In Q. Zhu & A. T. Azar (Eds.), Complex system modelling and control through intelligent soft computations (Vol. 319, pp. 1–44). Studies in fuzziness and soft computing. Germany: Springer. doi:10.1007/978-3-319-12883-2_1.

  18. Azar, A. T., & Vaidyanathan, S. (2016). Advances in chaos theory and intelligent control (Vol. 337). Studies in fuzziness and soft computing. Germany: Springer. ISBN: 978-3-319-30338-3.

    Google Scholar 

  19. Bai, E. W., & Lonngren, K. E. (1997). Synchronization of two Lorenz systems using active control. Chaos Solitons and Fractals, 8(1), 51–58.

    Article  MATH  Google Scholar 

  20. Bai, E. W., & Lonngren, K. E. (2000). Sequential synchronization of two Lorenz systems using active control. Chaos Solitons and Fractals, 11(7), 1041–1044.

    Article  MATH  Google Scholar 

  21. Barrett, J. H. (1954). Differential equations of non-integer order. Canadian Journal of Mathematics, 6(4), 529–541.

    Article  MathSciNet  MATH  Google Scholar 

  22. Erol, B. (Ed.). (1990). Chaos in brain function. New York: Springer.

    Google Scholar 

  23. Bhalekar, S., & Daftardar-Gejji, V. (2010). Synchronization of different fractional order chaotic systems using active control. Communications in Nonlinear Science and Numerical Simulations, 15(11), 3536–3546.

    Article  MATH  Google Scholar 

  24. Bhalekar, S., & Daftardar-Gejji, V. (2011). Anti-synchronization of non-identical fractional order chaotic systems using active control. International Journal of Differential Equations. Article ID 250763.

    Google Scholar 

  25. Bhalekar, S., & Daftardar-Gejji, V. (2011). A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order. Journal of Fractional Calculus and Applications, 1(5), 1–8.

    MATH  Google Scholar 

  26. Bhalekar, S., Daftardar-Gejji, V., Baleanu, D., & Magin, R. (2011). Fractional Bloch equation with delay. Computers and Mathematics with Applications, 61(5), 1355–1365.

    Article  MathSciNet  MATH  Google Scholar 

  27. Bhalekar, S. (2012). Dynamical analysis of fractional order Uçar prototype delayed system. Signal Image Video Processing, 6(3), 513–519.

    Article  Google Scholar 

  28. Bhalekar, S. (2014). On the Uçar prototype model with incommensurate delays. Signal Image Video Processing, 8(4), 635–639.

    Article  Google Scholar 

  29. Bhalekar, S. (2014). Synchronization of incommensurate non-identical fractional order chaotic systems using active control. The European Physical Journal—Special Topics, 223(8), 1495–1508.

    Article  Google Scholar 

  30. Bhalekar, S. (2014). Synchronization of non-identical fractional order hyperchaotic systems using active control. World Journal of Modelling and Simulation, 10(1), 60–68.

    Google Scholar 

  31. Bhalekar, S. (2016). Stability analysis of Uçar prototype delayed system. Signal Image Video Processing, 10(4), 777–781.

    Article  Google Scholar 

  32. Bhalekar, S. (2016). Stability and bifurcation analysis of a generalized scalar delay differential equation. Chaos: An Interdisciplinary. Journal of Nonlinear Science, 26(8), 084306. doi:10.1063/1.4958923.

    Article  MathSciNet  Google Scholar 

  33. Boulkroune, A., Bouzeriba, A., Bouden, T., & Azar, A. T. (2016). Fuzzy adaptive synchronization of uncertain fractional-order chaotic systems. In Advances in chaos theory and intelligent control (Vol. 337). Studies in fuzziness and soft computing. Germany: Springer.

    Google Scholar 

  34. Boulkroune. A., Hamel, S., & Azar, A. T. (2016). Fuzzy control-based function synchronization of unknown chaotic systems with dead-zone input. In Advances in chaos theory and intelligent control (Vol. 337). Studies in fuzziness and soft computing. Germany: Springer.

    Google Scholar 

  35. Campbell, S. A. (2007). Time delays in neural systems. Handbook of brain connectivity (pp. 65–90). Berlin: Springer.

    Chapter  Google Scholar 

  36. Chen, W. C. (2008). Nonlinear dynamics and chaos in a fractional-order financial system. Chaos Solitons and Fractals, 36(5), 1305–1314.

