A more chaotic and easily hardware implementable new 3-D chaotic system in comparison with 50 reported systems

Original Paper

Abstract

This paper attempts to construct a new 3-D chaotic system which is easily hardware realisable and fulfil the requirement of a real-life application. The proposed system is relatively more chaotic (based on the first Lyapunov exponent) and has larger bandwidth than 50 available chaotic systems. Lyapunov spectrum and bifurcation diagram of the system reveal that it has chaotic behaviour for a wider range of its parameters. Such characteristic is helpful for an easy hardware realisation of the system. It is to be noted that the reported systems with hidden attractors are not considered here for the comparison. The proposed system has more complexity and disorder due to several unique properties like asymmetry to principle coordinates, dissimilar and asymmetrical equilibria, and non-uniform contraction and expansion of volume in phase space. The proposed system also exhibits asymmetric pairs of coexisting attractors during its operation in two modes. The new system has different routes to chaos including crisis, an inverse crisis, period-doubling and reverse period-doubling routes to chaos with the variation of parameters. MATLAB simulation results confirm the claims, and the results of hardware circuit realisation validate the simulation results. An application of the new system is shown by masking and retrieving an information signal. It is also shown that the proposed system is better than a well-known Lorenz chaotic system for this application. A system with the above unique properties is rare in the literature.

Keywords

New chaotic system: more chaotic Wider spectrum Hardware implementation Crisis route to chaos Inverse crisis route to chaos Communication breaking 

References

  1. 1.
    Rajagopal, K., Karthikeyan, A., Srinivasan, A.K.: FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization. Nonlinear Dyn. 87(4), 2281–2304 (2017)CrossRefGoogle Scholar
  2. 2.
    Wang, Z., Akgul, A., Pham, V.T., Jafari, S.: Chaos-based application of a novel no-equilibrium chaotic system with coexisting attractors. Nonlinear Dyn. 89(3), 1–11 (2017)Google Scholar
  3. 3.
    Pham, V.-T., Volos, C., Kingni, S.T., Kapitaniak, T., Jafari, S.: Bistable hidden attractors in a novel chaotic system with hyperbolic sine equilibrium. Circuits Syst. Signal Process. 37(3), 1028–1043 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Aram, Z., Jafari, S., Ma, J., Sprott, J.C., Zendehrouh, S., Pham, V.T.: Using chaotic artificial neural networks to model memory in the brain. Commun. Nonlinear Sci. Numer. Simul. 44(1), 1–14 (2017)MathSciNetGoogle Scholar
  5. 5.
    Andrievskii, A.L., Fradkov, B.R.: Control of chaos: methods and applications. II. Applications. Autom. Remote Control 65(4), 505–533 (2004)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Xiong, L., Lu, Y.-J., Zhang, Y.-F., Zhang, X.-G., Gupta, P.: Design and hardware implementation of a new chaotic secure communication technique. PLoS ONE 11(8), 1–19 (2016)Google Scholar
  7. 7.
    Teh, J.S., Samsudin, A., Al-Mazrooie, M., Akhavan, A.: GPUs and chaos: a new true random number generator. Nonlinear Dyn. 82(4), 1913–1922 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Wang, Y., Liu, Z., Ma, J., He, H.: A pseudorandom number generator based on piecewise logistic map. Nonlinear Dyn. 83(4), 2373–2391 (2016)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Xue, Q., Leung, H., Wang, R., Liu, B., Huang, L., Guo, S.: The chaotic dynamics of drilling. Nonlinear Dyn. 83(4), 2003–2018 (2016)CrossRefGoogle Scholar
  10. 10.
    Qi, G., Chen, G., Zhang, Y.: On a new asymmetric chaotic system. Chaos Solitons Fractals 37(2), 409–423 (2008)CrossRefGoogle Scholar
  11. 11.
    Serajian, R.: Parameters changing influence with different lateral stiffnesses on nonlinear analysis of hunting behavior of a bogie. J. Meas. Eng. 1(4), 196–206 (2013)Google Scholar
  12. 12.
