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Synchronization of Fractional Chaotic and Hyperchaotic Systems Using an Extended Active Control

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Advances in Chaos Theory and Intelligent Control

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 337))

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Abstract

An extended active control technique is used to synchronize fractional order chaotic and hyperchaotic systems with and without delay. The coupling strength is set to the value less than one to achieve the complete synchronization more easily. Explicit formula for the error matrix is also proposed in this chapter. Numerical examples are given for the fractional order chaotic Liu system, hyperchaotic new system and Ucar delay system. The effect of fractional order and coupling strength on the synchronization time is studied for non-delayed cases. It is observed that the synchronization time decreases with increase in fractional order as well as with increase in coupling strength for the Liu system. For the new system, the synchronization time decreases with increase in fractional order as well as with decrease in coupling strength.

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References

  1. Vaidyanathan S, Azar AT (2015) Analysis, control and synchronization of a nine-term 3-d novel chaotic system. In: Azar AT, Vaidyanathan S (eds) Chaos modeling and control systems design. Studies in computational intelligence, vol 581. Springer-Verlag GmbH, Berlin, pp 3–17

    Google Scholar 

  2. Vaidyanathan S, Azar AT (2015) Analysis, control and synchronization of a nine-term 3-D novel chaotic system. In: Azar AT, Vaidyanathan S (eds) Chaos modeling and control systems design. Studies in computational intelligence, vol 581. Springer, Berlin, pp 3–17

    Google Scholar 

  3. Vaidyanathan S, Azar AT (2015) Analysis and control of a 4-D novel hyperchaotic system. In: Azar AT, Vaidyanathan S (eds) Chaos modeling and control systems design. Studies in computational intelligence, vol 581. Springer-Verlag GmbH, Berlin, pp 19–38

    Google Scholar 

  4. Vaidyanathan S, Azar AT (2015) Analysis and control of a 4-D novel hyperchaotic system. In: Azar AT, Vaidyanathan S (eds) Chaos modeling and control systems design. Studies in computational intelligence, vol 581. Springer, Berlin, pp 19–38

    Google Scholar 

  5. Azar AT, Vaidyanathan S (eds) (2015) Chaos modeling and control systems design. Studies in computational intelligence, vol 581. Springer, New York

    Google Scholar 

  6. Azar AT, Vaidyanathan S (2015) Handbook of research on advanced intelligent control engineering and automation. Advances in Computational Intelligence and Robotics (ACIR) Book Series. IGI Global, Hershey PA

    Google Scholar 

  7. Azar AT, Vaidyanathan S (2015) Computational intelligence applications in modeling and control. Studies in computational intelligence, vol 575. Springer, New York

    Google Scholar 

  8. Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64(8):821

    Article  MathSciNet  MATH  Google Scholar 

  9. Pecora LM, Carroll TL (1991) Driving systems with chaotic signals. Phys Rev A 44:2374

    Article  Google Scholar 

  10. Hilfer R (ed) (2001) Applications of fractional calculus in physics. World Scientific, Singapore

    MATH  Google Scholar 

  11. He R, Vaidya PG (1998) Implementation of chaotic cryptography with chaotic synchronization. Phys Rev E 57(2):1532

    Article  Google Scholar 

  12. Huang L, Feng R, Wang M (2004) Synchronization of chaotic systems via nonlinear control. Phys Lett A 320:271

    Article  MathSciNet  MATH  Google Scholar 

  13. Liao TL (1998) Adaptive synchronization of two Lorenz systems. Chaos Solitons Fractals 9:1555

    Article  MATH  Google Scholar 

  14. Yassen MT (2001) Adaptive control and synchronization of a modified Chua’s circuit system. Appl Math Comput 135(1):113

    Article  MathSciNet  MATH  Google Scholar 

  15. Bai EW, Lonngre KE (1997) Synchronization of two Lorenz systems using active control. Chaos Solitons Fractals 8:51–58