    Article  Google Scholar 

  37. Chen, Y., & Moore, K. L. (2002). Analytical stability bound for a class of delayed fractional-order dynamic systems. Nonlinear Dynamics, 29(1), 191–200.

    Article  MathSciNet  MATH  Google Scholar 

  38. Cooke, K. L., & Yorke, J. A. (1973). Some equations modelling growth processes and gonorrhea epidemics. Mathematical Biosciences, 16(1), 75–101.

    Article  MathSciNet  MATH  Google Scholar 

  39. Daftardar-Gejji, V., Sukale, Y., & Bhalekar, S. (2014). A new predictor-corrector method for fractional differential equations. Applied Mathematics and Computations, 244(2014), 158–182.

    Article  MathSciNet  MATH  Google Scholar 

  40. Daftardar-Gejji, V., Sukale, Y., & Bhalekar, S. (2015). Solving fractional delay differential equations: A new approach. Fractional Calculus and Applied Analysis, 18(2), 400–418.

    Article  MathSciNet  MATH  Google Scholar 

  41. Daftardar-Gejji, V., & Babakhani, A. (2004). Analysis of a system of fractional differential equations. Journal of Mathematical Analysis and Applications, 293(2), 511–522.

    Article  MathSciNet  MATH  Google Scholar 

  42. Daftardar-Gejji, V., & Jafari, H. (2006). Boundary value problems for fractional diffusion-wave equation. Australian Journal of Mathematical Analysis and Applications, 3(1), 8.

    MathSciNet  MATH  Google Scholar 

  43. Daftardar-Gejji, V., & Jafari, H. (2006). An iterative method for solving non linear functional equations. Journal of Mathematical Analysis and Applications, 316(2), 753–763.

    Article  MathSciNet  MATH  Google Scholar 

  44. Daftardar-Gejji, V., & Bhalekar, S. (2008). Boundary value problems for multi-term fractional differential equations. Journal of Mathematical Analysis and Applications, 345(2), 754–765.

    Google Scholar 

  45. Daftardar-Gejji, V., & Bhalekar, S. (2008). Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian method. Applied Mathematics and Computation, 202(1), 113–120.

    Article  MathSciNet  MATH  Google Scholar 

  46. Daftardar-Gejji, V., & Bhalekar, S. (2010). Chaos in fractional ordered Liu system. Computers and Mathematics with Applications, 59(3), 1117–1127.

    Article  MathSciNet  MATH  Google Scholar 

  47. Davis, L. C. (2003). Modification of the optimal velocity traffic model to include delay due to driver reaction time. Physica A, 319, 557–567.

    Article  MATH  Google Scholar 

  48. De Cesare, L., & Sportelli, M. (2005). A dynamic IS-LM model with delayed taxation revenues. Chaos Solitons and Fractals, 25(1), 233–244.

    Article  MathSciNet  MATH  Google Scholar 

  49. Delbosco, D., & Rodino, L. (1996). Existence and uniqueness for a nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications, 204(2), 609–625.

    Article  MathSciNet  MATH  Google Scholar 

  50. Deng, W., Li, C., & Lü, J. (2007). Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics, 48(4), 409–416.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  52. Deng, W. H., & Li, C. P. (2005). Synchronization of chaotic fractional Chen system. Journal of the Physical Society of Japan, 74(6), 1645–1648.

    Article  MATH  Google Scholar 

  53. Diethelm, K., & Ford, N. J. (2002). Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 265(2), 229–248.

    Article  MathSciNet  MATH  Google Scholar 

  54. Diethelm, K., Ford, N. J., & Freed, A. D. (2002). A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29(1–4), 3–22.

    Article  MathSciNet  MATH  Google Scholar 

  55. Dokoumetzidis, A., Iliadis, A., & Macheras, P. (2001). Nonlinear dynamics and chaos theory: Concepts and applications relevant to pharmacodynamics. Pharmaceutical Research, 18(4), 415–426.

    Article  Google Scholar 

  56. Eidukaitis, A., Varoneckas, G., & Emaitytee, D. (2004). Application of chaos theory in analyzing the cardiac rhythm in healthy subjects at different sleep stages. Human Physiology, 30(5), 551–555.

    Article  Google Scholar 

  57. Epstein, I., & Luo, Y. (1991). Differential delay equations in chemical kinetics. Nonlinear models: The cross-shaped phase diagram and the Oregonator. The Journal of Chemical Physics, 95(1), 244–254.

    Article  Google Scholar 

  58. Fanti, L., & Manfredi, P. (2007). Chaotic business cycles and fiscal policy: An IS-LM model with distributed tax collection lags. Chaos Solitons and Fractals, 32(2), 736–744.

