Skip to main content
SpringerLink
Log in

Further nonlinear dynamical analysis of simple jerk system with multiple attractors

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

Abstract

This paper presents an analytical framework to investigate the dynamical behavior of a recent chaotic jerk model with multiple attractors. The methods of analytical analysis are adopted to complement numerical approach employed previously. In order to accomplish this goal, first, we amend the form of original system to a more general form and then apply both normal form theory and perturbation methods in order to investigate various dynamical behaviors exhibited by the system. The codimension one and codimension two bifurcations including pitchfork, Hopf, Bogdanov–Takens and generalized Hopf bifurcations are examined. The stability of bifurcated limit cycles is studied. The approximate solutions of homoclinic orbit and unstable limit cycle arising in the system are attained. The changes in dynamics relative to new parameters introduced in the model are also traced. Finally, numerical simulations and proposed circuit realization of the model are presented so as to validate theoretical results. We demonstrate that the jerk model has rich dynamics that make it ideal in realization of various applications including secure communications systems.

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

Similar content being viewed by others

References

  1. Izhikevich, E.M.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, Cambridge (2007)

    Google Scholar 

  2. Stavroulakis, P.: Chaos Applications in Telecommunications. CRC Press, Boca Raton (2006)

    Google Scholar 

  3. Strogatz, S.H.: Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, Boulder (2001)

    MATH  Google Scholar 

  4. Tu, P.N.V.: Dynamical Systems—An Introduction with Applications in Economics and Biology. Springer, Berlin (1995)

    Google Scholar 

  5. El-Sayed, A.M.A., Nour, H.M., Elsaid, A., Matouk, A.E., Elsonbaty, A.: Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional order hyperchaotic system. Appl. Math. Model. 40(5–6), 3516–3534 (2016)

    Article  MathSciNet  Google Scholar 

  6. Elsonbaty, A., Hegazy, S.F., Obayya, S.S.A.: A new technique for ultrafast physical random number generation using optical chaos. SPIE Photonics Europe, 98921P-98921P-6

  7. Wang, Q., Yu, S., Li, C., Lu, J., Fang, X., Guyeux, C., Bahi, J.M.: Theoretical design and FPGA-based implementation of higher dimensional digital chaotic systems. IEEE Trans. Circuits 63(3), 401–412 (2016)

    Article  MathSciNet  Google Scholar 

  8. Hong, Q., Xie, Q., Shen, Y., Wang, X.: Generating multi-double-scroll attractors via nonautonomous approach. Chaos 26(8), 083110 (2016)

    Article  MathSciNet  Google Scholar 

  9. Elsonbaty, A., Hegazy, S.F., Obayya, S.S.A.: Simultaneous suppression of time-delay signature in intensity and phase of dual-channel chaos communication. IEEE J. Quantum Electron. 51(9), 1–9 (2015)

    Article  Google Scholar 

  10. El-Sayed, A.M.A., Nour, H.M., Elsaid, A., Matouk, A.E., Elsonbaty, A.: Circuit realization, bifurcations, chaos and hyperchaos in a new 4D system. Appl. Math. Comput. 239, 333–345 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Nour, H.M., Elsaid, A., Elsonbaty, A.: Circuit realization, chaos synchronization and estimation of parameters of a hyperchaotic system with unknown parameters. J. Egypt. Math. Soc. 22(3), 550–557 (2014)

    Article  Google Scholar 

  12. El-Sayed, A.M.A., Elsaid, A., Nour, H.M., Elsonbaty, A.: Synchronization of different dimensional chaotic systems with time varying parameters, disturbances and input nonlinearities. J. Appl. Anal. Comput. 4(4), 323–338 (2014)

  13. El-Sayed, A.M.A., Elsaid, A., Nour, H.M., Elsonbaty, A.: Dynamical behavior, chaos control and synchronization of a memristor-based ADVP circuit. Commun. Nonlinear Sci. Numer. Simul. 18, 148–170 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, C., Liu, Y., Xie, T., Chen, M.Z.Q.: Breaking a novel image encryption scheme based on improved hyperchaotic sequences. Nonlinear Dyn. 73(3), 2083–2089 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, C.: Cracking a hierarchical chaotic image encryption algorithm based on permutation. Signal Process. 118, 203–210 (2016)

    Article  Google Scholar 

  16. Kocarev, L., Lian, S.: Chaos-Based Cryptography. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  17. Muthuswamy, B., Chua, L.O.: Simplest chaotic circuit. Int. J. Bifurcat. Chaos. 20, 1567–1580 (2010)

    Article  Google Scholar 

  18. 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(4), 1550052 (2015)

    Article  MathSciNet  Google Scholar 

  19. 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 

  20. 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 

  21. 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 

  22. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Kuznetsov, N.V., Leonov, G.A., Vagaitsev, V.I.: Analytical-numerical method for attractor localization of generalized Chua’s system. IFAC Proc. 4, 29–33 (2010)

    Article  Google Scholar 

  25. Kuznetsov, A.P., Kuznetsov, S.P., Stankevich, N.V.: A simple autonomous quasi-periodic self oscillator. Commun. Nonlinear Sci. Numer. Simul. 15, 1676–1681 (2010)

    Article  Google Scholar 

  26. 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 

  27. Li, C.B., Sprott, J.C.: Coexisting hidden attractors in a 4-D simplified Lorenz system. Int. J. Bifurcat. Chaos 24, 1–12 (2014)

    MathSciNet  MATH  Google Scholar 

  28. 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 

  29. 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 

  30. Han, M., Yu, P.: Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  31. Meiss, J.D.: Differential Dynamical Systems. SIAM, Monographs on Mathematical Modeling and Computation (2007)

  32. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, Berlin (2003)

    MATH  Google Scholar 

  33. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York (1998)

    MATH  Google Scholar 

  34. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  35. Peng, G., Jiang, Y.L.: Computation of universal unfolding of the double-zero bifurcation in symmetric systems by a homological method. J. Differ. Equ. Appl. 19, 1501–1512 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Chen, Y.Y., Chen, S.H., Sze, K.Y.: A hyperbolic Lindstedt–Poincaré method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators. Acta Mech. Sin. 25, 721–729 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Chen, Y.Y., Chen, S.H.: Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method. Nonlinear Dyn. 58, 417–429 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, Hoboken (1993)

    MATH  Google Scholar 

  39. Nayfeh, A.H., Balachandran, B.: Motion near a Hopf bifurcation of three-dimensional system. Mech. Res. Commun. 17(4), 191–198 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the Editor and anonymous Reviewers for providing their useful comments which improve style, readability and clarity of this work.

Author information

Authors and Affiliations

  1. Department of Engineering Mathematics and Physics, Faculty of Engineering, Mansoura University, Mansoura, 35516, Egypt

    Amr R. Elsonbaty

  2. Mathematics Department, Faculty of Science, Alexandria University, Alexandria, Egypt

    A. M. A. El-Sayed

Authors
  1. Amr R. Elsonbaty
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. M. A. El-Sayed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amr R. Elsonbaty.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elsonbaty, A.R., El-Sayed, A.M.A. Further nonlinear dynamical analysis of simple jerk system with multiple attractors. Nonlinear Dyn 87, 1169–1186 (2017). https://doi.org/10.1007/s11071-016-3108-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-3108-3

Keywords

Search

Navigation