An Eight-Term 3-D Novel Chaotic System with Three Quadratic Nonlinearities, Its Adaptive Feedback Control and Synchronization

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

Abstract

This research work describes an eight-term 3-D novel polynomial chaotic system consisting of three quadratic nonlinearities. First, this work presents the 3-D dynamics of the novel chaotic system and depicts the phase portraits of the system. Next, the qualitative properties of the novel chaotic system are discussed in detail. The novel chaotic system has four equilibrium points. We show that two equilibrium points are saddle points and the other equilibrium points are saddle-foci. The Lyapunov exponents of the novel chaotic system are obtained as \(L_1 = 0.4715, L_2 = 0\) and \(L_3 = -2.4728\). The Lyapunov dimension of the novel chaotic system is obtained as \(D_{L} = 2.1907\). Next, we present the design of adaptive feedback controller for globally stabilizing the trajectories of the novel chaotic system with unknown parameters. Furthermore, we present the design of adaptive feedback controller for achieving complete synchronization of the identical novel chaotic systems with unknown parameters. The main adaptive control results are proved using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this research work for eight-term 3-D novel chaotic system.

Keywords

Chaos Chaotic 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.MathSciNetMATHCrossRefGoogle 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. (2016a). 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. (2016b). 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.MATHCrossRefGoogle Scholar
  19. 19.
    Chen, G., & Ueta, T. (1999). Yet another chaotic attractor. International Journal of Bifurcation and Chaos, 9(7), 1465–1466.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    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.Google Scholar
  21. 21.
    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.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    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
  23. 23.
    Feki, M. (2003). An adaptive chaos synchronization scheme applied to secure communication. Chaos, Solitons and Fractals, 18(1), 141–148.MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    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.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Genesio, R., & Tesi, A. (1992). Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica, 28(3), 531–548.MATHCrossRefGoogle Scholar
  26. 26.
    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
  27. 27.
    Henon, M., & Heiles, C. (1964). The applicability of the third integral of motion: Some numerical experiments. The Astrophysical Journal, 69, 73–79.MathSciNetGoogle Scholar
  28. 28.
    Huang, X., Zhao, Z., Wang, Z., & Li, Y. (2012). Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing, 94, 13–21.CrossRefGoogle Scholar
  29. 29.
    Jiang, G. P., Zheng, W. X., & Chen, G. (2004). Global chaos synchronization with channel time-delay. Chaos, Solitons & Fractals, 20(2), 267–275.MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    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
  31. 31.
    Kaslik, E., & Sivasundaram, S. (2012). Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks, 32, 245–256.MATHCrossRefGoogle Scholar
  32. 32.
    Khalil, H. K. (2001). Nonlinear systems. New Jersey, USA: Prentice Hall.Google Scholar
  33. 33.
    Kyriazis, M. (1991). Applications of chaos theory to the molecular biology of aging. Experimental Gerontology, 26(6), 569–572.CrossRefGoogle Scholar
  34. 34.
    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
  35. 35.
    Li, D. (2008). A three-scroll chaotic attractor. Physics Letters A, 372(4), 387–393.MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Li, N., Zhang, Y., & Nie, Z. (2011). Synchronization for general complex dynamical networks with sampled-data. Neurocomputing, 74(5), 805–811.CrossRefGoogle Scholar
  37. 37.
    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
  38. 38.
    Li, Z., & Chen, G. (2006). Integration of fuzzy logic and chaos theory. Studies in fuzziness and soft computing (Vol. 187). Germany: Springer.Google Scholar
  39. 39.
    Lian, S., & Chen, X. (2011). Traceable content protection based on chaos and neural networks. Applied Soft Computing, 11(7), 4293–4301.CrossRefGoogle Scholar
  40. 40.
    Liu, C., Liu, T., Liu, L., & Liu, K. (2004). A new chaotic attractor. Chaos, Solitions and Fractals, 22(5), 1031–1038.MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Lorenz, E. N. (1963). Deterministic periodic flow. Journal of the Atmospheric Sciences, 20(2), 130–141.CrossRefGoogle Scholar
  42. 42.
    Lü, J., & Chen, G. (2002). A new chaotic attractor coined. International Journal of Bifurcation and Chaos, 12(3), 659–661.MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    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
  44. 44.
    Murali, K., & Lakshmanan, M. (1998). Secure communication using a compound signal from generalized chaotic systems. Physics Letters A, 241(6), 303–310.MATHCrossRefGoogle Scholar
  45. 45.
    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
  46. 46.
    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
  47. 47.
    Pecora, L. M., & Carroll, T. L. (1990). Synchronization in chaotic systems. Physical Review Letters, 64(8), 821–824.MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    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
  49. 49.
    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
  50. 50.
    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
  51. 51.
    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
  52. 52.
    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
  53. 53.
    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
  54. 54.
    Qi, G., & Chen, G. (2006). Analysis and circuit implementation of a new 4D chaotic system. Physics Letters A, 352, 386–397.MATHCrossRefGoogle Scholar
  55. 55.
    Qu, Z. (2011). Chaos in the genesis and maintenance of cardiac arrhythmias. Progress in Biophysics and Molecular Biology, 105(3), 247–257.CrossRefGoogle Scholar