    Article  MathSciNet  MATH  Google Scholar 

  59. Feliu, V., Rivas, R., & Castillo, F. (2009). Fractional order controller robust to time delay for water distribution in an irrigation main canal pool. Computers and Electronics in Agriculture, 69(2), 185–197.

    Google Scholar 

  60. Field, R. J., & Gyorgyi, L. (1993). Chaos in chemistry and biochemistry. Singapore: World Scientific.

    Book  Google Scholar 

  61. Fridman, E., Fridman, L., & Shustin, E. (2000). Steady modes in relay control systems with time delay and periodic disturbances. Journal of Dynamic Systems, Measurement, and Control, 122(4), 732–737.

    Article  Google Scholar 

  62. Fu-Hong, M., Shu-Yi, S., Wen-Di, H., & En-Rong, W. (2015). Circuit implementations, bifurcations and chaos of a novel fractional-order dynamical system. Chinese Physics Letters, 32(3), 030503.

    Article  Google Scholar 

  63. Gao, Z., & Liao, X. (2014). Active disturbance rejection control for synchronization of different fractional-order chaotic systems. In 11th World Congress on Intelligent Control and Automation (WCICA). June 29 2014–July 4 2014, Shenyang (pp. 2699–2704). IEEE.

    Google Scholar 

  64. Gorenflo, R., & Mainardi, F. (1996). Fractional oscillations and Mittag-Leffler functions. In Kuwait University, Department of Mathematics and Computer Science, International Workshop on the Recent Advances in Applied Mathematics. May 4–7, 1996, State of Kuwait (pp. 193–208).

    Google Scholar 

  65. Grigorenko, I., & Grigorenko, E. (2003). Chaotic dynamics of the fractional Lorenz system. Physical Review Letters, 91(3), 034101.

    Article  Google Scholar 

  66. Hamamci, S.E. (2007). An algorithm for stabilization of fractional order time delay systems using fractional-order PID Controllers. IEEE Transactions on Automatic Control, 52(10), 1964–1969.

    Google Scholar 

  67. Hartley, T. T., Lorenzo, C. F., & Qammer, H. K. (1995). Chaos in a fractional order Chua’s system. IEEE Transactions on Circuits and Systems I, 42(8), 485–490.

    Article  Google Scholar 

  68. Hassell, M. P., Comins, H. N., & May, R. M. (1991). Spatial structure and chaos in insect population dynamics. Nature, 353(6341), 255–258.

    Article  Google Scholar 

  69. He, J. H. (1998). Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Methods in Applied Mechanical Engineering, 167(1), 57–68.

    Article  MathSciNet  MATH  Google Scholar 

  70. He, J. H. (1999). Homotopy perturbation technique. Computer Methods in Applied Mechanical Engineering, 178(3), 257–262.

    Article  MathSciNet  MATH  Google Scholar 

  71. He, R., & Vaidya, P. G. (1998). Implementation of chaotic cryptography with chaotic synchronization. Physical Review E, 57(2), 1532.

    Article  Google Scholar 

  72. Hilfer, R. (Ed.). (2001). Applications of fractional calculus in physics. Singapore: World Scientific.

    MATH  Google Scholar 

  73. Hotzel, R. (1998). Summary: Some stability conditions for fractional delay systems. Journal of Mathematical Systems Estimation and Control, 8, 499–502.

    MathSciNet  MATH  Google Scholar 

  74. Huang, L., Feng, R., & Wang, M. (2004). Synchronization of chaotic systems via nonlinear control. Physics Letters A, 320(4), 271–275.

    Article  MathSciNet  MATH  Google Scholar 

  75. Hwang, C., & Cheng, Y. C. (2006). A numerical algorithm for stability testing of fractional delay systems. Automatica, 42(5), 825–831.

    Google Scholar 

  76. Ingo, C., Magin, R. L., & Parrish, T. B. (2014). New insights into the fractional order diffusion equation using entropy and kurtosis. Entropy, 16(11), 5838–5852.

    Article  Google Scholar 

  77. Jesus, I. S., & Machado, J. A. T. (2008). Fractional control of heat diffusion systems. Nonlinear Dynamics, 54(3), 263–282.

    Article  MathSciNet  MATH  Google Scholar 

  78. Jesus, I. S., Machado, J. A. T., & Barbosa, R. S. (2010). Control of a heat diffusion system through a fractional order nonlinear algorithm. Computers and Mathematics with Applications, 59(5), 1687–1694.