    Article  Google Scholar 

  16. Bai EW, Lonngren KE (2000) Sequential synchronization of two Lorenz systems using active control. Chaos Solitons Fractals 11:1041–1044

    Article  MATH  Google Scholar 

  17. Vaidyanathan S, Azar AT, Rajagopal K, Alexander P (2015) Design and SPICE implementation of a 12-term novel hyperchaotic system and its synchronization via active control. Int J Model Identif Control 23(3):267–277

    Article  Google Scholar 

  18. Azar AT, Serrano FE (2015) Adaptive Sliding mode control of the Furuta pendulum. In: Azar AT, Zhu Q (eds) Advances and applications in sliding mode control systems. Studies in computational intelligence, vol 576. Springer, Berlin, pp 1–42

    Google Scholar 

  19. Azar AT, Zhu Q (2015) Advances and applications in sliding mode control systems. Studies in computational intelligence, vol 576. Springer, Germany

    Google Scholar 

  20. Vaidyanathan S, Azar AT (2015) Anti-synchronization of identical chaotic systems using sliding mode control and an application to Vaidyanathan-Madhavan chaotic systems. In: Azar AT, Zhu Q (eds) Advances and applications in sliding mode control systems. Studies in computational intelligence book series. Springer, Berlin, pp 527–547

    Google Scholar 

  21. Vaidyanathan S, Azar AT (2015) Hybrid synchronization of identical chaotic systems using sliding mode control and an application to Vaidyanathan chaotic systems. In: Azar AT, Zhu Q (eds) (2015) Advances and applications in sliding mode control systems. Studies in computational intelligence book series. Springer, Berlin, pp 549–569

    Google Scholar 

  22. Vaidyanathan S, Sampath S, Azar AT (2015) Global chaos synchronisation of identical chaotic systems via novel sliding mode control method and its application to Zhu system. Int J Model Identif Control 23(1):92–100

    Article  Google Scholar 

  23. Azar AT, Serrano FE (2015) Deadbeat control for multivariable systems with time varying delays. In: Azar AT, Vaidyanathan S (eds) Chaos modeling and control systems design. Studies in computational intelligence, vol 581. Springer, Berlin, pp 97–132

    Google Scholar 

  24. Vaidyanathan S, Idowu BA, Azar AT (2015) Backstepping controller design for the global chaos synchronization of Sprott’s jerk systems. In: Azar AT, Vaidyanathan S (eds) Chaos modeling and control systems design. Springer, Berlin, pp 39–58

    Google Scholar 

  25. Daftardar-Gejji V, Bhalekar S (2008) Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method. Appl Math Comput 202:113–120

    Article  MathSciNet  MATH  Google Scholar 

  26. Ingo C, Magin RL, Parrish TB (2014) New insights into the fractional order diffusion equation using entropy and kurtosis. Entropy 16(11):5838–5852

    Article  Google Scholar 

  27. Jesus IS, Machado JAT (2008) Fractional control of heat diffusion systems. Nonlinear Dyn 54(3):263–282

    Article  MathSciNet  MATH  Google Scholar 

  28. Jesus IS, Machado JAT, Barbosa RS (2010) Control of a heat diffusion system through a fractional order nonlinear algorithm. Comput Math Appl 59(5):1687–1694

    Article  MathSciNet  MATH  Google Scholar 

  29. Mainardi F, Luchko Y, Pagnini G (2001) The fundamental solution of the space-time fractional diffusion equation. Fract Calc Appl Anal 4(2):153–192

    MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  31. Magin RL (2006) Fractional calculus in bioengineering. Begll House Publishers, Redding

    Google Scholar 

  32. Anastasio TJ (1994) The fractional-order dynamics of Brainstem Vestibulo-Oculomotor neurons. Biol Cybern 72:69–79

    Article  Google Scholar 

  33. Ortigueira MD, Machado JAT (2006) Fractional calculus applications in signals and systems. Signal Process 86(10):2503–2504

