Skip to main content
SpringerLink
Log in

Chaos-based application of a novel no-equilibrium chaotic system with coexisting attractors

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Novel chaotic system designs and their engineering applications have received considerable critical attention. In this paper, a new three-dimensional chaotic system and its application are introduced. The interesting aspects of this chaotic system are the absence of equilibrium points and the coexisting of limit cycle and torus. Basic dynamics of the no-equilibrium system have been executed by means of phase portraits, bifurcation diagram, continuation, and Lyapunov exponents. Experimental results of the electronic circuit realizing the no-equilibrium system have been reported to show system’s feasibility. By using the chaoticity of the new system without equilibrium, we have developed a random bit generator for practical signal encryption application. Numerical results illustrate the usefulness of the random bit generator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Lorenz, E.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  2. Ma, J., Wu, X., Chu, R., Zhang, L.: Selection of multi-scroll attractors in Jerk circuits and their verification using Pspice. Nonlinear Dyn. 76, 1951–1962 (2014)

    Article  Google Scholar 

  3. Wang, C., Chu, R., Ma, J.: Controlling a chaotic resonator by means of dynamics track control. Complexity 21, 370–378 (2015)

    Article  MathSciNet  Google Scholar 

  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, 449–459 (2017)

    Article  MathSciNet  Google Scholar 

  5. Ma, J., Wu, F., Ren, G., Tang, J.: A class of initials-dependent dynamical systems. Appl. Math. Comput. 298, 65–76 (2017)

    MathSciNet  Google Scholar 

  6. Rössler, O.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976)

    Article  Google Scholar 

  7. Shaw, R.: Strange attractor, chaotic behavior and information flow. Z. Naturforsch. A 36, 60–112 (1981)

    Article  MathSciNet  Google Scholar 

  8. van der Schrier, G., Maas, L.R.M.: The diffusionless Lorenz equations; Shil’nikov bifurcations and reduction to an explicit map. Phys. D 141, 19–36 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Koyuncu, I., Ozcerit, A.T., Pehlivan, I.: An analog circuit design and FPGA-based implementation of the Burke-Shaw chaotic system. Optoelectron Adv. Mater. Rapid Commun 7, 635–638 (2013)

    Google Scholar 

  10. Chen, G.R., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lü, J., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, 659–661 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, Q.G., Chen, G.R.: A chaotic system with one saddle and two stable node-foci. Int. J. Bifurc. Chaos 18, 1393–1414 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yang, Q.G., Wei, Z.C., Chen, G.R.: An unusual 3D autonomous quadratic chaotic system with two stable node-foci. Int. J. Bifurc. Chaos 20, 1061–1083 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pehlivan, I., Uyaroglu, Y.: A new chaotic attractor from general Lorenz system family and its electronic experimental implementation. Turk. J. Electr. Eng. Comput. Sci. 18, 171–184 (2010)

    Google Scholar 

  15. Liu, Z., Zhu, X., Hu, W., Jiang, F.: Principles of chaotic signal radar. Int. J. Bifurc. Chaos 17, 1735–1739 (2007)

    Article  MATH  Google Scholar 

  16. Vidal, G., Baptista, M.S., Mancini, H.: A fast and light stream cipher for smartphones. Eur. Phys. J. Spec. Top 223, 1601–1610 (2014)

    Article  Google Scholar 

  17. Banerjee, S., Rondoni, L., Mukhopadhyay, S., Misra, A.: Synchronization of spatio-temporal semiconductor lasers and its application in color image encryption. Opt. Commun. 284, 2278–2291 (2011)

    Article  Google Scholar 

  18. Volos, C.K., Kyprianidis, I.M., Stouboulos, I.N.: A chaotic path planning generator for autonomous mobile robots. Robot. Auton. Syst. 60, 651–656 (2012)

    Article  Google Scholar 

  19. Abel, A., Schwarz, W.: Chaos communications-principles, schemes, and system analysis. Proc. IEEE 90, 691–710 (2002)

    Article  Google Scholar 

  20. Xi, F., Chen, S.Y., Liu, Z.: Chaotic analog-to-information conversion: principle and reconstructability with parameter identifiability. Int. J. Bifurc. Chaos 23, 1430,025 (2014)

    Google Scholar 

  21. Tlelo-Cuautle, E., Carbajal-Gomez, V.H., Obeso-Rodelo, P.J., Rangel-Magdaleno, J.J., Nunez-Perez, J.C.: FPGA realization of a chaotic communication system applied to image processing. Nonlinear Dyn. 82, 1879–1892 (2015)

