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Hyperchaos and Adaptive Control of a Novel Hyperchaotic System with Two Quadratic Nonlinearities

  • Sundarapandian Vaidyanathan
  • Ahmad Taher Azar
  • Adel Ouannas
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 688)

Abstract

Liu-Su-Liu chaotic system (2007) is one of the classical 3-D chaotic systems in the literature. By introducing a feedback control to the Liu-Su-Liu chaotic system,we obtain a novel hyperchaotic system in this work, which has two quadratic nonlinearities. The phase portraits of the novel hyperchaotic system are displayed and the qualitative properties of the novel hyperchaotic system are discussed. We show that the novel hyperchaotic system has a unique equilibrium point at the origin, which is unstable. The Lyapunov exponents of the novel 4-D hyperchaotic system are obtained as \(L_1 = 1.1097\), \(L_2 = 0.1584\), \(L_3 = 0\) and \(L_4 = -14.1666\). The maximal Lyapunov exponent (MLE) of the novel hyperchaotic system is obtained as \(L_1 = 1.1097\) and Lyapunov dimension as \(D_L = 3.0895\). Since the sum of the Lyapunov exponents of the novel hyperchaotic system is negative, it follows that the novel hyperchaotic system is dissipative. Next, we derive new results for the adaptive control design of the novel hyperchaotic system with unknown parameters. We also derive new results for the adaptive synchronization design of identical novel hyperchaotic systems with unknown parameters. The adaptive control results derived in this work for the novel hyperchaotic system are proved using Lyapunov stability theory. Numerical simulations in MATLAB are shown to validate and illustrate all the main results derived in this work.

Keywords

Chaos Chaotic systems Hyperchaos Hyperchaotic systems Adaptive control Feedback control Synchronization 

References

  1. 1.
    Akgul, A., Moroz, I., Pehlivan, I., & Vaidyanathan, S. (2016). A new four-scroll chaotic attractor and its engineering applications. Optik, 127, 5491–5499.CrossRefGoogle Scholar
  2. 2.
    Arneodo, A., Coullet, P., & Tresser, C. (1981). Possible new strange attractors with spiral structure. Communications in Mathematical Physics, 79, 573–579.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Azar, A. T. (2010). Fuzzy Systems. Vienna, Austria: IN-TECH.Google Scholar
  4. 4.
    Azar, A. T. (2012). Overview of type-2 fuzzy logic systems. International Journal of Fuzzy System Applications, 2(4), 1–28.CrossRefGoogle Scholar
  5. 5.
    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.CrossRefGoogle Scholar
  6. 6.
    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. Studies in computational intelligence (Vol. 576, pp. 1–42). Germany: Springer.Google Scholar
  7. 7.
    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. Studies in computational intelligence (Vol. 581, pp. 97–132). Germany: Springer.Google Scholar
  8. 8.
    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. Studies in fuzziness and soft computing (Vol. 319, pp. 1–44). Germany: Springer.Google Scholar
  9. 9.
    Azar, A. T., & Serrano, F. E. (2015). Stabilizatoin and control of mechanical systems with backlash. In A. T. Azar & S. Vaidyanathan (Eds.), Handbook of research on advanced intelligent control engineering and automation. Advances in computational intelligence and robotics (ACIR) (pp. 1–60). USA: IGI-Global.Google Scholar
  10. 10.
    Azar, A. T., & Vaidyanathan, S. (2015). Chaos modeling and control systems design. Studies in computational intelligence (Vol. 581). Germany: Springer.Google Scholar
  11. 11.
    Azar, A. T., & Vaidyanathan, S. (2015). Computational intelligence applications in modeling and control. Studies in computational intelligence (Vol. 575). Germany: Springer.Google Scholar
  12. 12.
    Azar, A. T., & Vaidyanathan, S. (2015) Handbook of research on advanced intelligent control engineering and automation. Advances in Computational Intelligence and Robotics (ACIR). USA: IGI-Global.Google Scholar
  13. 13.
    Azar, A. T., & Vaidyanathan, S. (2016). Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337). Germany: Springer.Google Scholar
  14. 14.
    Azar, A. T., & Zhu, Q. (2015). Advances and applications in sliding mode control systems. Studies in computational intelligence (Vol. 576). Germany: Springer.Google Scholar
  15. 15.
    Barrow-Green, J. (1997). Poincaré and the three body problem. American Mathematical Society.Google Scholar
  16. 16.
    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, pp. 681–697). Germany: Springer.Google Scholar
  17. 17.
    Boulkroune, A., Hamel, S., Azar, A. T., & Vaidyanathan, S. (2016). Fuzzy control-based function synchronization of unknown chaotic systems with dead-zone input. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 699–718). Germany: Springer.Google Scholar
  18. 18.
    Carroll, T. L., & Pecora, L. M. (1991). Synchronizing chaotic circuits. IEEE Transactions on Circuits and Systems, 38(4), 453–456.CrossRefzbMATHGoogle Scholar
  19. 19.
    Chen, A., Lu, J., Lü, J., & Yu, S. (2006). Generating hyperchaotic Lü attractor via state feedback control. Physica A, 364, 103–110.CrossRefGoogle Scholar
  20. 20.