    Younnesian, D., Jafari, A.A., Serajian, R.: Effects of bogie and body inertia on the nonlinear wheel-set hunting recognized by the Hopf bifurcation theory. Int. J. Autom. Eng. 1(3), 186–196 (2011)Google Scholar
  13. 13.
    Mohammadi, S., Serajian, R.: Effects of the change in auto coupler parameters on in-train longitudinal forces during brake application. Mech. Ind. 26(205), 186–196 (2011)Google Scholar
  14. 14.
    Lu, J., Chen, G., Cheng, D., Celikovsky, S.: Bridge the gap between the lorenz system and the chen system. Int. J. Bifurc. Chaos 12(12), 2917–2926 (2002)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Singh, P.P., Singh, J.P., Roy, B.K.: Synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems using nonlinear active control. Chaos Solitons Fractals 69, 31–39 (2014)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 09, 1465 (1999)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Liu, C., Liu, T., Liu, L., Liu, K.: A new chaotic attractor. Chaos Solitons Fractals 22(5), 1031–1038 (2004)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Tigan, G., Opri, D.: Analysis of a 3D chaotic system. Chaos Solitons Fractals 36(5), 1315–1319 (2008)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Pan, L., Zhou, W., Fang, J., Li, D.: A new three-scroll unified chaotic system coined. Int. J. Nonlinear Sci. 10(4), 462–474 (2010)MathSciNetGoogle Scholar
  20. 20.
    Pehlivan, I., Uyarolu, Y.: A new chaotic attractor from general Lorenz system family and its electronic experimental implementation. Turk. J. Electr. Eng. Comput. Sci. 18(2), 171–184 (2010)Google Scholar
  21. 21.
    Sportt, J.C.: Some simple chaotic flow. Phys. Rev. E 50(2), 647–650 (1994)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Qi, G., Du, S., Chen, G., Chen, Z., Yuan, Z.: Analysis of a new chaotic system. Phys. A 352(2–4), 295–308 (2005)CrossRefGoogle Scholar
  23. 23.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)MATHCrossRefGoogle Scholar
  24. 24.
    Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)MATHCrossRefGoogle Scholar
  25. 25.
    Singh, J.P., Roy, B.K.: Coexistence of asymmetric hidden chaotic attractors in a new simple 4-D chaotic system with curve of equilibria. Optik 145, 209–217 (2017).  https://doi.org/10.1007/s40435-017-0332-8 CrossRefGoogle Scholar
  26. 26.
    Singh, J.P., Roy, B.K., Jafari, S.: New family of 4-D hyperchaotic and chaotic systems with quadric surfaces of equilibria. Chaos Solitons Fractals 106, 243–257 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Singh, J.P., Roy, B.K.: Second order adaptive time varying sliding mode control for synchronization of hidden chaotic orbits in a new uncertain 4-D conservative chaotic system. Trans. Inst. Meas. Control. (2017).  https://doi.org/10.1177/0142331217727580 Google Scholar
  28. 28.
    Singh, J.P., Roy, B.K.: Hidden attractors in a new complex generalised Lorenz hyperchaotic system, its synchronisation using adaptive contraction theory, circuit validation and application. Nonlinear Dyn. (2018).  https://doi.org/10.1007/s11071-018-4062-z Google Scholar
  29. 29.
    Alsafasfeh, Q.H.: A new chaotic behavior from Lorenz and Rossler systems and its electronic circuit implementation. Circuits Syst. 02(02), 101–105 (2011)CrossRefGoogle Scholar
  30. 30.
    Jiang, S., Yin, J.: Global existence, uniqueness and pathwise property of solutions to a stochastic Rössler–Lorentz system. Chin. Ann. Math. Ser. B 36(1), 105–124 (2015)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Bovy, J.: Lyapunov exponents and strange attractors in discrete and continuous dynamical systems. Theoretical Physics Project (2004)Google Scholar
  32. 32.