    Article  MATH  Google Scholar 

  34. Tseng C, Lee SL (2014) Digital image sharpening using Riesz fractional order derivative and discrete hartley transform. In: 2014 IEEE Asia pacific conference on circuits and systems (APCCAS). IEEE, Ishigaki, pp 483–486

    Google Scholar 

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

    Article  Google Scholar 

  36. Wu GC, Baleanu D, Lin ZX (2015) Image encryption technique based on fractional chaotic time series. J Vibr Control. Article 1077546315574649

    Google Scholar 

  37. Zhao J, Chang Y, Li X (2015) A novel image encryption scheme based on an improper fractional-order chaotic system. Nonlinear Dyn 80(4):1721–1729

    Google Scholar 

  38. Wang S, Sun W, Ma CY, Wang D, Chen Z (2013) Secure communication based on a fractional order chaotic system. Int J Security Appl 7(5):205–216

    Article  Google Scholar 

  39. 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 Dyn 38:383–400

    Article  MATH  Google Scholar 

  40. Meilanov RP, Magomedov RA (2014) Thermodynamics in fractional calculus. J Eng Phys Thermophys 87(6):1521–1531

    Article  Google Scholar 

  41. 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. Chin Phys Lett 32(3):030503

    Article  Google Scholar 

  42. Liao H (2014) Optimization analysis of Duffing oscillator with fractional derivatives. Nonlinear Dyn 79(2):1311–1328

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. 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  MATH  Google Scholar 

  46. 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. Int J Dyn Control. doi:10.1007/s40435-015-0169-y (in press)

  47. Barrett JH (1954) Differential equations of non-integer order. Can J Math 64:529–541

    Article  MathSciNet  MATH  Google Scholar 

  48. Al-Bassam MA (1965) Some existence theorems on differential equations of generalized order. Journal fr die reine und angewandte Mathematik 2181:70–78

    MathSciNet  MATH  Google Scholar 

  49. Delbosco D, Rodino L (1996) Existence and uniqueness for a nonlinear fractional differential equation. J Math Anal Appl 2042:609–625

    Article  MathSciNet  MATH  Google Scholar 

  50. Diethelm K, Ford NJ (2002) Analysis of fractional differential equations. J Math Anal Appl 2652:229–248

    Article  MathSciNet  MATH  Google Scholar 

  51. Daftardar-Gejji V, Babakhani A (2004) Analysis of a system of fractional differential equations. J Math Anal Appl 2932:511–522

    Article  MathSciNet  MATH  Google Scholar 

  52. Gorenflo R, Mainardi F (1996) Fractional oscillations and Mittag-Leffler functions. In: International workshop on the recent advances in applied mathematics. Kuwait University, Department of Mathematics and Computer Science, State of Kuwait, pp 193–208

    Google Scholar 

  53. Luchko YF, Gorenflo R (1999) An operational method for solving fractional differential equations with the Caputo derivatives. Acta Mathematica Vietnamica 24:207–233

    MathSciNet  MATH  Google Scholar 

  54. Daftardar-Gejji V, Jafari H (2006) Boundary value problems for fractional diffusion-wave equation. Aust J Math Anal Appl 3:1–8

    MathSciNet  MATH  Google Scholar 

  55. Daftardar-Gejji V, Bhalekar S (2008) Boundary value problems for multi-term fractional differential equations. J Math Anal Appl 345:754–765

    Article  MathSciNet  MATH  Google Scholar 

  56. Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29:3–22

    Article  MathSciNet  MATH  Google Scholar 

  57. Daftardar-Gejji V, Sukale Y, Bhalekar S (2014) A new predictorcorrector method for fractional differential equations. Appl Math Comput 244:158–182