    Chen, G., & Ueta, T. (1999). Yet another chaotic attractor. International Journal of Bifurcation and Chaos, 9(7), 1465–1466.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chen, W. H., Wei, D., & Lu, X. (2014). Global exponential synchronization of nonlinear time-delay Lur’e systems via delayed impulsive control. Communications in Nonlinear Science and Numerical Simulation, 19(9), 3298–3312.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Chen, Z., Yang, Y., Qi, G., & Yuan, Z. (2007). A novel hyperchaos system only with one equilibrium. Physics Letters A, 360, 696–701.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Dadras, S., & Momeni, H. R. (2009). A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors. Physics Letters A, 373, 3637–3642.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Das, S., Goswami, D., Chatterjee, S., & Mukherjee, S. (2014). Stability and chaos analysis of a novel swarm dynamics with applications to multi-agent systems. Engineering Applications of Artificial Intelligence, 30, 189–198.CrossRefGoogle Scholar
  25. 25.
    Fang, J., Deng, W., Wu, Y., & Ding, G. (2014). A novel hyperchaotic system and its circuit implementation. Optik, 125(20), 6305–6311.CrossRefGoogle Scholar
  26. 26.
    Feki, M. (2003). An adaptive chaos synchronization scheme applied to secure communication. Chaos, Solitons and Fractals, 18(1), 141–148.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gan, Q., & Liang, Y. (2012). Synchronization of chaotic neural networks with time delay in the leakage term and parametric uncertainties based on sampled-data control. Journal of the Franklin Institute, 349(6), 1955–1971.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Genesio, R., & Tesi, A. (1992). Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica, 28(3), 531–548.CrossRefzbMATHGoogle Scholar
  29. 29.
    Gibson, W. T., & Wilson, W. G. (2013). Individual-based chaos: Extensions of the discrete logistic model. Journal of Theoretical Biology, 339, 84–92.MathSciNetCrossRefGoogle Scholar
  30. 30.
    Henon, M., & Heiles, C. (1964). The applicability of the third integral of motion: Some numerical experiments. The Astrophysical Journal, 69, 73–79.MathSciNetGoogle Scholar
  31. 31.
    Huang, X., Zhao, Z., Wang, Z., & Li, Y. (2012). Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing, 94, 13–21.CrossRefGoogle Scholar
  32. 32.
    Jia, Q. (2007). Hyperchaos generated from the Lorenz chaotic system and its control. Physics Letters A, 366, 217–222.CrossRefzbMATHGoogle Scholar
  33. 33.
    Jiang, G. P., Zheng, W. X., & Chen, G. (2004). Global chaos synchronization with channel time-delay. Chaos, Solitons and Fractals, 20(2), 267–275.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Karthikeyan, R., & Sundarapandian, V. (2014). Hybrid chaos synchronization of four-scroll systems via active control. Journal of Electrical Engineering, 65(2), 97–103.CrossRefGoogle Scholar
  35. 35.
    Kaslik, E., & Sivasundaram, S. (2012). Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks, 32, 245–256.CrossRefzbMATHGoogle Scholar
  36. 36.
    Khalil, H. K. (2001). Nonlinear systems. New Jersey, USA: Prentice Hall.Google Scholar
  37. 37.
    Kyriazis, M. (1991). Applications of chaos theory to the molecular biology of aging. Experimental Gerontology, 26(6), 569–572.CrossRefGoogle Scholar
  38. 38.
    Lang, J. (2015). Color image encryption based on color blend and chaos permutation in the reality-preserving multiple-parameter fractional Fourier transform domain. Optics Communications, 338, 181–192.CrossRefGoogle Scholar
  39. 39.
    Li, D. (2008). A three-scroll chaotic attractor. Physics Letters A, 372(4), 387–393.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Li, N., Zhang, Y., & Nie, Z. (2011). Synchronization for general complex dynamical networks with sampled-data. Neurocomputing, 74(5), 805–811.CrossRefGoogle Scholar
  41. 41.
    Li, N., Pan, W., Yan, L., Luo, B., & Zou, X. (2014). Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers. Communications in Nonlinear Science and Numerical Simulation, 19(6), 1874–1883.CrossRefGoogle Scholar
  42. 42.
    Li, Z., & Chen, G. (2006). Integration of fuzzy logic and chaos theory. Studies in fuzziness and soft computing (Vol. 187). Germany: Springer.Google Scholar
  43. 43.
    Lian, S., & Chen, X. (2011). Traceable content protection based on chaos and neural networks. Applied Soft Computing, 11(7), 4293–4301.CrossRefGoogle Scholar
  44. 44.
    Liu, L., Su, Y. C., & Liu, C. X. (2007). Experimental confirmation of a new reversed butterfly-shaped attractor. Chinese Physics B, 16, 1897–1900.CrossRefGoogle Scholar
  45. 45.
    Lorenz, E. N. (1963). Deterministic periodic flow. Journal of the Atmospheric Sciences, 20(2), 130–141.CrossRefGoogle Scholar
  46. 46.
    Lü, J., & Chen, G. (2002). A new chaotic attractor coined. International Journal of Bifurcation and Chaos, 12(3), 659–661.MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Mondal, S., & Mahanta, C. (2014). Adaptive second order terminal sliding mode controller for robotic manipulators. Journal of the Franklin Institute, 351(4), 2356–2377.MathSciNetCrossRefGoogle Scholar
  48. 48.
    Murali, K., & Lakshmanan, M. (1998). Secure communication using a compound signal from generalized chaotic systems. Physics Letters A, 241(6), 303–310.CrossRefzbMATHGoogle Scholar
  49. 49.