    Singh, J.P., Roy, B.K.: Crisis and inverse crisis route to chaos in a new 3-D chaotic system with saddle, saddle foci and stable node foci nature of equilibria. Optik 127(24), 11982–12002 (2016)CrossRefGoogle Scholar
  33. 33.
    Singh, J.P., Roy, B.K.: The simplest 4-D chaotic system with line of equilibria, chaotic 2-torus and 3-torus behaviour. Nonlinear Dyn. 89(3), 1845–1862 (2017)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Sprott, J.C.: Maximally complex simple attractors. Chaos 17(3), 1–6 (2007)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Xu, Y., Wang, Y.: A new chaotic system without linear term and its impulsive synchronization. Optik 125(11), 2526–2530 (2014)CrossRefGoogle Scholar
  36. 36.
    Kim, D., Chang, P.H.: A new butterfly-shaped chaotic attractor. Results Phys. 3, 14–19 (2013)CrossRefGoogle Scholar
  37. 37.
    Zhu, C.X.: Theoretic and numerical study of a new chaotic system. Intell. Inf. Manag. 02(02), 104–109 (2010)Google Scholar
  38. 38.
    Zhou, W., Xu, Y., Lu, H., Pan, L.: On dynamics analysis of a new chaotic attractor. Phys. Lett. A 372(36), 5773–5777 (2008)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Li, C., Li, H., Tong, Y.: Analysis of a novel three-dimensional chaotic system. Optik 124(13), 1516–1522 (2012)CrossRefGoogle Scholar
  40. 40.
    Li, X.F., Chlouverakis, K.E., Xu, D.L.: Nonlinear dynamics and circuit realization of a new chaotic flow: a variant of Lorenz, Chen and Lu. Nonlinear Anal. Real World Appl. 10(4), 2357–2368 (2009)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Cai, G., Tan, Z.: Chaos synchronization of a new chaotic system via nonlinear control. J. Uncertain Syst. 1(3), 235–240 (2007)Google Scholar
  42. 42.
    Genesio, R., Tesi, A.: Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28(3), 531–548 (1992)MATHCrossRefGoogle Scholar
  43. 43.
    Kingni, S.T., Keuninckx, L., Woafo, P., Van Der Sande, G., Danckaert, J.: Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: Theory and electronic implementation. Nonlinear Dyn. 73(1–2), 1111–1123 (2013)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Silva, C.P.: Shil’nikov’s theorem-a tutorial. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 10(10), 1057–7122 (1993)MATHGoogle Scholar
  45. 45.
    Majeed, N.M.: Implementation of differential chaos shift keying communication system using Matlab–Simulink. J. Am. Sci. 10(10), 240–244 (2014)Google Scholar
  46. 46.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D 16(3), 285–317 (1985)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Wang, G., Zhang, X., Zheng, Y., Li, Y.: A new modified hyperchaotic Lu system. Phys. A 371(2), 260–272 (2006)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Li, D.: Chaos, Lorenz attractor, Rossler attractor, three-scroll attractor. Phys. Lett. Sect. A Gen. Atom. Solid State Phys. 372(4), 387–393 (2008)MATHGoogle Scholar
  49. 49.
    Chen, Z., Yang, Y., Yuan, Z.: A single three-wing or four-wing chaotic attractor generated from a three-dimensional smooth quadratic autonomous system. Chaos Solitons Fractals 38(4), 1187–1196 (2008)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Li, X.F., Chu, Y.D., Zhang, J.G., Chang, Y.X.: Nonlinear dynamics and circuit implementation for a new Lorenz-like attractor. Chaos Solitons Fractals 41(5), 2360–2370 (2009)MATHCrossRefGoogle Scholar
  51. 51.