    Article  MathSciNet  Google Scholar 

  22. Wang, B., Zhong, S.M., Dong, X.C.: On the novel chaotic secure communication scheme design. Commun. Nonlinear Sci. Numer. Simul. 39, 108–117 (2006)

    Article  MathSciNet  Google Scholar 

  23. Sprott, J.C.: Elegant Chaos Algebraically Simple Chaotic Flows. World Scientific, Singapore (2010)

    Book  MATH  Google Scholar 

  24. Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N., Leonov, G., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wei, Z.: Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A 376, 102–108 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang, X., Chen, G.: Constructing a chaotic system with any number of equilibria. Nonlinear Dyn. 71, 429–436 (2013)

    Article  MathSciNet  Google Scholar 

  27. Jafari, S., Sprott, J.C., Golpayegani, S.M.R.H.: Elementary quadratic chaotic flows with no equilibria. Phys. Lett. A 377, 699–702 (2013)

    Article  MathSciNet  Google Scholar 

  28. Akgul, A., Pehlivan, I.: A new three dimensional chaotic system without equilibirium points, its dynamical analysis. Tech. Gaz. 23, 209–214 (2016)

    Google Scholar 

  29. Akgul, A., Calgan, H., Koyuncu, I., Pehlivan, I., Istanbullu, A.: Chaos-based engineering applications with a 3D chaotic system without equilibrium points. Nonlinear Dyn. 84, 481–495 (2016)

    Article  MathSciNet  Google Scholar 

  30. Jafari, S., Pham, V.T., Kapitaniak, T.: Multiscroll chaotic sea obtained from a simple 3D system without equilibrium. Int. J. Bifurc. Chaos 26, 1650,031 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hu, X., Liu, C., Liu, L., Ni, J., Li, S.: Multi-scroll hidden attractors in improved sprott a system. Nonlinear Dyn. 86, 1725–1734 (2016)

    Article  Google Scholar 

  32. Leonov, G.A., Kuznetsov, N.V., Kuznetsova, O.A., Seldedzhi, S.M., Vagaitsev, V.I.: Hidden oscillations in dynamical systems. Trans. Syst. Control 6, 54–67 (2011)

    Google Scholar 

  33. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Localization of hidden Chua’s attractors. Phys. Lett. A 375, 2230–2233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Hidden attractor in smooth Chua system. Phys. D 241, 1482–1486 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Leonov, G.A., Kuznetsov, N.V.: Hidden attractors in dynamical systems: from hidden oscillation in Hilbert-Kolmogorov, Aizerman and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurc. Chaos 23, 1330,002 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Chudzid, A., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Multistability and rare attractors in Van der Pol-Duffing oscillator. Int. J. Bifurc. Chaos 21, 1907–1912 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Brezetskyi, S., Dudkowski, D., Kapitaniak, T.: Rare and hidden attractors in Van der Pol-Duffing oscillators. Eur. Phys. J. Spec. Top. 224, 1459–1467 (2015)

    Article  Google Scholar 

  38. Zhusubaliyev, Z.T., Mosekilde, E., Churilov, A.N., Medvedev, A.: Multistability and hidden attractors in an impulsive Goodwin oscillator with time delay. Eur. Phys. J. Spec. Top. 224, 1519–1539 (2015)

    Article  Google Scholar 

  39. Zhusubaliyev, Z.T., Mosekilde, E.: Multistability and hidden attractors in a multilevel DC/DC converter. Math. Comput. Simul. 109, 32–45 (2015)

    Article  MathSciNet  Google Scholar 

  40. Leonov, G.A., Kuznetsov, N.V., Kiseleva, M.A., Solovyeva, E.P., Zaretskiy, A.M.: Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor. Nonlinear Dyn. 77, 277–288 (2014)

    Article  Google Scholar 

  41. Wei, Z., Wang, R., Liu, A.: A new finding of the existence of hidden hyperchaotic attractor with no equilibria. Math. Comput. Simul. 100, 13–23 (2014)

    Article  MathSciNet  Google Scholar 

  42. Leonov, G.A., Kuznetsov, N.V., Mokaev, T.N.: Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity. Commun. Nonlinear Sci. Numer. Simul. 28, 166–174 (2015)

    Article  MathSciNet  Google Scholar 

  43. Zuo, J., Li, C.: Multiple attractors and dynamic analysis of a no-equilibrium chaotic system. Optik 127, 7952–7957 (2016)

    Article  Google Scholar 

  44. Pham, V.T., Volos, C., Jafari, S., Kapitaniak, T.: Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dyn. 87, 2001–2010 (2017)