    Nehmzow, U., & Walker, K. (2005). Quantitative description of robot-environment interaction using chaos theory. Robotics and Autonomous Systems, 53(3–4), 177–193.CrossRefGoogle Scholar
  50. 50.
    Pandey, A., Baghel, R. K., & Singh, R. P. (2012). Synchronization analysis of a new autonomous chaotic system with its application in signal masking. IOSR Journal of Electronics and Communication Engineering, 1(5), 16–22.CrossRefGoogle Scholar
  51. 51.
    Pecora, L. M., & Carroll, T. L. (1990). Synchronization in chaotic systems. Physical Review Letters, 64(8), 821–824.MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Pehlivan, I., Moroz, I. M., & Vaidyanathan, S. (2014). Analysis, synchronization and circuit design of a novel butterfly attractor. Journal of Sound and Vibration, 333(20), 5077–5096.CrossRefGoogle Scholar
  53. 53.
    Pham, V. T., Vaidyanathan, S., Volos, C. K., & Jafari, S. (2015). Hidden attractors in a chaotic system with an exponential nonlinear term. European Physical Journal—Special Topics, 224(8), 1507–1517.Google Scholar
  54. 54.
    Pham, V. T., Volos, C. K., Vaidyanathan, S., Le, T. P., & Vu, V. Y. (2015). A memristor-based hyperchaotic system with hidden attractors: Dynamics, synchronization and circuital emulating. Journal of Engineering Science and Technology Review, 8(2), 205–214.Google Scholar
  55. 55.
    Pham, V. T., Jafari, S., Vaidyanathan, S., Volos, C., & Wang, X. (2016). A novel memristive neural network with hidden attractors and its circuitry implementation. Science China Technological Sciences, 59(3), 358–363.Google Scholar
  56. 56.
    Pham, V. T., Vaidyanathan, S., Volos, C., Jafari, S., & Kingni, S. T. (2016). A no-equilibrium hyperchaotic system with a cubic nonlinear term. Optik, 127(6), 3259–3265.Google Scholar
  57. 57.
    Pham, V. T., Vaidyanathan, S., Volos, C. K., Jafari, S., Kuznetsov, N. V., & Hoang, T. M. (2016). A novel memristive time-delay chaotic system without equilibrium points. European Physical Journal: Special Topics, 225(1), 127–136.Google Scholar
  58. 58.
    Qi, G., & Chen, G. (2006). Analysis and circuit implementation of a new 4D chaotic system. Physics Letters A, 352, 386–397.CrossRefzbMATHGoogle Scholar
  59. 59.
    Qu, Z. (2011). Chaos in the genesis and maintenance of cardiac arrhythmias. Progress in Biophysics and Molecular Biology, 105(3), 247–257.CrossRefGoogle Scholar
  60. 60.
    Rasappan, S., & Vaidyanathan, S. (2012). Global chaos synchronization of WINDMI and Coullet chaotic systems by backstepping control. Far East Journal of Mathematical Sciences, 67(2), 265–287.Google Scholar
  61. 61.
    Rasappan, S., & Vaidyanathan, S. (2012). Hybrid synchronization of n-scroll Chua and Lur’e chaotic systems via backstepping control with novel feedback. Archives of Control Sciences, 22(3), 343–365.Google Scholar
  62. 62.
    Rasappan, S., & Vaidyanathan, S. (2012). Synchronization of hyperchaotic Liu system via backstepping control with recursive feedback. Communications in Computer and Information Science, 305, 212–221.Google Scholar
  63. 63.
    Rasappan, S., & Vaidyanathan, S. (2013). Hybrid synchronization of \(n\)-scroll chaotic Chua circuits using adaptive backstepping control design with recursive feedback. Malaysian Journal of Mathematical Sciences, 7(2), 219–246.MathSciNetGoogle Scholar
  64. 64.
    Rasappan, S., & Vaidyanathan, S. (2014). Global chaos synchronization of WINDMI and Coullet chaotic systems using adaptive backstepping control design. Kyungpook Mathematical Journal, 54(1), 293–320.MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Rhouma, R., & Belghith, S. (2011). Cryptoanalysis of a chaos based cryptosystem on DSP. Communications in Nonlinear Science and Numerical Simulation, 16(2), 876–884.MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Rikitake, T. (1958). Oscillations of a system of disk dynamos. Mathematical Proceedings of the Cambridge Philosophical Society, 54(1), 89–105.MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Rössler, O. E. (1976). An equation for continuous chaos. Physics Letters A, 57(5), 397–398.CrossRefGoogle Scholar
  68. 68.
    Sampath, S., Vaidyanathan, S., Volos, C. K., & Pham, V. T. (2015). An eight-term novel four-scroll chaotic system with cubic nonlinearity and its circuit simulation. Journal of Engineering Science and Technology Review, 8(2), 1–6.Google Scholar
  69. 69.
    Sarasu, P., & Sundarapandian, V. (2011). Active controller design for the generalized projective synchronization of four-scroll chaotic systems. International Journal of Systems Signal Control and Engineering Application, 4(2), 26–33.Google Scholar
  70. 70.
    Sarasu, P., & Sundarapandian, V. (2011). The generalized projective synchronization of hyperchaotic Lorenz and hyperchaotic Qi systems via active control. International Journal of Soft Computing, 6(5), 216–223.Google Scholar
  71. 71.
