Skip to main content
SpringerLink
Log in

Coexistence of hidden chaotic attractors in a novel no-equilibrium system

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

Abstract

Hidden attractors have received considerable interest in physics, mechanics and other dynamical areas recently. This paper introduces a novel autonomous system with hidden attractor. In particular, there exists no-equilibrium point in this system. Although the new system is simple with six terms, it exhibits complex behavior such as chaos and multistability. In addition, the offset boosting of a variable is achieved by adding a single controlled constant. Dynamical properties of the no-equilibrium system have been discovered by using nonlinear dynamical tools as well as an electronic implementation.

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

Similar content being viewed by others

References

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

    Google Scholar 

  2. 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 

  3. Bragin, V.O., Vagaitsev, V.I., Kuznetsov, N.V., Leonov, G.A.: Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J. Comput. Syst. Sci. Int. 50, 511–543 (2011)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. 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. Bifurcat. Chaos 23, 1330002 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kapitaniak, T., Leonov, G.A.: Multistability: uncovering hidden attractors. Eur. Phys. J. Spec. Top. 224, 1405–1408 (2005)

    Article  Google Scholar 

  7. 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  Google Scholar 

  8. Jafari, S., Sportt, J.C., Nazarimehr, F.: Recent new examples of hidden attractors. Eur. Phys. J. Spec. Top. 224, 1469–1476 (2015)

    Article  Google Scholar 

  9. 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 

  10. 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 

  11. Leonov, G.A., Kuznetsov, N.V., Mokaev, T.N.: Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys. J. Spec. Top. 224, 1421–1458 (2015)

    Article  Google Scholar 

  12. 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 

  13. Llibre, J.: Centers: their integrability and relations with the divergence. Appl. Math. Nonlinear Sci. 1, 79–86 (2016)

    Article  Google Scholar 

  14. Li, Q., Zeng, H., Yang, X.-S.: On the hidden twin attractors and bifurcation in the Chua’s circuit. Nonlinear Dyn. 77, 255–266 (2014)

    Article  MathSciNet  Google Scholar 

  15. Wei, Z., Yu, P., Zhang, W.: Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system. Nonlinear Dyn. 82, 131–141 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, Z., Sun, W., Wei, Z., Zhang, S.: Dynamics and delayed feedback control for a 3D jerk system with hidden attractor. Nonlinear Dyn. 82, 577–588 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wei, Z., Yang, Q.: Dynamical analysis of a new autonomous 3-D system only with stable equilibria. Nonlinear Anal. Real World Appl. 12, 106–118 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Molaie, M., Jafari, S., Sprott, J.C., Golpayegani, S.: Simple chaotic flows with one stable equilibrium. Int. J. Bifurcat. Chaos 23, 1350188 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Hens, C., Dana, S.K., Feudel, U.: Extreme multistability: attractors manipulation and robustness. Chaos 25, 053112 (2015)

    Article  MathSciNet  Google Scholar 

  22. Li, C., Sprott, J.C.: Finding coexisting attractors using amplitude control. Nonlinear Dyn. 78, 2059–2064 (2014)

    Article  MathSciNet  Google Scholar 

  23. 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 

  24. Masoller, C.: Coexistence of attractors in a laser diode with optical feedback from a large external cavity. Phys. Rev. A 50, 2569–2578 (1994)

    Article  Google Scholar 

  25. 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 

  26. Upadhyay, R.K.: Multiple attractors and crisis route to chaos in a model of food-chain. Chaos Solitons Fractals 16, 737–747 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Cushing, J.M., Henson, S.M., Blackburn, C.C.: Multiple mixed attractors in a competition model. J. Biol. Dyn. 1, 347–362 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Massoudi, A., Mahjani, M.G., Jafarian, M.: Multiple attractors in Koper–Gaspard model of electrochemical. J. Electroanal. Chem. 647, 74–86 (2010)

    Article  Google Scholar 

  29. Pivka, L., Wu, C.W., Huang, A.: Chua’s oscillator: a compendium of chaotic phenomena. J. Franklin Inst. 331, 705–741 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kengne, J., Chedjou, J.C., Fonzin Fozin, T., Kyamakya, K., Kenne, G.: On the analysis of semiconductor diode based chaotic and hyperchaotic chaotic generators—a case study. Nonlinear Dyn. 77, 373–386 (2014)

    Article  MathSciNet  Google Scholar 

  31. Kengne, J.: Coexistence of chaos with hyperchaos, period-3 doubling bifurcation, and transient chaos in the hyperchaotic oscillator with gyrators. Int. J. Bifurcat. Chaos 25, 1550052 (2015)