    Google Scholar 

  79. Jun, D., Guangjun, Z., Shaoying, W., & Qiongyao, L. (2014). Chaotic synchronization between fractional-order financial system and financial system of integer orders. In Control and Decision Conference (2014 CCDC), The 26th Chinese IEEE. May 31, 2014–June 2, 2014, Changsha (pp. 4924–4928). IEEE.

    Google Scholar 

  80. Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations. Amsterdam: Elsevier.

    MATH  Google Scholar 

  81. Kuang, Y. (1993). Delay differential equations with applications in population biology. Boston: Academic Press.

    MATH  Google Scholar 

  82. Lakshmanan, M., & Senthilkumar, D. V. (2010). Dynamics of nonlinear time-delay systems. Heidelberg: Springer.

    MATH  Google Scholar 

  83. Li, C., & Chen, G. (2004). Chaos and hyperchaos in the fractional order Rossler equations. Physica A: Statistical Mechanics and its Applications, 341, 55–61.

    Article  MathSciNet  Google Scholar 

  84. Li, C. P., & Deng, W. H. (2006). Chaos synchronization of fractional order differential system. International Journal of Modern Physics B, 20(7), 791–803.

    Article  MathSciNet  MATH  Google Scholar 

  85. Li, C. P., Deng, W. H., & Xu, D. (2006). Chaos synchronization of the Chua system with a fractional order. Physica A, 360(2), 171–185.

    Article  MathSciNet  Google Scholar 

  86. Li, M., Li, D., Wang, J., & Zhao, C. (2013). Active disturbance rejection control for fractional-order system. ISA Transactions, 52(3), 365–374.

    Article  Google Scholar 

  87. Liao, T. L. (1998). Adaptive synchronization of two Lorenz systems. Chaos Solitons and Fractals, 9(9), 1555–1561.

    Article  MATH  Google Scholar 

  88. Liao, H. (2014). Optimization analysis of Duffing oscillator with fractional derivatives. Nonlinear Dynamics, 79(2), 1311–1328.

    Article  MathSciNet  MATH  Google Scholar 

  89. Luchko, Y. F., & Gorenflo, R. (1999). An operational method for solving fractional differential equations with the Caputo derivatives. Acta Mathematica Vietnamica, 24(2), 207–233.

    MathSciNet  MATH  Google Scholar 

  90. Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20(2), 130–141.

    Article  Google Scholar 

  91. Magin, R. L. (2006). Fractional calculus in bioengineering. Redding: Begll House Publishers.

    Google Scholar 

  92. Mainardi, F., Luchko, Y., & Pagnini, G. (2001). The fundamental solution of the space-time fractional diffusion equation. Fractional Calculus and Applied Analysis, 4(2), 153–192.

    MathSciNet  MATH  Google Scholar 

  93. Mainardi, F. (2010). Fractional calculus and waves in linear viscoelasticity: An introduction to mathematical models. London: Imperial College Press.

    Book  MATH  Google Scholar 

  94. Mainieri, R., & Rehacek, J. (1999). Projective synchronization in three-dimensional chaotic systems. Physical Review Letters, 82(15), 3042–3045.

    Article  Google Scholar 

  95. Matsumoto, T. (1984). A chaotic attractor from Chua’s circuit. IEEE Transactions on Circuits and Systems, 31(12), 1055–1058.

    Article  MathSciNet  MATH  Google Scholar 

  96. Matignon, D. (1996). Stability results for fractional differential equations with applications to control processing. In Computational engineering in systems and application multiconference (pp. 963–968). Lille: Gerf EC Lille, Villeneuve d’Ascq.

    Google Scholar 

  97. Meilanov, R. P., & Magomedov, R. A. (2014). Thermodynamics in fractional calculus. Journal of Engineering Physics and Thermophysics, 87(6), 1521–1531.

    Article  Google Scholar 

  98. Mekki, H., Boukhetala, D., & Azar, A. T. (2015). Sliding modes for fault tolerant control. In Advances and applications in sliding mode control systems (Vol. 576, pp. 407–433). Studies in computational intelligence book series. Berlin/Heidelberg: Springer. doi:10.1007/978-3-319-11173-5_15.

  99. Muthukumar, P., & Balasubramaniam, P. (2013). Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography. Nonlinear Dynamics, 74(4), 1169–1181.