    Sarasu, P., & Sundarapandian, V. (2012). Adaptive controller design for the generalized projective synchronization of 4-scroll systems. International Journal of Systems Signal Control and Engineering Application, 5(2), 21–30.Google Scholar
  72. 72.
    Sarasu, P., & Sundarapandian, V. (2012). Generalized projective synchronization of three-scroll chaotic systems via adaptive control. European Journal of Scientific Research, 72(4), 504–522.Google Scholar
  73. 73.
    Sarasu, P., & Sundarapandian, V. (2012). Generalized projective synchronization of two-scroll systems via adaptive control. International Journal of Soft Computing, 7(4), 146–156.Google Scholar
  74. 74.
    Shahverdiev, E. M., & Shore, K. A. (2009). Impact of modulated multiple optical feedback time delays on laser diode chaos synchronization. Optics Communications, 282(17), 3568–2572.CrossRefGoogle Scholar
  75. 75.
    Shimizu, T., & Morioka, N. (1980). On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model. Physics Letters A, 76(3–4), 201–204.MathSciNetCrossRefGoogle Scholar
  76. 76.
    Smaoui, N., Karouma, A., & Zribi, M. (2013). Adaptive synchronization of hyperchaotic Chen systems with application to secure communication. International Journal of Innovative Computing, Information and Control, 9(3), 1127–1144.Google Scholar
  77. 77.
    Sprott, J. C. (1994). Some simple chaotic flows. Physical Review E, 50(2), 647–650.MathSciNetCrossRefGoogle Scholar
  78. 78.
    Sprott, J. C. (2010). Elegant chaos. World Scientific.Google Scholar
  79. 79.
    Suérez, I. (1999). Mastering chaos in ecology. Ecological Modelling, 117(2–3), 305–314.CrossRefGoogle Scholar
  80. 80.
    Sundarapandian, V. (2010). Output regulation of the Lorenz attractor. Far East Journal of Mathematical Sciences, 42(2), 289–299.MathSciNetzbMATHGoogle Scholar
  81. 81.
    Sundarapandian, V. (2013). Analysis and anti-synchronization of a novel chaotic system via active and adaptive controllers. Journal of Engineering Science and Technology Review, 6(4), 45–52.Google Scholar
  82. 82.
    Sundarapandian, V., & Karthikeyan, R. (2011). Anti-synchronization of hyperchaotic Lorenz and hyperchaotic Chen systems by adaptive control. International Journal of Systems Signal Control and Engineering Application, 4(2), 18–25.Google Scholar
  83. 83.
    Sundarapandian, V., & Karthikeyan, R. (2011). Anti-synchronization of Lü and Pan chaotic systems by adaptive nonlinear control. European Journal of Scientific Research, 64(1), 94–106.Google Scholar
  84. 84.
    Sundarapandian, V., & Karthikeyan, R. (2012). Adaptive anti-synchronization of uncertain Tigan and Li systems. Journal of Engineering and Applied Sciences, 7(1), 45–52.Google Scholar
  85. 85.
    Sundarapandian, V., & Karthikeyan, R. (2012). Hybrid synchronization of hyperchaotic Lorenz and hyperchaotic Chen systems via active control. Journal of Engineering and Applied Sciences, 7(3), 254–264.Google Scholar
  86. 86.
    Sundarapandian, V., & Pehlivan, I. (2012). Analysis, control, synchronization, and circuit design of a novel chaotic system. Mathematical and Computer Modelling, 55(7–8), 1904–1915.MathSciNetCrossRefzbMATHGoogle Scholar
  87. 87.
    Sundarapandian, V., & Sivaperumal, S. (2011). Sliding controller design of hybrid synchronization of four-wing chaotic systems. International Journal of Soft Computing, 6(5), 224–231.CrossRefGoogle Scholar
  88. 88.
    Suresh, R., & Sundarapandian, V. (2013). Global chaos synchronization of a family of \(n\)-scroll hyperchaotic Chua circuits using backstepping control with recursive feedback. Far East Journal of Mathematical Sciences, 73(1), 73–95.zbMATHGoogle Scholar
  89. 89.
    Tacha, O. I., Volos, C. K., Kyprianidis, I. M., Stouboulos, I. N., Vaidyanathan, S., & Pham, V. T. (2016). Analysis, adaptive control and circuit simulation of a novel nonlinear finance system. Applied Mathematics and Computation, 276, 200–217.MathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    Usama, M., Khan, M. K., Alghatbar, K., & Lee, C. (2010). Chaos-based secure satellite imagery cryptosystem. Computers and Mathematics with Applications, 60(2), 326–337.MathSciNetCrossRefzbMATHGoogle Scholar
  91. 91.
    Vaidyanathan, S. (2011). Hybrid chaos synchronization of Liu and Lu systems by active nonlinear control. Communications in Computer and Information Science, 204, 1–10.CrossRefGoogle Scholar
  92. 92.
    Vaidyanathan, S. (2012). Analysis and synchronization of the hyperchaotic Yujun systems via sliding mode control. Advances in Intelligent Systems and Computing, 176, 329–337.Google Scholar
  93. 93.
    Vaidyanathan, S. (2012). Anti-synchronization of Sprott-L and Sprott-M chaotic systems via adaptive control. International Journal of Control Theory and Applications, 5(1), 41–59.Google Scholar
  94. 94.