    Article  Google Scholar 

  45. Huang, A., Pivka, L., Wu, C.W., Franz, M.: Chua’s equation with cubic nonlinearity. Int. J. Bifurc. Chaos 6, 2175–2222 (1996)

    Article  MATH  Google Scholar 

  46. Jang, M.J., Chen, C.L., Chen, C.K.: Sliding mode control of chaos in the cubic Chua’s circuit system. Int. J. Bifurc. Chaos 12, 1437–1449 (2002)

    Article  Google Scholar 

  47. Zhong, G.Q.: Implementation of Chua’s circuit with a cubic nonlinearity. IEEE Trans. Circuits Syst.-I 41, 934–941 (1994)

    Article  Google Scholar 

  48. Zhang, M., Han, Q.: Dynamic analysis of an autonomous chaotic system with cubic nonlinearity. Optik 127, 4315–4319 (2016)

    Article  Google Scholar 

  49. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  50. Kuznetsov, N.V.: The Lyapunov dimension and its estimation via the Leonov method. Phys. Lett. A 380, 2142–2149 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  51. Kuznetsov, N.V., Alexeeva, T.A., Leonov, G.A.: Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations. Nonlinear Dyn. 85, 195–201 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  52. Leonov, G.A., Kuznetsov, N.V., Korzhemanova, N.A., Kusakin, D.V.: Lyapunov dimension formula for the global attractor of the Lorenz system. Commun. Nonlinear Sci. Numer. Simul. 41, 84–103 (2016)

    Article  MathSciNet  Google Scholar 

  53. Hens, C., Dana, S.K., Feudel, U.: Extreme multistability: attractors manipulation and robustness. Chaos 25, 053,112 (2015)

    Article  MathSciNet  Google Scholar 

  54. Vaithianathan, V., Veijun, J.: Coexistence of four different attractors in a fundamental power system model. IEEE Trans. Circuits Syst. I(46), 405–409 (1999)

    Google Scholar 

  55. Zeng, Z., Huang, T., Zheng, W.: Multistability of recurrent networks with time-varying delays and the piecewise linear activation function. IEEE Trans. Neural Netw. 21, 1371–7 (2010)

    Article  Google Scholar 

  56. Zeng, Z., Zheng, W.: Multistability of neural networks with time-varying delays and concave-convex characteristic. IEEE Trans. Neural Netw. Learn. Syst. 23, 293–305 (2012)

    Article  Google Scholar 

  57. Kengne, J., Chedjou, J.C., Kom, M., Kyamakya, K., Tamba, V.K.: Regular oscillations, chaos, and multistability in a system of two coupled van der pol oscillators: numerical and experimental studies. Nonlinear Dyn. 76, 1119–1132 (2014)

    Article  MathSciNet  Google Scholar 

  58. Leipnik, R.B., Newton, T.A.: Double strange attractors in rigid body motion with linear feedback control. Phys. Lett. A 86, 63–87 (1981)

    Article  MathSciNet  Google Scholar 

  59. Kengne, J., Njitacke, Z.T., Negou, A.N., Tsostop, M.F., Fotsin, H.B.: Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit. Int. J. Bifurcat. Chaos 26, 1650,081 (2016)

    Article  MATH  Google Scholar 

  60. Schot, S.: Jerk: the time rate of change of acceleration. Am. J. Phys. 46, 1090–1094 (1978)

    Article  Google Scholar 

  61. Linz, S.J.: Nonlinear dynamical models and jerky motion. Am. J. Phys. 65, 523–526 (1997)

    Article  Google Scholar 

  62. Sprott, J.C.: Some simple chaotic jerk functions. Am. J. Phys. 65, 537–543 (1997)

    Article  Google Scholar 

  63. Malasoma, J.M.: What is the simplest dissipative chaotic jerk equation which is parity invariant. Phys. Lett. A 264, 383–389 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  64. Eichhorn, R., Linz, S.J., Hanggi, P.: Simple polynomial classes of chaotic jerky dynamics. Chaos Solitons Fract. 13, 1–15 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  65. Sun, K.H., Sprott, J.C.: A simple jerk system with piecewise exponential nonlinearity. Int. J. Nonlinear Sci. Numer. Simul 10, 1443–1450 (2009)

    Article  Google Scholar 

  66. Sprott, J.C.: A new chaotic jerk circuit. IEEE Trans. Circuits Syst.-II: Exp Briefs 58, 240–243 (2011)

    Article  Google Scholar 

  67. Louodop, P., Kountchou, M., Fotsin, H., Bowong, S.: Practical finite-time synchronization of jerk systems: theory and experiment. Nonlinear Dyn. 78, 597–607 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  68. Liu, C., Yi, J., Xi, X., An, L., Fu, Y.: Research on the multi-scroll chaos generation based on Jerk mode. Procedia Eng. 29, 957–961 (2012)