    Munmuangsaen, B., Srisuchinwong, B.: A new five-term simple chaotic attractor. Phys. Lett. A 373(44), 4038–4043 (2009)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Liu, Y., Yang, Q.: Dynamics of a new Lorenz-like chaotic system. Nonlinear Anal. Real World Appl. 11(4), 2563–2572 (2010)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Dadras, S., Momeni, H.R.: A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors. Phys. Lett. A 373(40), 3637–3642 (2009)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Wei, Z.: Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A 376(2), 102–108 (2011)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Li, X., Ou, Q.: Dynamical properties and simulation of a new Lorenz-like chaotic system. Nonlinear Dyn. 65(3), 255–270 (2011)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    San-Um, W., Srisuchinwong, B.: Highly complex chaotic system with piecewise linear nonlinearity and compound structures. J. Comput. 7(4), 1041–1047 (2012)Google Scholar
  57. 57.
    Liu, J., Zhang, W.: A new three-dimensional chaotic system with wide range of parameters. Optik 124(22), 5528–5532 (2013)CrossRefGoogle Scholar
  58. 58.
    Abooee, A., Yaghini-Bonabi, H., Jahed-Motlagh, M.R.: Chaotic system, circuitry realization, Lyapunov exponent. Commun. Nonlinear Sci. Numer. Simul. 18(5), 1235–1245 (2013)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Qiao, Z., Li, X.: Dynamical analysis and numerical simulation of a new Lorenz-type chaotic system. Math. Comput. Modell. Dyn. Syst. 20(3), 264–283 (2014)MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Deng, K., Li, J., Yu, S.: Dynamics analysis and synchronization of a new chaotic attractor. Optik 125(13), 3071–3075 (2014)CrossRefGoogle Scholar
  61. 61.
    Wang, H., Yu, Y., Wen, G.: Dynamical analysis of the Lorenz-84 atmospheric circulation model. J. Appl. Math. 2014, 1–15 (2014)MathSciNetGoogle Scholar
  62. 62.
    Zhou, W.-H., Wang, Z.-P., Wu, M.-W., Zheng, W.-H.: Dynamics analysis and circuit implementation of a new three-dimensional chaotic system. Optik 126, 765–768 (2015)CrossRefGoogle Scholar
  63. 63.
    Liu, J., Qu, Q., Li, G.: A new six-term 3-D chaotic system with fan-shaped Poincaré maps. Nonlinear Dyn. 82(4), 2069–2079 (2015)CrossRefGoogle Scholar
  64. 64.
    Gholizadeh, A., Nik, H.S., Jajarmi, A.: Analysis and control of a three-dimensional autonomous chaotic system. Appl. Math. Inf. Sci. 747(2), 739–747 (2015)MathSciNetGoogle Scholar
  65. 65.
    Wu, X., He, Y., Yu, W., Yin, B.: A new chaotic attractor and its synchronization implementation. Circuits Syst. Signal Process. 34(6), 1747–1768 (2015)CrossRefGoogle Scholar
  66. 66.
    Bhalekar, S.B.: Forming mechanizm of Bhalekar–Gejji chaotic dynamical system. Am. J. Comput. Appl. Math. 2(6), 257–259 (2013)CrossRefGoogle Scholar
  67. 67.
    Su, K.: Dynamic analysis of a chaotic system. Optik 126(24), 4880–4886 (2015)CrossRefGoogle Scholar
  68. 68.
    Çiçek, S., Ferikolu, A., Pehlivan, I.: A new 3D chaotic system: dynamical analysis, electronic circuit design, active control synchronization and chaotic masking communication application. Optik 127(8), 4024–4030 (2016)CrossRefGoogle Scholar
  69. 69.
    Akgul, A., Hussain, S., Pehlivan, I.: A new three-dimensional chaotic system without equilibrium points, its dynamical analyses and electronic circuit application. Optik 127, 7062–7071 (2016)CrossRefGoogle Scholar
  70. 70.
    Zhang, M., Han, Q.: Dynamic analysis of an autonomous chaotic system with cubic nonlinearity. Optik 127(10), 4315–4319 (2016)CrossRefGoogle Scholar
  71. 71.