    Google Scholar 

  58. Adomian G (1994) Solving Frontier problems of physics: the decomposition method. Kluwer Academic, Dordrecht

    Book  MATH  Google Scholar 

  59. He JH (1998) Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput Methods Appl Mech Eng 167:57–68

    Article  MathSciNet  MATH  Google Scholar 

  60. He JH (1999) Homotopy perturbation technique. Comput Methods Appl Mech Eng 178:257–262

    Article  MathSciNet  MATH  Google Scholar 

  61. Daftardar-Gejji V, Jafari H (2006) An iterative method for solving non linear functional equations. J Math Anal Appl 316:753–763

    Article  MathSciNet  MATH  Google Scholar 

  62. Tang RA, Liu YL, Xue JK (2009) An extended active control for chaos synchronization. Phys Lett A 373:1449–1454

    Article  MathSciNet  MATH  Google Scholar 

  63. Deng WH, Li CP (2005) Chaos synchronization of the fractional Lü system. Phys A 353:61–72

    Article  Google Scholar 

  64. Deng WH, Li CP (2005) Synchronization of chaotic fractional Chen system. J Phys Soc Jpn 74:1645–1648

    Article  MATH  Google Scholar 

  65. Li CP, Deng WH (2006) Chaos synchronization of fractional order differential system. Int J Mod Phys B 20(7):791–803

    Article  MathSciNet  MATH  Google Scholar 

  66. Li CP, Deng WH, Xu D (2006) Chaos synchronization of the Chua system with a fractional order. Phys A 360:171–185

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  68. Yu Y, Li H (2008) The synchronization of fractional-order Rossler hyperchaotic systems. Phys A 387:1393–1403

    Article  MathSciNet  Google Scholar 

  69. Wang J, Xionga X, Zhang Y (2006) Extending synchronization scheme to chaotic fractional-order Chen systems. Phys A 370:279–285

    Article  Google Scholar 

  70. 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. IEEE, Changsha, pp 4924–4928

    Google Scholar 

  71. Mohadeszadeh M, Delavari H (2015) Synchronization of fractional-order hyper-chaotic systems based on a new adaptive sliding mode control. Int J Dyn Control. doi:10.1007/s40435-015-0177-y (in press)

  72. Gao L, Wang Z, Zhou K, Zhu W, Wu Z, Ma T (2015) Modified sliding mode synchronization of typical three-dimensional fractional-order chaotic systems. Neurocomputing 166:53–58

    Article  Google Scholar 

  73. Tian X, Fei S, Chai L (2015) On modified second-order sliding mode synchronization of two different fractional order hyperchaotic systems. Int J Multimed Ubiquitous Eng 10(4):387–398

    Google Scholar 

  74. Guanand J, Wang K (2015) Sliding mode control and modified generalized projective synchronization of a new fractional-order chaotic system. Math Probl Eng. ID 941654

    Google Scholar 

  75. Tavazoei MS, Haeri M (2008) Synchronization of chaotic fractional-order systems via active sliding mode controller. Phys A 387:57–70

    Article  MathSciNet  Google Scholar 

  76. Xu Y, Wang H (2013) Synchronization of fractional-order chaotic systems with Gaussian fluctuation by sliding mode control. Abstr Appl Anal. Article ID 948782, 7 pages

    Google Scholar 

  77. Liu H, Yang J (2015) Sliding-mode synchronization control for uncertain fractional-order chaotic systems with time delay. Entropy 17:4202–4214

    Article  Google Scholar 

  78. Li J, Guo X, Yao L (2014) Adaptive synchronization of fractional-order general complex dynamical networks. In: 2014 11th world congress on intelligent control and automation (WCICA). IEEE, Shenyang, pp 4367–4372

    Google Scholar 

  79. Rad P, Nikdadian M, Bahadorzadeh M (2015) Synchronizing the fractional-order Genesio-Tesi chaotic system using adaptive control. Int J Sci Eng Res 6:1699–1702