    Article  MathSciNet  MATH  Google Scholar 

  100. Muthukumar, P., Balasubramaniam, P., & Ratnavelu, K. (2014). Synchronization and an application of a novel fractional order King Cobra chaotic system. Chaos, 24(3), 033105.

    Article  MathSciNet  Google Scholar 

  101. Muthukumar, P., Balasubramaniam, P., & Ratnavelu, K. (2015). Sliding mode control design for synchronization of fractional order chaotic systems and its application to a new cryptosystem. International Journal of Dynamics and Control. In Press. doi:10.1007/s40435-015-0169-y.

  102. Ortigueira, M. D., & Machado, J. A. T. (2006). Fractional calculus applications in signals and systems. Signal Processing, 86(10), 2503–2504.

    Article  MATH  Google Scholar 

  103. Pecora, L. M., & Carroll, T. L. (1990). Synchronization in chaotic systems. Physical Review Letters, 64(8), 821.

    Article  MathSciNet  MATH  Google Scholar 

  104. Pecora, L. M., & Carroll, T. L. (1991). Driving systems with chaotic signals. Physical Review A, 44(4), 2374.

    Article  Google Scholar 

  105. Podlubny, I. (1999). Fractional differential equations. San Diego: Academic Press.

    MATH  Google Scholar 

  106. Pyragas, K. (1992). Continuous control of chaos by self-controlling feedback. Physics letters A, 170(6), 421–428.

    Article  Google Scholar 

  107. Ran, Q., Yuan, L., & Zhao, T. (2015). Image encryption based on nonseparable fractional Fourier transform and chaotic map. Optics Communications, 348, 43–49.

    Google Scholar 

  108. Sabatier, J., Poullain, S., Latteux, P., Thomas, J., & Oustaloup, A. (2004). Robust speed control of a low damped electromechanical system based on CRONE control: Application to a four mass experimental test bench. Nonlinear Dynamics, 38(1–4), 383–400.

    Article  MATH  Google Scholar 

  109. Samko, S. G., Kilbas, A. A., & Maricev, O. I. (1993). Fractional integrals and derivatives. Yverdon: Gordon and Breach.

    Google Scholar 

  110. Serletic, A. (1996). Is there chaos in economic series? Canadian Journal of Economics, 29, S210–S212.

    Article  Google Scholar 

  111. 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 and Fractals, 36(1), 98–103.

    Article  MathSciNet  MATH  Google Scholar 

  112. Si-Ammour, A., Djennoune, S., & Bettayeb, M. (2009). A sliding mode control for linear fractional systems with input and state delays. Communications in Nonlinear Science and Numerical Simulation, 14(5), 2310–2318.

    Article  MathSciNet  MATH  Google Scholar 

  113. Tseng, C., & Lee, S. L. (2014). Digital image sharpening using Riesz fractional order derivative and discrete hartley transform. In 2014 IEEE Asia Pacific Conference on Circuits and Systems (APCCAS) (pp. 483–486). Ishigaki: IEEE.

    Google Scholar 

  114. Uçar, A. (2002). A prototype model for chaos studies. International Journal of Engineering Science, 40(3), 251–258.

    Article  MathSciNet  MATH  Google Scholar 

  115. Uçar, A. (2003). On the chaotic behaviour of a prototype delayed dynamical system. Chaos Solitons and Fractals, 16(2), 187–194.

    Article  MATH  Google Scholar 

  116. 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. International Journal of Modelling Identification and Control, 23(3), 267–277.

    Article  Google Scholar 

  117. 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 (pp. 527–547). Studies in computational intelligence book series. Berlin: Springer.

    Google Scholar 

  118. 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 (pp. 549–569). Studies in computational intelligence book series. Berlin: Springer.

    Google Scholar 

  119. 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, 23(1), 92–100.

    Article  Google Scholar 

  120. 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 (pp. 39–58). Berlin: Springer.

    Google Scholar 

  121. 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. In Advances in chaos theory and intelligent control (Vol. 337). Studies in fuzziness and soft computing. Germany: Springer.

    Google Scholar 

  122. Vaidyanathan, S., & Azar, A. T. (2016). Qualitative study and adaptive control of a novel 4-D hyperchaotic system with three quadratic nonlinearities. In Advances in chaos theory and intelligent control (Vol. 337). Studies in fuzziness and soft computing. Germany: Springer.

    Google Scholar 

  123. 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. In Advances in chaos theory and intelligent control (Vol. 337). Studies in fuzziness and soft computing. Germany: Springer.

