Skip to main content
SpringerLink
Log in

Melnikov analysis of chaos and heteroclinic bifurcation in Josephson system driven by an amplitude-modulated force

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

In this paper, the heteroclinic bifurcation and chaos in Josephson system subjected to an amplitude modulated force is discussed. By applying Melnikov method, it will be obtained that the heteroclinic bifurcation conditions of existence of chaotic motion. The effects of parameters of system on dynamical behaviors is also studied by using numerical simulation. Numerical simulation including bifurcation diagram of fixed points, phase portraits, bifurcation diagrams of system, not only show the consistent with the theoretical analysis but also exhibit the interesting bifurcation diagrams and the more new complex dynamical behaviors.

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

Similar content being viewed by others

References

  1. Konishi K (2003) Generating chaotic behaviours in an oscillator driven by periodic forces. Phys Lett A 320:200–206

    Article  MathSciNet  MATH  Google Scholar 

  2. Veukatesan A, Parthasarathy S, Lakshman M (2003) Occurrence of multiple period-doubling bifurcation route to chaos in periodically pulsed chaotic dynamical systems. Chaos Slitons and Fractals 18:891–898

    Article  MATH  Google Scholar 

  3. Ge ZM, Leu WY (2004) Anti-control of chaos of two-degrees-of-freedom loudspeaker system and chaos synchronization of different order systems. Chaos Slitons and Fractals 20:503–521

    Article  MATH  Google Scholar 

  4. Gandhimathi VM, Murali K, Rajasekar S (2006) Stochastic resonance with different periodic forces in overdamped two coupled anharmonic oscillators. Chaos Slitons and Fractals 30:1034–1047

    Article  Google Scholar 

  5. Belyky VN, Pedersen NF, Soerensen OH (1977) Shunted-Josephson model, I.The autonomous, II.The non-autonomous case. Phys Rev B 16:4853–4871

    Article  Google Scholar 

  6. Schlup WA (1974) I-V characteristics and stationary dynamics of a Josephson junction including the interference term in the current phase relation. J Phys C Solid State Phys 7:736–748

    Article  Google Scholar 

  7. Bartuccelli M, Christiansen PL, Pedesen NF, Soerensen MP (1986) Prediction of chaos in a Josephson junction by the Melnikov function technique. Phys Rev B 33:4686–4691

    Article  Google Scholar 

  8. Salam FMA, Satry S (1985) Dynamics of the forced Josephson junction circuit: the regions of chaos. IEEE Trans Circuits Syst 32:784–796

    Article  MathSciNet  MATH  Google Scholar 

  9. Jing ZJ (1983) Application of qualitative methods of differential equation to study phase-locked loops. SIAM J Appl Math 43:1245–1258

    Article  MathSciNet  Google Scholar 

  10. Jing ZJ (1989) Chaotic behavior in the Josephson equation with periodic force. SIAM J Appl Math 49:1749–1758

    Article  MathSciNet  MATH  Google Scholar 

  11. Jing ZJ, Chan KY, Xu DS, Cao HJ (2001) Bifurcation of periodic solutions and chaos in Josephson system. Discrete Contin Dyn Syst 7:573–592

    Article  MathSciNet  MATH  Google Scholar 

  12. Yang JP, Feng W, Jing ZJ (2006) Complex dynamics in Josephson system with two external forcing terms. Chaos Solitons Fractals 30:235–256

    Article  MathSciNet  MATH  Google Scholar 

  13. Jing ZJ, Cao HJ (2002) Bifurcation of periodic orbits in a Josephson equation with a phase shift. Int J Bifurcat Chaos Appl Sci Eng 12:1515–1530

    Article  MathSciNet  MATH  Google Scholar 

  14. Cao HJ, Jing XJ (2001) Chaotic dynamics of Josephson equation driven by constant dc and ac forcing. Chaos Solitons Fractals 12:1887–1895

    Article  MathSciNet  MATH  Google Scholar 

  15. Jing ZJ, Huang JC, Deng J (2007) Complex dynamics in three-well duffing system with two external forcings. Chaos Solitons Fractals 33:795–812

    Article  MathSciNet  MATH  Google Scholar 

  16. Jing ZJ, Yang ZY, Jiang T (2006) Complex dynamics in Duffing-Van der Pol equation. Chaos Solitons Fractals 27:722–747

    Article  MathSciNet  MATH  Google Scholar 

  17. Jing ZJ, Wang RQ (2005) Complex dynamics in duffing system with two external forcing. Chaos Solitons Fractals 23:399–411

    Article  MathSciNet  MATH  Google Scholar 

  18. Ravichandran V, Chinnathambi V, Rajassekar S (2007) Homoclinic bifurcation and chaos in Duffing oscillator driven by an amplitude-modulated force. Phys A 376:223–236

    Article  MathSciNet  Google Scholar 

  19. Wiggins S (1988) Global bifurcation and chaos: analytical methods. Springer, New York

    Book  MATH  Google Scholar 

  20. Wiggins S (1990) Introduction to applied nonlinear dunamical systems and chaos. Springer, New York

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The author is very thankful to the editors and referees for careful reading of the paper and valuable suggestions and useful comments that improved the presentation of the paper. This work is supported by the National Natural Science Foundation of China (Nos. 11471197 and 11402139) and the Youth Science and Technology Research Fund of Shanxi Province (No. 201601D202002).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Shanxi University, Taiyuan, China

    Yanxiang Shi

Authors
  1. Yanxiang Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yanxiang Shi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shi, Y. Melnikov analysis of chaos and heteroclinic bifurcation in Josephson system driven by an amplitude-modulated force. Int. J. Dynam. Control 6, 589–600 (2018). https://doi.org/10.1007/s40435-017-0340-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-017-0340-8

Keywords

Mathematics Subject Classification

Search

Navigation