    Article  MathSciNet  Google Scholar 

  32. 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 

  33. 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 

  34. Chudzik, A., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Multistability and rare attractors in van der Pol-Duffing oscillator. Int. J. Bifurcat. Chaos 21, 1907–1912 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sprott, J.C., Wang, X., Chen, G.R.: Coexistence of point, periodic and strange attractors. Int. J. Bifurcat. Chaos 23, 1350093 (2013)

    Article  MathSciNet  Google Scholar 

  36. Li, C., Sprott, J.C.: Coexisting hidden attractors in a 4-D simplified Lorenz system. Int. J. Bifurcat. Chaos 24, 1450034 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Dudkowski, D., Prasad, A., Kapitaniak, T.: Perpetual points and hidden attractors in dynamical systems. Phys. Lett. A 379, 2591–2596 (2015)

    Article  MathSciNet  Google Scholar 

  38. Prasad, A.: Existence of perpetual points in nonlinear dynamical systems and its applications. Int. J. Bifurcat. Chaos 25, 1530005 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kuznetsov, A.P., Kuznetsov, S.P., Mosekilde, E., Stankevich, N.V.: Co-existing hidden attractors in a radio-physical oscillator. J. Phys. A Math. Theor. 48, 125101 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50, R647 (1994)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  42. Li, C., Sprott, J.C., Yuan, Z., Li, H.: Constructing chaotic systems with total amplitude control. Int. J. Bifurcat. Chaos 25, 1530025 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Munmuangsaen, B., Sprott, J.C., Thio, W.J.-C., Buscarino, A., Fortuna, L.: A simple chaotic flow with a continuously adjustable attractor dimension. Int. J. Bifurcat. Chaos 25, 1530036 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Kengne, J., Chedjou, J.C., Kom, M., Kyamakya, K., Kamdoum Tamba, V.: 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 

  45. Kengne, J., Njitacke, Z.T., Fotsin, H.B.: Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn. 77, 373–386 (2014)

    Article  MathSciNet  Google Scholar 

  46. Ott, E.: Chaos Dyn. Syst. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  47. Buscarino, A., Fortuna, L., Frasca, M., Gambuzza, L.V.: A chaotic circuit based on Hewlett-Packard memristor. Chaos 22, 023136 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. Lü, J., Chen, G.: Generating multiscroll chaotic attractors: theories, methods and applications. Int. J. Bifurcat. Chaos 16, 775–858 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wang, L.: 3-scroll and 4-scroll chaotic attractors generated form a new 3-D quadratic autonomous system. Nonlinear Dyn. 56, 453–462 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  50. Buscarino, A., Fortuna, L., Frasca, M., Sciuto, G.: Design of time-delay chaotic electronic circuits. IEEE Trans. Circuits Syst. I Regul. Pap. 58, 1888–1896 (2011)

    Article  MathSciNet  Google Scholar 

  51. Banerjee, T., Biswas, D., Sarkar, B.C.: Design and analysis of a first order time-delayed chaotic system. Nonlinear Dyn. 70, 721–734 (2012)

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

    Article  Google Scholar 

  53. Volos, C.K., Kyprianidis, I.M., Stouboulos, I.N.: Image encryption process based on chaotic synchronization phenomena. Signal Process. 93, 1328–1390 (2013)

    Article  Google Scholar 

  54. Khan, M., Shah, T., Gondal, M.A.: An efficient technique for the construction of substitution box with chaotic partial diffirential equation. Nonlinear Dyn. 73, 1795–1801 (2013)

    Article  MathSciNet  Google Scholar 

  55. Ma, J., Wu, X.Y., Qin, H.X.: Realization of synchronization between hyperchaotic systems by using a scheme of intermittent linear coupling. Acta Phys. Sin. 62, 170502 (2013)

    Google Scholar 

  56. 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 

  57. Liu, G., Yang, W., Liu, W., Dai, Y.: Designing S-boxes based on -D four-wing autonomous chaotic system. Nonlinear Dyn. 82, 1867–1877 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank Prof. GuanRong Chen, Department of Electronic Engineering, City University of Hong Kong, for suggesting helpful references. This work has been supported by the Polish National Science Centre, MAESTRO Programme—Project No 2013/08/A/ST8/00/780.

Author information

Authors and Affiliations

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

    Viet-Thanh Pham

  2. Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924, Lodz, Poland

    Viet-Thanh Pham & Tomasz Kapitaniak

  3. Department of Physics, Aristotle University of Thessaloniki, 54124, Thessaloníki, Greece

    Christos Volos

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

    Sajad Jafari

Authors
  1. Viet-Thanh Pham
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christos Volos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sajad Jafari
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tomasz Kapitaniak
    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

Pham, VT., Volos, C., Jafari, S. et al. Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dyn 87, 2001–2010 (2017). https://doi.org/10.1007/s11071-016-3170-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-3170-x

Keywords

Search

Navigation