Skip to main content
SpringerLink
Log in

Multi-scroll hidden attractors in improved Sprott A system

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

Abstract

In this paper, two simple 3D chaotic systems are constructed by introducing nonlinear functions into Sprott A system, which all have no equilibrium, and multi-scroll hidden attractors can be obtained. In the case of simple sine function without restricting its nonlinear dynamical range, the number of multi-scroll attractors has nothing to do with the system equilibria, which is only determined by the transient simulation time. In the other case, the introduced nonlinear function is composed of nonlinear part and linear part; thus, the nonlinearity of improved Sprott A system is limited in certain range, as a result, the number of multi-scroll hidden attractors can be selected arbitrarily within finite transient simulation time. Furthermore, the dynamical properties of two systems are studied through phase plane, time series, Poincaré map and frequency spectra. Finally, an electronic circuit of improved Sprott A system is implemented in Pspice, and the results of electronic circuit are consistent with that of the numerical simulation.

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

Similar content being viewed by others

References

  1. Shilnikov, L.P.: A case of the existence of a denumerable set of periodic motions. Sov. Math. Dockl. 6, 163–166 (1965)

    Google Scholar 

  2. Molaie, M., Jafari, S., Sprott, J.C., Golpayegani, S.M.R.H.: Simple chaotic flows with one stable equilibrium. Int. J. Bifurcation Chaos. 23(11), 1350188 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Wang, X., Chen, G.: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17(3), 1264–1272 (2011)

    Article  MathSciNet  Google Scholar 

  4. Jafari, S., Sprott, J.C., Golpayegani, S.M.R.H.: Elementary quadratic chaotic flows with no equilibria. Phys. Lett. A. 377(9), 699–702 (2013)

    Article  MathSciNet  Google Scholar 

  5. Wei, Z.: Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A. 376(2), 102–108 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jafari, S., Sprott, J.C.: Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals 57, 79–84 (2013)

    Article  MathSciNet  Google Scholar 

  7. 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. Bifurcation Chaos. 23(1), 1330002 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Localization of hidden chua’s attractors. Phys. Lett. A. 375(23), 2230–2233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Hidden attractor in smooth chua systems. Phys. D. 241(18), 1482–1486 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. (2016). doi:10.1016/j.physrep.2016.05.002

    MathSciNet  Google Scholar 

  11. Kuznetsov, N.V.: Hidden Attractors in Fundamental Problems and Engineering Models: A Short Survey. Lecture Notes in Electrical Engineering, vol. 371, pp. 13–25. Springer, Switzerland (2015)

  12. Wang, X., Chen, G.: Constructing a chaotic system with any number of equilibria. Nonlinear Dyn. 71(3), 429–436 (2012)

    Article  MathSciNet  Google Scholar 

  13. Gmez, G.L., Cruz, H.C., Lpez, G.R.M., Garca, G.E.E.: Synchronization of chuas circuits with multi-scroll attractors: application to communication. Commun. Nonlinear Sci. Numer. Simul. 14(6), 2765–2775 (2009)

    Article  Google Scholar 

  14. Orue, A.B., Alvarez, G., Pastor, G., Romera, M., Montoya, F., Li, S.: A new parameter determination method for some double-scroll chaotic systems and its applications to chaotic cryptanalysis. Commun. Nonlinear Sci. Numer. Simul. 15(11), 3471–3483 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Han, F., Hu, J., Yu, X., Wang, Y.: Fingerprint images encryption via multi-scroll chaotic attractors. Appl. Math. Comput. 185(2), 931–939 (2007)

    Article  MATH  Google Scholar 

  16. Gmez, G.L., Cruz, H.C., Lpez, R.M., Garca, G.E.E.: Synchronization of multi-scroll chaos generators: application to private communication. Revista Mexicana De Fisica 54(4), 299–305 (2008)

    Google Scholar 

  17. YalIn, M.E., Suykens, J.A.K., Vandewalle, J., ZoUz, S.: Families of scroll grid attractors. Int. J. Bifurcation Chaos. 12(01), 23–41 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Deng, W., L, J.: Generating multi-directional multi-scroll chaotic attractors via a fractional differential hysteresis system. Phys. Lett. A. 369(5–6), 438–443 (2007)

    Article  MATH  Google Scholar 

  19. Yu, S., L, J., Chen, G.: A family of n-scroll hyperchaotic attractors and their realization. Phys. Lett. A. 364(3–4), 244–251 (2007)

    Article  Google Scholar 

  20. Lu, J., Murali, K., Sinha, S., Leung, H., Aziz-Alaoui, M.A.: Generating multi-scroll chaotic attractors by thresholding. Phys. Lett. A. 372(18), 3234–3239 (2008)

    Article  MATH  Google Scholar 

  21. Dadras, S., Momeni, H.R.: Four-scroll hyperchaos and four-scroll chaos evolved from a novel 4d nonlinear smooth autonomous system. Phys. Lett. A. 374(11–12), 1368–1373 (2010)