  56. 56.
    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
  57. 57.
    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
  58. 58.
    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
  59. 59.
    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
  60. 60.
    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.MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Rhouma, R., & Belghith, S. (2011). Cryptoanalysis of a chaos based cryptosystem on DSP. Communications in Nonlinear Science and Numerical Simulation, 16(2), 876–884.MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Rikitake, T. (1958). Oscillations of a system of disk dynamos. Mathematical Proceedings of the Cambridge Philosophical Society, 54(1), 89–105.MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Rössler, O. E. (1976). An equation for continuous chaos. Physics Letters A, 57(5), 397–398.CrossRefGoogle Scholar
  64. 64.
    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
  65. 65.
    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
  66. 66.
    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
  67. 67.
    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
  68. 68.
    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
  69. 69.
    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
  70. 70.
    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
  71. 71.
    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
  72. 72.
    Sprott, J. C. (1994). Some simple chaotic flows. Physical Review E, 50(2), 647–650.MathSciNetCrossRefGoogle Scholar
  73. 73.
    Sprott, J. C. (2010). Elegant chaos. World Scientific.Google Scholar
  74. 74.
    Suérez, I. (1999). Mastering chaos in ecology. Ecological Modelling, 117(2–3), 305–314.CrossRefGoogle Scholar
  75. 75.
    Sundarapandian, V. (2010). Output regulation of the Lorenz attractor. Far East Journal of Mathematical Sciences, 42(2), 289–299.MathSciNetMATHGoogle Scholar
  76. 76.
    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
  77. 77.
    Sundarapandian, V., & Karthikeyan, R. (2011). Anti-synchronization of hyperchaotic Lorenz and hyperchaotic Chen systems by adaptive control. International Journal of Systmes Signal Control and Engineering Application, 4(2), 18–25.Google Scholar
  78. 78.
    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
  79. 79.
    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
  80. 80.
    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
  81. 81.
    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.MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    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
  83. 83.
    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.MATHGoogle Scholar
  84. 84.
    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.MathSciNetMATHCrossRefGoogle Scholar
  85. 85.
    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.MathSciNetMATHCrossRefGoogle Scholar
  86. 86.
    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
  87. 87.
    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
  88. 88.
    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
  89. 89.
    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
  90. 90.
    Vaidyanathan, S. (2012). Output regulation of the Liu chaotic system. Applied Mechanics and Materials, 110–116, 3982–3989.Google Scholar
  91. 91.
    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
  92. 92.
    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
  93. 93.
    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
  94. 94.
    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
  95. 95.
    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.MathSciNetMATHGoogle Scholar
  96. 96.
    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.CrossRefGoogle Scholar
  97. 97.
    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.CrossRefGoogle Scholar
  98. 98.
    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.CrossRefGoogle Scholar
  99. 99.
    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.MathSciNetCrossRefGoogle Scholar
  100. 100.
    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
  101. 101.
    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.MathSciNetGoogle Scholar
  102. 102.
    Vaidyanathan, S. (2015). A novel chemical chaotic reactor system and its adaptive control. International Journal of ChemTech Research, 8(7), 146–158.MathSciNetGoogle Scholar
  103. 103.
    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
  104. 104.
    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.MathSciNetGoogle Scholar
  105. 105.
    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
  106. 106.
    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
  107. 107.
    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
  108. 108.
    Vaidyanathan, S. (2015). Adaptive control of a chemical chaotic reactor. International Journal of PharmTech Research, 8(3), 377–382.MathSciNetGoogle Scholar
  109. 109.
    Vaidyanathan, S. (2015). Adaptive synchronization of chemical chaotic reactors. International Journal of ChemTech Research, 8(2), 612–621.MathSciNetGoogle Scholar
  110. 110.
    Vaidyanathan, S. (2015). Adaptive synchronization of generalized Lotka-Volterra three-species biological systems. International Journal of PharmTech Research, 8(5), 928–937.Google Scholar
  111. 111.
    Vaidyanathan, S. (2015). Adaptive synchronization of novel 3-D chemical chaotic reactor systems. International Journal of ChemTech Research, 8(7), 159–171.MathSciNetGoogle Scholar
  112. 112.
    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.MathSciNetCrossRefGoogle Scholar
  113. 113.
    Vaidyanathan, S. (2015). Anti-synchronization of Brusselator chemical reaction systems via adaptive control. International Journal of ChemTech Research, 8(6), 759–768.Google Scholar
  114. 114.
    Vaidyanathan, S. (2015). Anti-synchronization of chemical chaotic reactors via adaptive control method. International Journal of ChemTech Research, 8(8), 73–85.Google Scholar
  115. 115.
    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
  116. 116.
