Skip to main content
SpringerLink
Log in

Stability and boundary equilibrium bifurcations of modified Chua’s circuit with smooth degree of 3

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

Chua’s circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua’s circuit system with the smooth degree of 3 are studied. The parametric areas of stability are specified in detail. It is found that the bifurcation graphs of the supercritical and irregular pitchfork bifurcations are similar to those of the piecewise-smooth continuous (PWSC) systems caused by piecewise smoothness. However, the bifurcation graph of the supercritical Hopf bifurcation is similar to those of smooth systems. Therefore, the boundary equilibrium bifurcations of the non-smooth systems with the smooth degree of 3 should receive more attention due to their special features.

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

Similar content being viewed by others

References

  1. Freire, E., Ponce, E., and Ros, J. A bi-parametric bifurcation in 3D continuous piecewise linear systems with two zones, application to Chua’s circuit. International Journal of Bifurcation and Chaos, 17, 445–457 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Dana, S. K., Chakraborty, S., and Ananthakrishina, G. Homoclinic bifurcation in Chua’s circuit. Pramana—Journal of Physics, 64, 443–454 (2005)

    Article  Google Scholar 

  3. Zhang, Z. D. and Bi, Q. S. Bifurcation in a piecewise linear circuit with switching boundaries. International Journal of Bifurcation and Chaos, 22, 1250034 (2012)

    Article  Google Scholar 

  4. Carmona, V., Freire, E., Ponce, E., Ros, J., and Torres, F. Limit cycle bifurcation in 3D continuous piecewise linear systems with two zones, application to Chua’s circuit. International Journal of Bifurcation and Chaos, 15, 3153–3164 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fu, S. H. and Lu, Q. S. Set stability of controlled Chua’s circuit under a non-smooth controller with the absolute value. International Journal of Control, Automation, and Systems, 12, 1–11 (2014)

    Article  Google Scholar 

  6. Tang, K. S., Man, K. F., Zhong, G. Q., and Chen, G. R. Modified Chua’s circuit with x|x|. Control Theory and Applications, 20, 223–227 (2003)

    MATH  MathSciNet  Google Scholar 

  7. Tang, F. and Wang, L. An adaptive active control for the modified Chua’s circuit. Physics Letters A, 346, 342–346 (2005)

    Article  MATH  Google Scholar 

  8. Iooss, G. and Joseph, D. Elementary Stability and Bifurcation Theory, Springer, New York (1980)

    Book  MATH  Google Scholar 

  9. Chow, S. N. and Hale, J. Methods of Bifurcation Theory, Springer, New York (1982)

    Book  MATH  Google Scholar 

  10. Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York (1983)

    Book  MATH  Google Scholar 

  11. Heemels, W. P. M. H. and Brogliato, B. B. The complementarity class of hybrid dynamical systems. European Journal of Control, 9, 311–319 (2003)

    Article  Google Scholar 

  12. Leine, R. I., van Campen, D. H., and vande Vrande, B. L. Bifurcations in nonlinear discontinuous systems. Nonlinear Dynamics, 23, 105–164 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Di Bernardo, M. and Budd C. J. Bifurcations in nonsmooth dynamical systems. SIAM Review, 50, 629–701 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Di Bernardo, M. and Hogan, S. J. Discontinuity-induced bifurcations of piecewise smooth dynamical systems. Philosophical Transactions of the Royal Society, A: Mathematical, Physical and Engineering Sciences, 368, 4915–4935 (2010)

    Article  MATH  Google Scholar 

  15. Di Bernardo, M., Nordmarkc, A., and Olivard, G. Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems. Physica D: Nonlinear Phenomena, 237, 119–136 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kuznetsov, Y. A. Elements of Bifurcation Theory, Applied Mathematical Sciences, Springer, New York (2004)

    Book  Google Scholar 

  17. Marsden, J. E. The Hopf Bifurcation and Its Applications, Springer, New York (1976)

    Book  MATH  Google Scholar 

  18. Chen, Z. Y., Zhang, X. F., and Bi, Q. S. Bursting phenomenon and the bifurcation mechanism in generalized Chua’s circuit. Acta Physica Sinica, 59, 2326–2333 (2010)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Shihui Fu

  2. Department of Biology, University of Maryland, College Park, MD, 20742, USA

    Xiangying Meng

  3. Department of Dynamics and Control, Beihang University, Beijing, 100191, China

    Qishao Lu

Authors
  1. Shihui Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiangying Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qishao Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qishao Lu.

Additional information

Project supported by the National Natural Science Foundation of China (Nos.U1204106, 11372282, 11272024, and 11371046) and the National Basic Research Program of China (973 Program) (Nos. 2012CB821200 and 2012CB821202)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fu, S., Meng, X. & Lu, Q. Stability and boundary equilibrium bifurcations of modified Chua’s circuit with smooth degree of 3. Appl. Math. Mech.-Engl. Ed. 36, 1639–1650 (2015). https://doi.org/10.1007/s10483-015-2009-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-015-2009-6

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation