Backstepping Controller Design for the Global Chaos Synchronization of Sprott’s Jerk Systems

  • Sundarapandian Vaidyanathan
  • Babatunde A. Idowu
  • Ahmad Taher Azar
Chapter
First Online:
Part of the Studies in Computational Intelligence book series (SCI, volume 581)

Abstract

This research work investigates the global chaos synchronization of Sprott’s jerk chaotic system using backstepping control method. Sprott’s jerk system (1997) is algebraically the simplest dissipative chaotic system consisting of five terms and a quadratic nonlinearity. Sprott’s chaotic system involves only five terms and one quadratic nonlinearity, while Rössler’s chaotic system (1976) involves seven terms and one quadratic nonlinearity. This work first details the properties of the Sprott’s jerk chaotic system. The phase portraits of the Sprott’s jerk system are described. The Lyapunov exponents of the Sprott’s jerk system are obtained as L 1 = 0.0525, L 2 = 0 and L 3 = −2.0727. The Lyapunov dimension of the Sprott’s jerk system is obtained as D L  = 2.0253. Next, an active backstepping controller is designed for the global chaos synchronization of identical Sprott’s jerk systems with known parameters. The backstepping control method is a recursive procedure that links the choice of a Lyapunov function with the design of a controller and guarantees global asymptotic stability of strict-feedback chaotic systems. Finally, an adaptive backstepping controller is designed for the global chaos synchronization of identical Sprott’s jerk systems with unknown parameters. MATLAB simulations are provided to validate and demonstrate the effectiveness of the proposed active and adaptive chaos synchronization schemes for the Sprott’s jerk systems.

Keywords

Chaos Backstepping control Sprott’s jerk system Active control Adaptive control 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Alligood, K. T., Sauer, T., and Yorke, J. A:. Chaos: An Introduction to Dynamical Systems. Springer, Berlin. (1997)Google Scholar
  2. 2.
    Arneodo, A., Coullet, P., Tresser, C.: Possible new strange attractors with spiral structure. Common. Math. Phys. 79(4), 573–576 (1981)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Cai, G., Tan, Z.: Chaos synchronization of a new chaotic system via nonlinear control. J. Uncertain Syst. 1(3), 235–240 (2007)Google Scholar
  4. 4.
    Carroll, T.L., Pecora, L.M.: Synchronizing chaotic circuits. IEEE Trans. Circ. Syst. 38(4), 453–456 (1991)CrossRefGoogle Scholar
  5. 5.
    Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurcat. Chaos 9(7), 1465–1466 (1999)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Chen, H.K., Lee, C.I.: Anti-control of chaos in rigid body motion. Chaos Solitons Fractals 21(4), 957–965 (2004)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chen, W.-H., Wei, D., Lu, X.: Global exponential synchronization of nonlinear time-delay Lure systems via delayed impulsive control. Commun. Nonlinear Sci. Numer. Simul. 19(9), 3298–3312 (2014)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Das, S., Goswami, D., Chatterjee, S., Mukherjee, S.: Stability and chaos analysis of a novel swarm dynamics with applications to multi-agent systems. Eng. Appl. Artif. Intell. 30, 189–198 (2014)CrossRefGoogle Scholar
  9. 9.
    Feki, M.: An adaptive chaos synchronization scheme applied to secure communication. Chaos Solitons Fractals 18(1), 141–148 (2003)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Gan, Q., Liang, Y.: Synchronization of chaotic neural networks with time delay in the leakage term and parametric uncertainties based on sampled-data control. J. Franklin Inst. 349(6), 1955–1971 (2012)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Gaspard, P.: Microscopic chaos and chemical reactions. Phys. A 263(1–4), 315–328 (1999)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Gibson, W.T., Wilson, W.G.: Individual-based chaos: Extensions of the discrete logistic model. J. Theor. Biol. 339, 84–92 (2013)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Guégan, D.: Chaos in economics and finance. Ann. Rev. Control 33(1), 89–93 (2009)CrossRefGoogle Scholar
  14. 14.
    Huang, J.: Adaptive synchronization between different hyperchaotic systems with fully uncertain parameters. Phys. Lett. A 372(27–28), 4799–4804 (2008)CrossRefMATHGoogle Scholar
  15. 15.