    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
  117. 117.
    Vaidyanathan, S. (2015). Dynamics and control of Brusselator chemical reaction. International Journal of ChemTech Research, 8(6), 740–749.Google Scholar
  118. 118.
    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
  119. 119.
    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.MathSciNetGoogle Scholar
  120. 120.
    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.MathSciNetGoogle Scholar
  121. 121.
    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
  122. 122.
    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
  123. 123.
    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
  124. 124.
    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
  125. 125.
    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
  126. 126.
    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
  127. 127.
    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
  128. 128.
    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.MathSciNetCrossRefGoogle Scholar
  129. 129.
    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
  130. 130.
    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
  131. 131.
    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
  132. 132.
    Vaidyanathan, S. (2016). Analysis, adaptive control and synchronization of a novel 3-D chaotic system with a quartic 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
  133. 133.
    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
  134. 134.
    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
  135. 135.
    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
  136. 136.
    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
  137. 137.
    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
  138. 138.
    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
  139. 139.
    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
  140. 140.
    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
  141. 141.
    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
  142. 142.
    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
  143. 143.
    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
  144. 144.
    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
  145. 145.
    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
  146. 146.
    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
  147. 147.
    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
  148. 148.
    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
  149. 149.
    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
  150. 150.
    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
  151. 151.
    Vaidyanathan, S., & Rajagopal, K. (2011a). Anti-synchronization of Li and T chaotic systems by active nonlinear control. Communications in Computer and Information Science, 198, 175–184.CrossRefGoogle Scholar
  152. 152.
    Vaidyanathan, S., & Rajagopal, K. (2011b). Global chaos synchronization of hyperchaotic Pang and Wang systems by active nonlinear control. Communications in Computer and Information Science, 204, 84–93.CrossRefGoogle Scholar
  153. 153.
    Vaidyanathan, S., & Rajagopal, K. (2011c). Global chaos synchronization of Lü and Pan systems by adaptive nonlinear control. Communications in Computer and Information Science, 205, 193–202.CrossRefGoogle Scholar
  154. 154.
    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.MATHCrossRefGoogle Scholar
  155. 155.
    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
  156. 156.
    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
  157. 157.
    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
  158. 158.
    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
  159. 159.
    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
  160. 160.
    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.MathSciNetMATHGoogle Scholar
  161. 161.
    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.MathSciNetMATHCrossRefGoogle Scholar
  162. 162.
    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
  163. 163.
    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
  164. 164.
    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.MathSciNetCrossRefGoogle Scholar
  165. 165.
    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
  166. 166.
    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
  167. 167.
    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
  168. 168.
    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.CrossRefGoogle Scholar
  169. 169.
    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
  170. 170.
    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
  171. 171.
    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.MathSciNetMATHCrossRefGoogle Scholar
  172. 172.
    Witte, C. L., & Witte, M. H. (1991). Chaos and predicting varix hemorrhage. Medical Hypotheses, 36(4), 312–317.CrossRefGoogle Scholar
  173. 173.
    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.Google Scholar
  174. 174.
    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
  175. 175.
    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.MathSciNetMATHCrossRefGoogle Scholar
  176. 176.
    Zhang, H., & Zhou, J. (2012). Synchronization of sampled-data coupled harmonic oscillators with control inputs missing. Systems & Control Letters, 61(12), 1277–1285.MathSciNetMATHCrossRefGoogle Scholar
  177. 177.
    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
  178. 178.
    Zhou, W., Xu, Y., Lu, H., & Pan, L. (2008). On dynamics analysis of a new chaotic attractor. Physics Letters A, 372(36), 5773–5777.MathSciNetMATHCrossRefGoogle Scholar
  179. 179.
    Zhu, C., Liu, Y., & Guo, Y. (2010). Theoretic and numerical study of a new chaotic system. Intelligent Information Management, 2, 104–109.CrossRefGoogle Scholar
  180. 180.
    Zhu, Q., & Azar, A. T. (2015). Complex system modelling and control through intelligent soft computations. Studies in fuzzines 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 UniversityAvadi, ChennaiIndia
  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 TebessiTébessaAlgeria

Personalised recommendations