    Google Scholar 

  80. 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 Dyn 80:753–765

    Article  MathSciNet  MATH  Google Scholar 

  81. Leung A, Li X, Chu Y, Rao X (2015) Synchronization of fractional-order chaotic systems using unidirectional adaptive full-state linear error feedback coupling. Nonlinear Dyn 82(1–2):185–199

    Google Scholar 

  82. El-Sayed AMA, Nour HM, Elsaid A, Matouk AE, Elsonbaty A (2015) Dynamical behaviors, circuit realization, chaos control and synchronization of a new fractional order hyperchaotic system. Appl Math Model doi:10.1016/j.apm.2015.10.010 (in press)

  83. Xingyuan W, Yijie H (2008) Projective synchronization of fractional order chaotic system based on linear separation. Phys Lett A 372:435–441

    Article  MATH  Google Scholar 

  84. Agrawal S, Das S (2014) Projective synchronization between different fractional-order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique. Math Methods Appl Sci 37:2164–2176

    Article  MathSciNet  MATH  Google Scholar 

  85. Liu J, Liu S, Yuan C (2013) Modified generalized projective synchronization of fractional-order chaotic Lü systems. Adv Diff Equ 2013(1). Article 374

    Google Scholar 

  86. Zhou P, Ding R, Cao Y (2012) Hybrid projective synchronization for two identical fractional-order chaotic systems. Discrete Dyn Nat Soc. Article ID 768587, 11 pages

    Google Scholar 

  87. Chen L, Chai Y, Wu R (2011) Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems. Phys Lett A 375(21):2099–2110

    Article  MATH  Google Scholar 

  88. Sun Z (2015) Lag projective synchronization of two chaotic systems with different fractional orders. J Korean Phys Soc 66:1192–1199

    Article  Google Scholar 

  89. Bhalekar S, Daftardar-Gejji V (2010) Synchronization of different fractional order chaotic systems using active control. Commun Nonlinear Sci Numer Simul 15(11):3536–3546

    Article  MathSciNet  MATH  Google Scholar 

  90. Bhalekar S, Daftardar-Gejji V (2011) Anti-synchronization of non-identical fractional order chaotic systems using active control. Int J Differ Equ. Article ID 250763

    Google Scholar 

  91. Bhalekar S (2014) Synchronization of incommensurate non-identical fractional order chaotic systems using active control. Eur Phys J Special Topics 223(8):1495–1508

    Article  Google Scholar 

  92. Bhalekar S (2014) Synchronization of non-identical fractional order hyperchaotic systems using active control. World J Model Simul 10(1):60–68

    Google Scholar 

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

    Google Scholar 

  94. 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). IEEE, Shenyang, pp 2699–2704

    Google Scholar 

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

    Article  Google Scholar 

  96. Li C, Zhang J (2015) Synchronisation of a fractional-order chaotic system using finite-time input-to-state stability. Int J Syst Sci doi:10.1080/00207721.2014.998741 (in press)

  97. Ran D, Caoyuan M, Yongyi Z, Yanfang L, Jianhua L (2014) Anti-synchronization of a class of fractional-order chaotic system with uncertain parameters. Comput Model New Technol 18(11):108–112

    Google Scholar 

  98. Noghredani N, Balochian S (2015) Synchronization of fractional-order uncertain chaotic systems with input nonlinearity. Int J General Syst 44:485–498

    Article  MathSciNet  MATH  Google Scholar 

  99. Balasubramaniam P, Muthukumar P, Ratnavelu K (2015) Theoretical and practical applications of fuzzy fractional integral sliding mode control for fractional-order dynamical system. Nonlinear Dyn 80:249–267

    Article  MathSciNet  MATH  Google Scholar 

  100. Chen L, Chai Y, Wu R, Ma T, Zhai H (2013) Dynamic analysis of a class of fractional-order neural networks with delay. Neurocomputing 111:190–194