    Huang, X., Zhao, Z., Wang, Z., Li, Y.: Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing 94, 13–21 (2012)CrossRefGoogle Scholar
  16. 16.
    Jiang, G.-P., Zheng, W.X., Chen, G.: Global chaos synchronization with channel time-delay. Chaos Solitons Fractals 20(2), 267–275 (2004)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012)CrossRefMATHGoogle Scholar
  18. 18.
    Kengne, J., Chedjou, J.C., Kenne, G., Kyamakya, K.: Dynamical properties and chaos synchronization of improved Colpitts oscillators. Commun. Nonlinear Sci. Numer. Simul. 17(7), 2914–2923 (2012)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Khalil, H. K.:. Nonlinear Systems. Prentice Hall, Englewood Cliffs (2001)Google Scholar
  20. 20.
    Kyriazis, M.: Applications of chaos theory to the molecular biology of aging. Exp. Gerontol. 26(6), 569–572 (1991)CrossRefGoogle Scholar
  21. 21.
    Li, D.: A three-scroll chaotic attractor. Phys. Lett. A 372(4), 387–393 (2008)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Li, N., Pan, W., Yan, L., Luo, B., Zou, X.: Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers. Commun. Nonlinear Sci. Numer. Simul. 19(6), 1874–1883 (2014)CrossRefGoogle Scholar
  23. 23.
    Li, N., Zhang, Y., Nie, Z.: Synchronization for general complex dynamical networks with sampled-data. Neurocomputing 74(5), 805–811 (2011)CrossRefGoogle Scholar
  24. 24.
    Lin, W.: Adaptive chaos control and synchronization in only locally Lipschitz systems. Phys. Lett. A 372(18), 3195–3200 (2008)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Liu, C., Liu, T., Liu, L., Liu, K.: A new chaotic attractor. Chaos Solitions and Fractals 22(5), 1031–1038 (2004)CrossRefMATHGoogle Scholar
  26. 26.
    Liu, L., Zhang, C., Guo, Z.A.: Synchronization between two different chaotic systems with nonlinear feedback control. Chin. Phys. 16(6), 1603–1607 (2007)CrossRefGoogle Scholar
  27. 27.
    Lorenz, E.N.: Deterministic periodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)CrossRefGoogle Scholar
  28. 28.
    Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurcat. Chaos 12(3), 659–661 (2002)CrossRefMATHGoogle Scholar
  29. 29.
    Murali, K., Lakshmanan, M.: Secure communication using a compound signal from generalized chaotic systems. Phys. Lett. A 241(6), 303–310 (1998)CrossRefMATHGoogle Scholar
  30. 30.
    Nehmzow, U., Walker, K.: Quantitative description of robot environment interaction using chaos theory. Robot. Autono. Syst. 53(3–4), 177–193 (2005)CrossRefGoogle Scholar
  31. 31.
    Njah, A.N., Ojo, K.S., Adebayo, G.A., Obawole, A.O.: Generalized control and synchronization of chaos in RCL-shunted Josephson junction using backstepping design. Phys. C 470(13–14), 558–564 (2010)CrossRefGoogle Scholar
  32. 32.
    Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Petrov, V., Gaspar, V., Masere, J., Showalter, K.: Controlling chaos in Belousov-Zhabotinsky reaction. Nature 361, 240–243 (1993)CrossRefGoogle Scholar
  34. 34.
    Qu, Z.: Chaos in the genesis and maintenance of cardiac arrhythmias. Prog. Biophys. Mol. Biol. 105(3), 247–257 (2011)CrossRefGoogle Scholar
  35. 35.
    Rafikov, M., Balthazar, J.M.: On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commun. Nonlinear Sci. Numer. Simul. 13(7), 1246–1255 (2007)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Rasappan, S., Vaidyanathan, S.: Global chaos synchronization of WINDMI and Coullet chaotic systems by backstepping control. Far East J. Math. Sci. 67(2), 265–287 (2012)MATHMathSciNetGoogle Scholar
  37. 37.