    Vaidyanathan, S. (2012). Global chaos control of hyperchaotic Liu system via sliding control method. International Journal of Control Theory and Applications, 5(2), 117–123.Google Scholar
  95. 95.
    Vaidyanathan, S. (2012). Output regulation of the Liu chaotic system. Applied Mechanics and Materials, 110–116, 3982–3989.Google Scholar
  96. 96.
    Vaidyanathan, S. (2012). Sliding mode control based global chaos control of Liu-Liu-Liu-Su chaotic system. International Journal of Control Theory and Applications, 5(1), 15–20.Google Scholar
  97. 97.
    Vaidyanathan, S. (2013). A new six-term 3-D chaotic system with an exponential nonlinearity. Far East Journal of Mathematical Sciences, 79(1), 135–143.Google Scholar
  98. 98.
    Vaidyanathan, S. (2013). Analysis and adaptive synchronization of two novel chaotic systems with hyperbolic sinusoidal and cosinusoidal nonlinearity and unknown parameters. Journal of Engineering Science and Technology Review, 6(4), 53–65.Google Scholar
  99. 99.
    Vaidyanathan, S. (2013). Analysis, control and synchronization of hyperchaotic Zhou system via adaptive control. Advances in Intelligent Systems and Computing, 177, 1–10.Google Scholar
  100. 100.
    Vaidyanathan, S. (2014). A new eight-term 3-D polynomial chaotic system with three quadratic nonlinearities. Far East Journal of Mathematical Sciences, 84(2), 219–226.Google Scholar
  101. 101.
    Vaidyanathan, S. (2014). Analysis and adaptive synchronization of eight-term 3-D polynomial chaotic systems with three quadratic nonlinearities. European Physical Journal: Special Topics, 223(8), 1519–1529.Google Scholar
  102. 102.
    Vaidyanathan, S. (2014). Analysis, control and synchronisation of a six-term novel chaotic system with three quadratic nonlinearities. International Journal of Modelling, Identification and Control, 22(1), 41–53.Google Scholar
  103. 103.
    Vaidyanathan, S. (2014). Generalized projective synchronisation of novel 3-D chaotic systems with an exponential non-linearity via active and adaptive control. International Journal of Modelling, Identification and Control, 22(3), 207–217.Google Scholar
  104. 104.
    Vaidyanathan, S. (2014). Global chaos synchronization of identical Li-Wu chaotic systems via sliding mode control. International Journal of Modelling, Identification and Control, 22(2), 170–177.Google Scholar
  105. 105.
    Vaidyanathan, S. (2015). 3-cells cellular neural network (CNN) attractor and its adaptive biological control. International Journal of PharmTech Research, 8(4), 632–640.Google Scholar
  106. 106.
    Vaidyanathan, S. (2015). A 3-D novel highly chaotic system with four quadratic nonlinearities, its adaptive control and anti-synchronization with unknown parameters. Journal of Engineering Science and Technology Review, 8(2), 106–115.Google Scholar
  107. 107.
    Vaidyanathan, S. (2015). A novel chemical chaotic reactor system and its adaptive control. International Journal of ChemTech Research, 8(7), 146–158.Google Scholar
  108. 108.
    Vaidyanathan, S. (2015). A novel chemical chaotic reactor system and its output regulation via integral sliding mode control. International Journal of ChemTech Research, 8(11), 669–683.Google Scholar
  109. 109.
    Vaidyanathan, S. (2015). Adaptive backstepping control of enzymes-substrates system with ferroelectric behaviour in brain waves. International Journal of PharmTech Research, 8(2), 256–261.Google Scholar
  110. 110.
    Vaidyanathan, S. (2015). Adaptive biological control of generalized Lotka-Volterra three-species biological system. International Journal of PharmTech Research, 8(4), 622–631.Google Scholar
  111. 111.
    Vaidyanathan, S. (2015). Adaptive chaotic synchronization of enzymes-substrates system with ferroelectric behaviour in brain waves. International Journal of PharmTech Research, 8(5), 964–973.Google Scholar
  112. 112.
    Vaidyanathan, S. (2015). Adaptive control design for the anti-synchronization of novel 3-D chemical chaotic reactor systems. International Journal of ChemTech Research, 8(11), 654–668.Google Scholar
  113. 113.
    Vaidyanathan, S. (2015). Adaptive control of a chemical chaotic reactor. International Journal of PharmTech Research, 8(3), 377–382.Google Scholar
  114. 114.
    Vaidyanathan, S. (2015). Adaptive synchronization of chemical chaotic reactors. International Journal of ChemTech Research, 8(2), 612–621.Google Scholar
  115. 115.
    Vaidyanathan, S. (2015). Adaptive synchronization of generalized Lotka-Volterra three-species biological systems. International Journal of PharmTech Research, 8(5), 928–937.Google Scholar
  116. 116.
    Vaidyanathan, S. (2015). Adaptive synchronization of novel 3-D chemical chaotic reactor systems. International Journal of ChemTech Research, 8(7), 159–171.Google Scholar
  117. 117.
    Vaidyanathan, S. (2015). Analysis, properties and control of an eight-term 3-D chaotic system with an exponential nonlinearity. International Journal of Modelling, Identification and Control, 23(2), 164–172.Google Scholar
  118. 118.