    Article  MATH  Google Scholar 

  22. Dong, G., Du, R., Tian, L., Jia, Q.: A novel 3d autonomous system with different multilayer chaotic attractors. Phys. Lett. A. 373(42), 3838–3845 (2009)

    Article  MATH  Google Scholar 

  23. Yu, S., Tang, W.K.S.: Generation of nm-scroll attractors in a two-port rcl network with hysteresis circuits. Chaos Solitons Fractals 39(2), 821–830 (2009)

    Article  MATH  Google Scholar 

  24. Ai, X., Sun, K., He, S., Wang, H.: Design of grid multiscroll chaotic attractors via transformations. Int. J. Bifurcation Chaos. 25(10), 1530027 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yu, S.M., Lin, Q.H., Qiu, S.S.: Simulation investigation on multi-scroll chaotic and hyperchaotic attractors for four-dimensional systems. Acta Phys. Sin-Ch Ed. 52(1), 25–33 (2003)

    Google Scholar 

  26. Han, F.L., Lu, J.H., Yu, X.H., Chen, G.R.: Generating multi-scroll chaotic attractors via a linear second-order hysteresis system. Dyn. Contin. Discret. Impuls. Syst.-Ser. B-Appl. Algorithms 12(1), 95–110 (2005)

    MathSciNet  MATH  Google Scholar 

  27. Jafari, S., Pham, V.T., Kapitaniak, T.: Multiscroll chaotic sea obtained from a simple 3d system without equilibrium. Int. J. Bifurcation Chaos 26(02), 1650031 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E. 50(2), R647–R650 (1994)

    Article  MathSciNet  Google Scholar 

  29. L, J.: Generating multiscroll chaotic attractors: theories, methods and applications. Int. J. Bifurcation Chaos 16(4), 775–848 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. L, J., Han, F., Yu, X., Chen, G.: Generating 3-d multi-scroll chaotic attractors: a hysteresis series switching method. Automatica 40(10), 1677–1687 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xu, F., Yu, P.: Chaos control and chaos synchronization for multi-scroll chaotic attractors generated using hyperbolic functions. J. Math. Anal. Appl. 362(1), 252–274 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhong, G.Q., Man, K.-F., Chen, G.: A systematic approach to generating n-scroll attractors. Int. J. Bifurcat Chaos 12(12), 2907–2915 (2002)

    Article  Google Scholar 

  33. Zhang, C., Yu, S.: Generation of grid multi-scroll chaotic attractors via switching piecewise linear controller. Phys. Lett. A. 374(30), 3029–3037 (2010)

    Article  MATH  Google Scholar 

  34. Yu, S.M.: Circuit implementation for generating three-dimensional multi-scroll chaotic attractors via triangular wave series. Acta Phys. Sin-Ch Ed. 54(4), 1500–1509 (2005)

    Google Scholar 

  35. Chen, S.-B., Zeng, Y.-C., Xu, M.-L., Chen, J.-S.: Construction of grid multi-scroll chaotic attractors and its circuit implementation with polynomial and step function. Acta Phys. Sin-Ch Ed. 60(2), 020507 (2011)

  36. Ma, J., Wu, X., Chu, R., Zhang, L.: Selection of multi-scroll attractors in jerk circuits and their verification using pspice. Nonlinear Dyn. 76(4), 1951–1962 (2014)

    Article  Google Scholar 

  37. Li, F., Yao, C.: The infinite-scroll attractor and energy transition in chaotic circuit. Nonlinear Dyn. 83(4), 1–11 (2016)

    Article  MathSciNet  Google Scholar 

  38. Tang, W.K.S., Zhong, G.Q., Chen, G., Man, K.F.: Generation of n-scroll attractors via sine function. IEEE Trans. Circuits Syst. I Reg. 48(11), 1369–1372 (2010)

    Article  Google Scholar 

  39. Yalin, M.E.: Multi-scroll and hypercube attractors from a general jerk circuit using josephson junctions. Chaos Solitons Fractals 34(5), 1659–1666 (2007)

    Article  Google Scholar 

  40. Hoover, W.G.: Remark on some simple chaotic flows. Phys. Rev. E. 51(1), 759–760 (1995)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank the anonymous referees for their constructive and helpful suggestions. This work is partially supported by the National Nature Science Foundation of China under the Grant Nos. 51177117 and 51307130.

Author information

Authors and Affiliations

  1. State Key Laboratory of Electrical Insulation and Power Equipment, Xian Jiaotong University, Xian, 710049, China

    Xiaoyu Hu, Chongxin Liu, Ling Liu, Junkang Ni & Shilei Li

  2. School of Electrical Engineering, Xian Jiaotong University, Xian, 710049, China

    Xiaoyu Hu, Chongxin Liu, Ling Liu, Junkang Ni & Shilei Li

Authors
  1. Xiaoyu Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chongxin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ling Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Junkang Ni
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Shilei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoyu Hu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, X., Liu, C., Liu, L. et al. Multi-scroll hidden attractors in improved Sprott A system. Nonlinear Dyn 86, 1725–1734 (2016). https://doi.org/10.1007/s11071-016-2989-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2989-5

Keywords

Search

Navigation