    Gholamin, P., Sheikhani, A.H.R.: A new three-dimensional chaotic system: dynamical properties and simulation. Chin. J. Phys. 55(4), 1300–1309 (2017)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Lai, Q., Huang, J., Xu, G.: Coexistence of multiple attractors in a new chaotic system. Acta Phys. Pol. B 47(10), 2315–2323 (2016)MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    Tuna, M., Fidan, C.B.: Electronic circuit design, implementation and FPGA-based realization of a new 3D chaotic system with single equilibrium point. Optik 127(24), 11786–11799 (2016)CrossRefGoogle Scholar
  74. 74.
    Vaidyanathan, S.: Mathematical analysis, adaptive control and synchronization of a ten-term novel three-scroll chaotic system with four quadratic nonlinearities. Int. J. Control Theory Appl. 9(1), 1–20 (2016)MathSciNetGoogle Scholar
  75. 75.
    Vaidyanathan, S., Karthikeyan, R.: Analysis, control, synchronization, and LabView implementation of a seven-term novel chaotic system. Int. J. Control Theory Appl. 9(1), 151–174 (2016)Google Scholar
  76. 76.
    Volos, C., Akgul, A., Pham, V.T., Stouboulos, I., Kyprianidis, I.: A simple chaotic circuit with a hyperbolic sine function and its use in a sound encryption scheme. Nonlinear Dyn. 89(2), 1047–1061 (2017)CrossRefGoogle Scholar
  77. 77.
    Kengne, J., Jafari, S., Njitacke, Z.T., Yousefi Azar Khanian, M., Cheukem, A.: Dynamic analysis and electronic circuit implementation of a novel 3D autonomous system without linear terms. Commun. Nonlinear Sci. Numer. Simul. 52, 62–76 (2017)CrossRefGoogle Scholar
  78. 78.
    Wu, X., Wang, K., Wang, X., Kan, H.: Lossless chaotic color image cryptosystem based on DNA encryption and entropy. Nonlinear Dyn. 90, 855–875 (2017)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Ge, X., Lu, B., Liu, F., Luo, X.: Cryptanalyzing an image encryption algorithm with compound chaotic stream cipher based on perturbation. Nonlinear Dyn. 90, 1141–1150 (2017)MathSciNetCrossRefGoogle Scholar
  80. 80.
    Abd, M.H., Tahir, F.R., Al-Suhail, G.A., Pham, V.-T.: An adaptive observer synchronization using chaotic time-delay system for secure communication. Nonlinear Dyn. 90, 2583–2598 (2017)MathSciNetCrossRefGoogle Scholar
  81. 81.
    Bartolini, G., Fridman, L., Pisano, A., Usai, E.: Modern Sliding Mode Control Theory. Springer, Berlin, Heidelberg (2008)MATHCrossRefGoogle Scholar
  82. 82.
    Lvarez, G., Montoya, F., Romera, M., Pastor, G.: Breaking two secure communication systems based on chaotic masking. IEEE Trans. Circuits Syst. Express Briefs 51, 505–506 (2004)CrossRefGoogle Scholar
  83. 83.
    Boutayeb, M., Darouach, M., Rafaralahy, H.: Generalized state-space observers for chaotic synchronization and secure communication. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49, 345–349 (2002)MathSciNetMATHCrossRefGoogle Scholar
  84. 84.
    Lvarez, G., Montoya, F., Romera, M., Pastor, G.: Breaking parameter modulated chaotic secure communication system. Chaos Solitons Fractals 21, 783–787 (2004)MATHCrossRefGoogle Scholar
  85. 85.
    Lvarez, G., Montoya, F., Pastor, G., Romera, M.: Breaking a secure communication scheme based on the phase synchronization of chaotic systems. Chaos Interdiscip. J. Nonlinear Sci. 14, 274–278 (2004)CrossRefGoogle Scholar
  86. 86.
    Lvarez, G., Li, S., Montoya, F., Pastor, G., Romera, M.: Breaking projective chaos synchronization secure communication using filtering and generalized synchronization. Chaos Solitons Fractals 24, 775–783 (2005)MATHCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Electrical EngineeringNational Institute of Technology SilcharSilcharIndia

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