    Vaidyanathan, S. (2015). Anti-synchronization of Brusselator chemical reaction systems via adaptive control. International Journal of ChemTech Research, 8(6), 759–768.Google Scholar
  119. 119.
    Vaidyanathan, S. (2015). Anti-synchronization of chemical chaotic reactors via adaptive control method. International Journal of ChemTech Research, 8(8), 73–85.Google Scholar
  120. 120.
    Vaidyanathan, S. (2015). Anti-synchronization of Mathieu-Van der Pol chaotic systems via adaptive control method. International Journal of ChemTech Research, 8(11), 638–653.Google Scholar
  121. 121.
    Vaidyanathan, S. (2015). Chaos in neurons and adaptive control of Birkhoff-Shaw strange chaotic attractor. International Journal of PharmTech Research, 8(5), 956–963.Google Scholar
  122. 122.
    Vaidyanathan, S. (2015). Dynamics and control of Brusselator chemical reaction. International Journal of ChemTech Research, 8(6), 740–749.Google Scholar
  123. 123.
    Vaidyanathan, S. (2015). Dynamics and control of Tokamak system with symmetric and magnetically confined plasma. International Journal of ChemTech Research, 8(6), 795–803.Google Scholar
  124. 124.
    Vaidyanathan, S. (2015). Global chaos control of Mathieu-Van der pol system via adaptive control method. International Journal of ChemTech Research, 8(9), 406–417.Google Scholar
  125. 125.
    Vaidyanathan, S. (2015). Global chaos synchronization of chemical chaotic reactors via novel sliding mode control method. International Journal of ChemTech Research, 8(7), 209–221.Google Scholar
  126. 126.
    Vaidyanathan, S. (2015). Global chaos synchronization of Duffing double-well chaotic oscillators via integral sliding mode control. International Journal of ChemTech Research, 8(11), 141–151.Google Scholar
  127. 127.
    Vaidyanathan, S. (2015). Global chaos synchronization of Mathieu-Van der Pol chaotic systems via adaptive control method. International Journal of ChemTech Research, 8(10), 148–162.Google Scholar
  128. 128.
    Vaidyanathan, S. (2015). Global chaos synchronization of novel coupled Van der Pol conservative chaotic systems via adaptive control method. International Journal of ChemTech Research, 8(8), 95–111.Google Scholar
  129. 129.
    Vaidyanathan, S. (2015). Global chaos synchronization of the forced Van der Pol chaotic oscillators via adaptive control method. International Journal of PharmTech Research, 8(6), 156–166.Google Scholar
  130. 130.
    Vaidyanathan, S. (2015). Hyperchaos, qualitative analysis, control and synchronisation of a ten-term 4-D hyperchaotic system with an exponential nonlinearity and three quadratic nonlinearities. International Journal of Modelling, Identification and Control, 23(4), 380–392.Google Scholar
  131. 131.
    Vaidyanathan, S. (2016). A novel 2-D chaotic enzymes-substrates reaction system and its adaptive backstepping control. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 507–528). Germany: Springer.Google Scholar
  132. 132.
    Vaidyanathan, S. (2016). A novel 3-D conservative jerk chaotic system with two quadratic nonlinearities and its adaptive control. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 349–376). Germany: Springer.Google Scholar
  133. 133.
    Vaidyanathan, S. (2016). A novel 3-D jerk chaotic system with three quadratic nonlinearities and its adaptive control. Archives of Control Sciences, 26(1), 19–47.Google Scholar
  134. 134.
    Vaidyanathan, S. (2016). A novel 4-D hyperchaotic thermal convection system and its adaptive control. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 75–100). Germany: Springer.Google Scholar
  135. 135.
    Vaidyanathan, S. (2016). A novel double convecton system, its analysis, adaptive control and synchronization. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 553–579). Germany: Springer.Google Scholar
  136. 136.
    Vaidyanathan, S. (2016). A seven-term novel 3-D jerk chaotic system with two quadratic nonlinearities and its adaptive backstepping control. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 581–607). Germany: Springer.Google Scholar
  137. 137.
    Vaidyanathan, S. (2016). Analysis, adaptive control and synchronization of a novel 3-D chaotic system with a quadratic nonlinearity and two quadratic nonlinearities. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 429–453). Germany: Springer.Google Scholar
  138. 138.
    Vaidyanathan, S. (2016). Analysis, control and synchronization of a novel 4-D highly hyperchaotic system with hidden attractors. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 529–552). Germany: Springer.Google Scholar
  139. 139.
    Vaidyanathan, S. (2016). Anti-synchronization of Duffing double-well chaotic oscillators via integral sliding mode control. International Journal of ChemTech Research, 9(2), 297–304.Google Scholar
  140. 140.
    Vaidyanathan, S. (2016). Dynamic analysis, adaptive control and synchronization of a novel highly chaotic system with four quadratic nonlinearities. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control Studies in fuzziness and soft computing (Vol. 337, pp. 405–428). Germany: Springer.Google Scholar
  141. 141.
    Vaidyanathan, S. (2016). Global chaos synchronization of a novel 3-D chaotic system with two quadratic nonlinearities via active and adaptive control. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 481–506). Germany: Springer.Google Scholar
  142. 142.
    Vaidyanathan, S. (2016). Qualitative analysis and properties of a novel 4-D hyperchaotic system with two quadratic nonlinearities and its adaptive control. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 455–480). Germany: Springer.Google Scholar
  143. 143.
    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). Germany: Springer.Google Scholar
  144. 144.