    Rhouma, R., Belghith, S.: Cryptoanalysis of a chaos based cryptosystem on DSP. Commun. Nonlinear Sci. Numer. Simul. 16(2), 876–884 (2011)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Rössler, O.E.: An equation for continuous chaos. Phys. Lett. 57A(5), 397–398 (1976)CrossRefGoogle Scholar
  39. 39.
    Sarasu, P., Sundarapandian, V.: Adaptive controller design for the generalized projective synchronization of 4-scroll systems. Int. J. Syst. Sig. Control Eng. Appl. 5(2), 21–30 (2012)Google Scholar
  40. 40.
    Sarasu, P., Sundarapandian, V.: Generalized projective synchronization of two-scroll systems via adaptive control. Int. J. Soft Comput. 7(4), 146–156 (2012)CrossRefGoogle Scholar
  41. 41.
    Sarasu, P., Sundarapandian, V.: Generalized projective synchronization of two-scroll systems via adaptive control. Eur. J. Sci. Res. 72(4), 504–522 (2012)Google Scholar
  42. 42.
    Shahverdiev, E.M., Bayramov, P.A., Shore, K.A.: Cascaded and adaptive chaos synchronization in multiple time-delay laser systems. Chaos Solitons Fractals 42(1), 180–186 (2009)CrossRefGoogle Scholar
  43. 43.
    Shahverdiev, E.M., Shore, K.A.: Impact of modulated multiple optical feedback time delays on laser diode chaos synchronization. Opt. Commun. 282(17), 3568–3572 (2009)CrossRefGoogle Scholar
  44. 44.
    Sharma, A., Patidar, V., Purohit, G., Sud, K.K.: Effects on the bifurcation and chaos in forced Duffing oscillator due to nonlinear damping. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2254–2269 (2012)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50(2), 647–650 (1994)CrossRefMathSciNetGoogle Scholar
  46. 46.
    Sprott, J.C.: Simplest dissipative chaotic flow. Phys. Lett. A 228(4–5), 271–274 (1997)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Sprott, J.C.: Competition with evolution in ecology and finance. Phys. Lett. A 325(5–6), 329–333 (2004)CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    Suérez, I.: Mastering chaos in ecology. Ecol. Model. 117(2–3), 305–314 (1999)CrossRefGoogle Scholar
  49. 49.
    Sundarapandian, V.: Output regulation of the Lorenz attractor. Far East J. Math. Sci. 42(2), 289–299 (2010)MATHMathSciNetGoogle Scholar
  50. 50.
    Sundarapandian, V., Pehlivan, I.: Analysis, control, synchronization, and circuit design of a novel chaotic system. Math. Comput. Model. 55(7–8), 1904–1915 (2012)CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    Suresh, R., Sundarapandian, V.: Global chaos synchronizatoin of a family of n-scroll hyperchaotic Chua circuits using backstepping control with recursive feedback. Far East J. Math. Sci. 73(1), 73–95 (2013)MATHGoogle Scholar
  52. 52.
    Tigan, G., Opris, D.: Analysis of a 3D chaotic system. Chaos Solitons Fractals 36, 1315–1319 (2008)CrossRefMATHMathSciNetGoogle Scholar
  53. 53.
    Tu, J., He, H., Xiong, P.: Adaptive backstepping synchronization between chaotic systems with unknown Lipschitz constant. Appl. Math. Comput. 236, 10–18 (2014)CrossRefMathSciNetGoogle Scholar
  54. 54.
    Ucar, A., Lonngren, K.E., Bai, E.W.: Chaos synchronization in RCL-shunted Josephson junction via active control. Chaos, Solitons Fractals 31(1), 105–111 (2007)CrossRefGoogle Scholar
  55. 55.
    Usama, M., Khan, M.K., Alghatbar, K., Lee, C.: Chaos-based secure satellite imagery cryptosystem. Comput. Math Appl. 60(2), 326–337 (2010)CrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    Vaidyanathan, S.: Adaptive backstepping controller and synchronizer design for Arneodo chaotic system with unknown parameters. Int. J. Comp. Sci. Inf. Technol. 4(6), 145–159 (2012)Google Scholar
  57. 57.
    Vaidyanathan, S.: Output regulation of the Liu chaotic system. Appl. Mech. Mater. 110–116, 3982–3989 (2012)Google Scholar
  58. 58.