    Google Scholar 

  124. Vaidyanathan, S., & Azar, A. T. (2016). Adaptive control and synchronization of halvorsen circulant chaotic systems. In Advances in chaos theory and intelligent control (Vol. 337). Studies in fuzziness and soft computing. Germany: Springer.

    Google Scholar 

  125. Vaidyanathan, S., & Azar, A. T. (2016). Adaptive backstepping control and synchronization of a novel 3-D jerk system with an exponential nonlinearity. In Advances in chaos theory and intelligent control (Vol. 337). Studies in fuzziness and soft computing. Germany: Springer.

    Google Scholar 

  126. Vaidyanathan, S., & Azar, A. T. (2016). Generalized projective synchronization of a novel hyperchaotic four-wing system via adaptive control method. In Advances in chaos theory and intelligent control (Vol. 337). Studies in fuzziness and soft computing. Germany: Springer.

    Google Scholar 

  127. Wang, D., & Yu, J. (2008). Chaos in the fractional order logistic delay system. Journal of Electronic Science and Technology of China, 6(3), 225–229.

    MathSciNet  Google Scholar 

  128. Wang, J., Xionga, X., & Zhang, Y. (2006). Extending synchronization scheme to chaotic fractional-order Chen systems. Physica A, 370(2), 279–285.

    Article  Google Scholar 

  129. Wang, J., & Zhang, Y. (2006). Designing synchronization schemes for chaotic fractional-order unified systems. Chaos Solitons and Fractals, 30(5), 1265–1272.

    Article  MathSciNet  MATH  Google Scholar 

  130. Wang, S., Sun, W., Ma, C. Y., Wang, D., & Chen, Z. (2013). Secure communication based on a fractional order chaotic system. International Journal of Security and Its Applications, 7(5), 205–216.

    Article  Google Scholar 

  131. Wang, S., Yu, Y., & Wen, G. (2014). Hybrid projective synchronization of time-delayed fractional order chaotic systems. Nonlinear Analysis: Hybrid Systems, 11, 129–138.

    MathSciNet  MATH  Google Scholar 

  132. Wu, G. C., Baleanu, D., & Lin, Z. X. (2016). Image encryption technique based on fractional chaotic time series. Journal of Vibration and Control, 22(8), 2092–2099.

    Article  MathSciNet  Google Scholar 

  133. Xi, H., & Gunton, J. D. (1995). Spatiotemporal chaos in a model of Rayleigh-Benard convection. Physical Review E, 52(4), 4963–4975.

    Article  Google Scholar 

  134. Xu, B., Chen, D., Zhang, H., & Wang, F. (2015). Modeling and stability analysis of a fractional-order Francis hydro-turbine governing system. Chaos Solitons and Fractals, 75, 50–61.

    Article  MathSciNet  MATH  Google Scholar 

  135. Xu, Y., & Wang, H. (2013). Synchronization of fractional-order chaotic systems with Gaussian fluctuation by sliding mode control. Abstract and Applied Analysis. Article ID 948782.

    Google Scholar 

  136. Xu, Y., Wang, H., Li, Y., & Pei, B. (2014). Image encryption based on synchronization of fractional chaotic systems. Communications in Nonlinear Science and Numerical Simulation, 19(10), 3735–3744.

    Article  MathSciNet  Google Scholar 

  137. Yassen, M. T. (2001). Adaptive control and synchronization of a modified Chua’s circuit system. Applied Mathematics and Computation, 135(1), 113–128.

    Article  MathSciNet  MATH  Google Scholar 

  138. Zhao, J., Wang, S., Chang, Y., & Li, X. (2015). A novel image encryption scheme based on an improper fractional-order chaotic system. Nonlinear Dynamics, 80(4), 1721–1729.

    Article  MathSciNet  Google Scholar 

  139. Zhu, Q., & Azar, A. T. (2015). Complex system modelling and control through intelligent soft computations. In Studies in fuzziness and soft computing (Vol. 319). Germany: Springer. ISBN: 978-3-319-12882-5.

    Google Scholar 

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Acknowledgements

Author acknowledges Council of Scientific and Industrial Research, New Delhi, India for funding through Research Project [25(0245)/15/EMR-II]. The author is grateful to Prof. Ahmad Taher Azar for his encouragement and support.

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  1. Department of Mathematics, Shivaji University, Vidyanagar, Kolhapur, 416004, India

    Sachin Bhalekar

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  1. Sachin Bhalekar
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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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Bhalekar, S. (2017). Dynamics of Fractional Order Complex Uçar System. 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_26

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