    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 modelling and control systems design. Studies in computational intelligence (Vol. 581, pp. 19–38). Germany: Springer.Google Scholar
  145. 145.
    Vaidyanathan, S., & Azar, A. T. (2015). Anti-synchronization of identical chaotic systems using sliding mode control and an application to Vaidhyanathan-Madhavan chaotic systems. Studies in Computational Intelligence, 576, 527–547.Google Scholar
  146. 146.
    Vaidyanathan, S., & Azar, A. T. (2015). Hybrid synchronization of identical chaotic systems using sliding mode control and an application to Vaidhyanathan chaotic systems. Studies in Computational Intelligence, 576, 549–569.Google Scholar
  147. 147.
    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 A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 203–224). Germany: Springer.Google Scholar
  148. 148.
    Vaidyanathan, S., & Azar, A. T. (2016). Adaptive backstepping control and synchronization of a novel 3-D jerk system with an exponential nonlinearity. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 249–274). Germany: Springer.Google Scholar
  149. 149.
    Vaidyanathan, S., & Azar, A. T. (2016). Adaptive control and synchronization of Halvorsen circulant chaotic systems. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 225–247). Germany: Springer.Google Scholar
  150. 150.
    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 A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 155–178). Germany: Springer.Google Scholar
  151. 151.
    Vaidyanathan, S., & Azar, A. T. (2016). Generalized projective synchronization of a novel hyperchaotic four-wing system via adaptive control method. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 275–296). Germany: Springer.Google Scholar
  152. 152.
    Vaidyanathan, S., & Azar, A. T. (2016). Qualitative study and adaptive control of a novel 4-D hyperchaotic system with three quadratic nonlinearities. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 179–202). Germany: Springer.Google Scholar
  153. 153.
    Vaidyanathan, S., & Madhavan, K. (2013). Analysis, adaptive control and synchronization of a seven-term novel 3-D chaotic system. International Journal of Control Theory and Applications, 6(2), 121–137.Google Scholar
  154. 154.
    Vaidyanathan, S., & Pakiriswamy, S. (2013). Generalized projective synchronization of six-term Sundarapandian chaotic systems by adaptive control. International Journal of Control Theory and Applications, 6(2), 153–163.Google Scholar
  155. 155.
    Vaidyanathan, S., & Pakiriswamy, S. (2015). A 3-D novel conservative chaotic System and its generalized projective synchronization via adaptive control. Journal of Engineering Science and Technology Review, 8(2), 52–60.Google Scholar
  156. 156.
    Vaidyanathan, S., & Rajagopal, K. (2011). Anti-synchronization of Li and T chaotic systems by active nonlinear control. Communications in Computer and Information Science, 198, 175–184.Google Scholar
  157. 157.
    Vaidyanathan, S., & Rajagopal, K. (2011). Global chaos synchronization of hyperchaotic Pang and Wang systems by active nonlinear control. Communications in Computer and Information Science, 204, 84–93.Google Scholar
  158. 158.
    Vaidyanathan, S., & Rajagopal, K. (2011). Global chaos synchronization of Lü and Pan systems by adaptive nonlinear control. Communications in Computer and Information Science, 205, 193–202.Google Scholar
  159. 159.
    Vaidyanathan, S., & Rajagopal, K. (2012). Global chaos synchronization of hyperchaotic Pang and hyperchaotic Wang systems via adaptive control. International Journal of Soft Computing, 7(1), 28–37.CrossRefzbMATHGoogle Scholar
  160. 160.
    Vaidyanathan, S., & Rasappan, S. (2011). Global chaos synchronization of hyperchaotic Bao and Xu systems by active nonlinear control. Communications in Computer and Information Science, 198, 10–17.CrossRefGoogle Scholar
  161. 161.
    Vaidyanathan, S., & Rasappan, S. (2014). Global chaos synchronization of \(n\)-scroll Chua circuit and Lur’e system using backstepping control design with recursive feedback. Arabian Journal for Science and Engineering, 39(4), 3351–3364.CrossRefGoogle Scholar
  162. 162.
    Vaidyanathan, S., & Sampath, S. (2011). Global chaos synchronization of hyperchaotic Lorenz systems by sliding mode control. Communications in Computer and Information Science, 205, 156–164.CrossRefGoogle Scholar
  163. 163.
    Vaidyanathan, S., & Sampath, S. (2012). Anti-synchronization of four-wing chaotic systems via sliding mode control. International Journal of Automation and Computing, 9(3), 274–279.CrossRefGoogle Scholar
  164. 164.
    Vaidyanathan, S., & Volos, C. (2015). Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system. Archives of Control Sciences, 25(3), 333–353.MathSciNetCrossRefGoogle Scholar
  165. 165.
    Vaidyanathan, S., Volos, C., & Pham, V. T. (2014). Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation. Archives of Control Sciences, 24(4), 409–446.Google Scholar
  166. 166.
    Vaidyanathan, S., Volos, C., Pham, V. T., Madhavan, K., & Idowu, B. A. (2014). Adaptive backstepping control, synchronization and circuit simulation of a 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities. Archives of Control Sciences, 24(3), 375–403.Google Scholar
  167. 167.
    Vaidyanathan, S., Idowu, B. A., & Azar, A. T. (2015). Backstepping controller design for the global chaos synchronization of Sprott’s jerk systems. Studies in Computational Intelligence, 581, 39–58.Google Scholar
  168. 168.