    Vaidyanathan, S.: A new six-term 3-D chaotic system with an exponential nonlinearity. Far East J. Math. Sci. 79(1), 135–143 (2013)MATHGoogle Scholar
  59. 59.
    Vaidyanathan, S.: Analysis and adaptive synchronization of two novel chaotic systems with hyperbolic sinusoidal and cosinusoidal nonlinearity and unknown parameters. J. Eng. Sci. Technol. Rev. 6(4), 53–65 (2013)MathSciNetGoogle Scholar
  60. 60.
    Vaidyanathan, S.: A new eight-term 3-D polynomial chaotic system with three quadratic nonlinearities. Far East J. Math. Sci. 84(2), 219–226 (2014)MATHMathSciNetGoogle Scholar
  61. 61.
    Vaidyanathan, S., Rajagopal, K.: Global chaos synchronization of four-scroll chaotic systems by active nonlinear control. Int. J. Control Theor. Appl. 4(1), 73–83 (2011)Google Scholar
  62. 62.
    Volos, C.K., Kyprianidis, I.M., Stouboulos, I.N.: Experimental investigation on coverage performance of a chaotic autonomous mobile robot. Robot. Auton. Syst. 61(12), 1314–1322 (2013)CrossRefGoogle Scholar
  63. 63.
    Wang, F., Liu, C.: A new criterion for chaos and hyperchaos synchronization using linear feedback control. Phys. Lett. A 360(2), 274–278 (2006)CrossRefMATHGoogle Scholar
  64. 64.
    Witte, C.L., Witte, M.H.: Chaos and predicting varix hemorrhage. Med. Hypotheses 36(4), 312–317 (1991)CrossRefMathSciNetGoogle Scholar
  65. 65.
    Wu, X., Guan, Z.-H., Wu, Z.: Adaptive synchronization between two different hyperchaotic systems. Nonlinear Anal. Theor. Methods Appl. 68(5), 1346–1351 (2008)CrossRefMATHMathSciNetGoogle Scholar
  66. 66.
    Xiao, X., Zhou, L., Zhang, Z.: Synchronization of chaotic Lure systems with quantized sampled-data controller. Commun. Nonlinear Sci. Numer. Simul. 19(6), 2039–2047 (2014)CrossRefMathSciNetGoogle Scholar
  67. 67.
    Yuan, G., Zhang, X., Wang, Z.: Generation and synchronization of feedback-induced chaos in semiconductor ring lasers by injection-locking. Optik—Int. J. Light Electron Opt. 125(8), 1950–1953 (2014)CrossRefGoogle Scholar
  68. 68.
    Zaher, A.A., Abu-Rezq, A.: On the design of chaos-based secure communication systems. Commun. Nonlinear Syst. Numer. Simul. 16(9), 3721–3727 (2011)CrossRefMATHMathSciNetGoogle Scholar
  69. 69.
    Zhang, H., Zhou, J.: Synchronization of sampled-data coupled harmonic oscillators with control inputs missing. Syst. Control Lett. 61(12), 1277–1285 (2012)CrossRefMATHGoogle Scholar
  70. 70.
    Zhang, J., Li, C., Zhang, H., Yu, J.: Chaos synchronization using single variable feedback based on backstepping method. Chaos Solitons Fractals 21(5), 1183–1193 (2004)CrossRefMATHMathSciNetGoogle Scholar
  71. 71.
    Zhou, W., Xu, Y., Lu, H., Pan, L.: On dynamics analysis of a new chaotic attractor. Phys. Lett. A 372(36), 5773–5777 (2008)CrossRefMATHMathSciNetGoogle Scholar
  72. 72.
    Zhu, C., Liu, Y., Guo, Y.: Theoretic and numerical study of a new chaotic system. Intell. Inf. Manage. 2, 104–109 (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sundarapandian Vaidyanathan
    • 1
  • Babatunde A. Idowu
    • 2
  • Ahmad Taher Azar
    • 3
  1. 1.Research and Development CentreVel Tech UniversityChennaiIndia
  2. 2.Department of PhysicsLagos State UniversityLagosNigeria
  3. 3.Faculty of Computers and InformationBenha UniversityBanhaEgypt

Personalised recommendations