    Article  Google Scholar 

  101. Dang HG, He WS, Yang XY (2014) Investigation of synchronization for a fractional-order delayed system. Appl Mech Mater 687:447–450

    Article  Google Scholar 

  102. Xiaohong Z, Peng C (2015) Different-lags synchronization in time-delay and circuit simulation of fractional-order chaotic system based on parameter identification. Open Electr Electr Eng J 9:117–126

    Article  Google Scholar 

  103. Velmurugan G, Rakkiyappan R, Cao J (2015) Finite-time synchronization of fractional-order memristor-based neural networks with time delays. Neural Netw doi:10.1016/j.neunet.2015.09.012 (in press)

  104. Liang S, Wu R, Chen L Adaptive pinning synchronization in fractional-order uncertain complex dynamical networks with delay. Phys A: Stat Mech Appl doi:10.1016/j.physa.2015.10.011 (in press)

  105. Kiani-B A, Fallahi K, Pariz N, Leung H (2009) A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter. Commun Nonlinear Sci Numer Simul 14(3):863–879

    Article  MathSciNet  MATH  Google Scholar 

  106. Sheu LJ, Chen WC, Chen YC, Weng WT (2010) A two-channel secure communication using fractional chaotic systems. World Acad Sci Eng Technol 65:1057–1061

    Google Scholar 

  107. Huang L, Zhang J, Shi S (2015) Circuit simulation on control and synchronization of fractional order switching chaotic system. Math Comput Simul 113:28–39

    Article  MathSciNet  Google Scholar 

  108. Zhen W, Xia H, Ning L, Xiao-Na S (2012) Image encryption based on a delayed fractional-order chaotic logistic system. Chin Phys B 21. Article ID 050506

    Google Scholar 

  109. Zhen W, Xia H, Yu-Xia L, Xiao-Na S (2013) A new image encryption algorithm based on the fractional-order hyperchaotic Lorenz system. Chin Phys B 22(1). Article ID 010504

    Google Scholar 

  110. Xu Y, Wang H, Li Y, Pei B (2014) Image encryption based on synchronization of fractional chaotic systems. Commun Nonlinear Sci Numer Simul 19(10):3735–3744

    Article  MathSciNet  Google Scholar 

  111. Huang X, Sun T, Li Y, Liang J (2015) A color image encryption algorithm based on a fractional-order hyperchaotic system. Entropy 17(1):28–38

    Article  Google Scholar 

  112. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  115. Bhalekar S (2013) Stability analysis of fractional differential systems with delay. In: Daftardar-Gejji V (ed) Fractional calculus: theory and applications. Narosa Publishing House, New Delhi, pp 60–68

    Google Scholar 

  116. Daftardar-Gejji V, Bhalekar S (2010) Chaos in fractional ordered Liu system. Comput Math Appl 59(3):1117–1127

    Article  MathSciNet  MATH  Google Scholar 

  117. Liang CG, Song Z, Xin TL (2008) A new hyperchaotic system and its linear feedback control. Chin Phys B 17:4039–4046

    Article  Google Scholar 

  118. Bhalekar S (2012) Dynamical analysis of fractional order Ucar prototype delayed system. Signals Image Video Process 6(3):513–519

    Article  Google Scholar 

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Acknowledgments

Author acknowledges Shivaji University, Kolhapur, India for the research grant provided under the Innovative Research Activities (2014–2016). 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. Benha University, Benha, Egypt

    Ahmad Taher Azar

  2. Chennai, India

    Sundarapandian Vaidyanathan

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Bhalekar, S. (2016). Synchronization of Fractional Chaotic and Hyperchaotic Systems Using an Extended Active Control. In: Azar, A., Vaidyanathan, S. (eds) Advances in Chaos Theory and Intelligent Control. Studies in Fuzziness and Soft Computing, vol 337. Springer, Cham. https://doi.org/10.1007/978-3-319-30340-6_3

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