    Vaidyanathan, S., Rajagopal, K., Volos, C. K., Kyprianidis, I. M., & Stouboulos, I. N. (2015). Analysis, adaptive control and synchronization of a seven-term novel 3-D chaotic system with three quadratic nonlinearities and its digital implementation in LabVIEW. Journal of Engineering Science and Technology Review, 8(2), 130–141.Google Scholar
  169. 169.
    Vaidyanathan, S., Pham, V. T., & Volos, C. K. (2015). A 5-D hyperchaotic Rikitake dynamo system with hidden attractors. European Physical Journal: Special Topics, 224(8), 1575–1592.Google Scholar
  170. 170.
    Vaidyanathan, S., Volos, C., Pham, V. T., & Madhavan, K. (2015). Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation. Archives of Control Sciences, 25(1), 5–28.Google Scholar
  171. 171.
    Vaidyanathan, S., Volos, C. K., Kyprianidis, I. M., Stouboulos, I. N., & Pham, V. T. (2015). Analysis, adaptive control and anti-synchronization of a six-term novel jerk chaotic system with two exponential nonlinearities and its circuit simulation. Journal of Engineering Science and Technology Review, 8(2), 24–36.Google Scholar
  172. 172.
    Vaidyanathan, S., Volos, C. K., & Pham, V. T. (2015). Analysis, adaptive control and adaptive synchronization of a nine-term novel 3-D chaotic system with four quadratic nonlinearities and its circuit simulation. Journal of Engineering Science and Technology Review, 8(2), 174–184.Google Scholar
  173. 173.
    Vaidyanathan, S., Volos, C. K., & Pham, V. T. (2015). Global chaos control of a novel nine-term chaotic system via sliding mode control. In A. T. Azar & Q. Zhu (Eds.), Advances and applications in sliding mode control systems. Studies in computational intelligence (Vol. 576, pp. 571–590). Germany: Springer.Google Scholar
  174. 174.
    Volos, C. K., Kyprianidis, I. M., & Stouboulos, I. N. (2013). Experimental investigation on coverage performance of a chaotic autonomous mobile robot. Robotics and Autonomous Systems, 61(12), 1314–1322.CrossRefGoogle Scholar
  175. 175.
    Volos, C. K., Kyprianidis, I. M., Stouboulos, I. N., Tlelo-Cuautle, E., & Vaidyanathan, S. (2015). Memristor: A new concept in synchronization of coupled neuromorphic circuits. Journal of Engineering Science and Technology Review, 8(2), 157–173.Google Scholar
  176. 176.
    Wei, Z., & Yang, Q. (2010). Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci. Applied Mathematics and Computation, 217(1), 422–429.MathSciNetCrossRefzbMATHGoogle Scholar
  177. 177.
    Witte, C. L., & Witte, M. H. (1991). Chaos and predicting varix hemorrhage. Medical Hypotheses, 36(4), 312–317.CrossRefGoogle Scholar
  178. 178.
    Xiao, X., Zhou, L., & Zhang, Z. (2014). Synchronization of chaotic Lur’e systems with quantized sampled-data controller. Communications in Nonlinear Science and Numerical Simulation, 19(6), 2039–2047.MathSciNetCrossRefGoogle Scholar
  179. 179.
    Yuan, G., Zhang, X., & Wang, Z. (2014). Generation and synchronization of feedback-induced chaos in semiconductor ring lasers by injection-locking. Optik—International Journal for Light and Electron Optics, 125(8), 1950–1953.CrossRefGoogle Scholar
  180. 180.
    Zaher, A. A., & Abu-Rezq, A. (2011). On the design of chaos-based secure communication systems. Communications in Nonlinear Systems and Numerical Simulation, 16(9), 3721–3727.MathSciNetCrossRefzbMATHGoogle Scholar
  181. 181.
    Zhang, H., & Zhou, J. (2012). Synchronization of sampled-data coupled harmonic oscillators with control inputs missing. Systems and Control Letters, 61(12), 1277–1285.MathSciNetCrossRefzbMATHGoogle Scholar
  182. 182.
    Zhang, X., Zhao, Z., & Wang, J. (2014). Chaotic image encryption based on circular substitution box and key stream buffer. Signal Processing: Image Communication, 29(8), 902–913.Google Scholar
  183. 183.
    Zhou, W., Xu, Y., Lu, H., & Pan, L. (2008). On dynamics analysis of a new chaotic attractor. Physics Letters A, 372(36), 5773–5777.MathSciNetCrossRefzbMATHGoogle Scholar
  184. 184.
    Zhu, C., Liu, Y., & Guo, Y. (2010). Theoretic and numerical study of a new chaotic system. Intelligent Information Management, 2, 104–109.CrossRefGoogle Scholar
  185. 185.
    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.Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Sundarapandian Vaidyanathan
    • 1
  • Ahmad Taher Azar
    • 2
    • 3
  • Adel Ouannas
    • 4
  1. 1.Research and Development CentreVel Tech UniversityChennaiIndia
  2. 2.Faculty of Computers and InformationBenha UniversityBenhaEgypt
  3. 3.Nanoelectronics Integrated Systems Center (NISC)Nile UniversityCairoEgypt
  4. 4.Laboratory of Mathematics, Informatics and Systems (LAMIS)University of Larbi TebessiTebessaAlgeria

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