    Article  Google Scholar 

  69. Yu, S., Lü, J., Leung, H., Chen, G.: Design and implementation of n-scroll chaotic attractors from a general Jerk circuit. IEEE Trans. Circuits. Syst. I(52), 1459–1476 (2005)

    MathSciNet  Google Scholar 

  70. Kengne, J., Njitacke, Z.T., Fotsin, H.: Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn. 83, 751–765 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  71. Bouali, S., Buscarino, A., Fortuna, L., Frasca, M., Gambuzza, L.V.: Emulating complex business cycles by using an electronic analogue. Nonlinear Anal. Real World Appl. 13, 2459–2465 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  72. Zhou, W., Wang, Z., Wu, M., Zheng, W., Weng, J.: Dynamics analysis and circuit implementation of a new three-dimensional chaotic system. Optik 126, 765–768 (2015)

    Article  Google Scholar 

  73. Lai, Q., Yang, L.: Chaos, bifurcation, coexisting attractors and circuit design of a three-dimensional continuous autonomous system. Optik 127, 5400–5406 (2016)

    Article  Google Scholar 

  74. Bianchi, G., Kuznetsov, N.V., Leonov, G.A., Yuldashev, M.V., Yuldashev, R.V.: Limitations of PLL simulation: hidden oscillations in MatLab and SPICE. Ultra modern telecommunications and control systems and workshops (ICUMT). 2015 7th International Congress on, Czech Republic, Brno, pp. 79–84 (2015)

  75. Bianchi, G., Kuznetsov, N.V., Leonov, G.A., Seledzhi, S.M., Yuldashev, M.V., Yuldashev, R.V.: Hidden oscillations in SPICE simulation of two-phase Costas loop with non-linear VCO. IFAC-PapersOnLine 49, 45–50 (2016)

    Article  Google Scholar 

  76. Kuznetsov, N.V., Leonov, G.A., Mokaev, T.N., Seledzhi, S.M.: Hidden attractor in the Rabinovich system, Chua circuits and PLL. In: AIP Conf. Proc., p. 210008. Rhodes, Greece (2016)

  77. Kuznetsov, N.V., Leonov, G.A., Yuldashev, M.V., Yuldashev, R.V.: Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE. Commun. Nonlinear Sci. Numer. Simul. 51, 39–49 (2017)

    Article  Google Scholar 

  78. Yalcin, M.E., Suykens, J.A.K., Vandewalle, J.: True random bit generation from a double–scroll attractor. IEEE Trans. Circuits Syst. I Regul. Pap. 51, 1395–1404 (2004)

    Article  MathSciNet  Google Scholar 

  79. Mansingka, A.S., Zidan, M.A., Barakat, M.L., Radwan, A.G., Salama, K.N.: Fully digital jerk–based chaotic oscillators for high throughput pseudo–random number generators up to 8.77 Gbit/s. Microelectronics J 44, 744–752 (2013)

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank Prof. GuanRong Chen, Department of Electronic Engineering, City University of Hong Kong, for suggesting helpful references. Zhen Wang is supported by the Natural Science Foundation of China (No.61473237), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2016JM1024), the Scientific Research Program Funded by Shaanxi Provincial Education Department (No.15JK2181), and the Scientific Research Foundation of Xijing University (Grant No.XJ160142). This work was partially supported by Sakarya University Scientific Research Projects Unit under Grants 2016-09-00-008, 2016-50-01-026.

Author information

Authors and Affiliations

  1. Department of Applied Sciences, Xijing University, Xi’an, 710123, China

    Zhen Wang

  2. Department of Electric and Electronic Engineering, Faculty of Technology, University of Sakarya, Sakarya, Turkey

    Akif Akgul

  3. School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi, Vietnam

    Viet-Thanh Pham

  4. Biomedical Engineering Department, Amirkabir University of Technology, Tehran, 15875-4413, Iran

    Sajad Jafari

Authors
  1. Zhen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akif Akgul
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Viet-Thanh Pham
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sajad Jafari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Viet-Thanh Pham.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Z., Akgul, A., Pham, VT. et al. Chaos-based application of a novel no-equilibrium chaotic system with coexisting attractors. Nonlinear Dyn 89, 1877–1887 (2017). https://doi.org/10.1007/s11071-017-3558-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3558-2

Keywords

Search

Navigation