Skip to main content
Log in

A special type of codimension two bifurcation and unusual dynamics in a phase-modulated system with switched strategy

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

Abstract

We investigate a new type of codimension two bifurcation and related dynamics in a phase-modulated system with switched strategy. Two curves intersect at a point and are called the crisis–Hopf bifurcation. At the critical crisis–Hopf vertex, a boundary crisis and Hopf bifurcation coincide. Metamorphosis of coexisting attractors can be observed in the vicinity of the vertex. Another novelty is that we discover some sets of measure zero with riddled holes in the neighborhood of the bifurcation point. These sets display some qualitative and quantitative features of riddled basins but they are essentially different from the riddled basin. It may provide a more comprehensive picture for unusual dynamical features.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Algaba, A., Gamero, E., Garcia, C., Merino, M.: A degenerate Hopf-saddle-node bifurcation analysis in a family of electronic circuits. Nonlinear Dyn. 48, 55–76 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Rashidi, R., Karami Mohammadi, A., Rashidi, R., Bakhtiari Nejad, F.: Bifurcation and nonlinear dynamic analysis of a rigid rotor supported by two-lobe noncircular gas-lubricated journal bearing system. Nonlinear Dyn. 61, 783–802 (2010)

    Article  MATH  Google Scholar 

  3. Hong, L., Sun, J.Q.: A fuzzy blue sky catastrophe. Nonlinear Dyn. 55, 261–267 (2009)

    Article  MATH  Google Scholar 

  4. Robert, C., Alligood, K.T., Ott, E., Yorke, J.A.: Explosions of chaotic sets. Physica D 144, 44–61 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hong, L., Xu, J.: Chaotic saddles in Wada basin boundaries and their bifurcations by the generalized cell-mapping digraph (GCMD) method. Nonlinear Dyn. 32, 371–385 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (1998)

    MATH  Google Scholar 

  7. Gallas, J.A.C., Grebogi, C., Yorke, J.A.: Vertices in parameter space: double crises which destroy chaotic attractors. Phys. Rev. Lett. 71, 1359–1362 (1993)

    Article  Google Scholar 

  8. Stewart, H.B., Ueda, Y., Grebogi, C., Yorke, J.A.: Double crises in two-parameter dynamical systems. Phys. Rev. Lett. 75, 2478–2481 (1995)

    Article  Google Scholar 

  9. Ashwin, P., Aston, P.J.: Blowout bifurcation of codimension two. Phys. Lett. A 244, 261–270 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hong, L., Sun, J.Q.: Codimennsion two bifurcation of nonlinear systems driven by fuzzy noise. Physica D 213, 181–189 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Harmer, G.P., Abbott, D.: Losing strategies can win by Parrondo’s paradox. Nature 402, 864 (1999)

    Article  Google Scholar 

  12. Liberzon, D.: Switching in System and Control. Birkhauser, Cambridge (2003)

    Book  Google Scholar 

  13. Almeida, J., Peralta-Salas, D., Romera, M.: Can two chaotic systems give rise to order? Physica D 200, 124–132 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Maier, M.P.S., Peacock-Lopez, E.: Switching induced oscillations in the logistic map. Phys. Lett. A 374, 1028–1032 (2010)

    Article  Google Scholar 

  15. Zhang, Y., Kong, G.: Multifarious intertwined basin boundaries of strange nonchaotic attractors in a quasiperiodically forced system. Phys. Lett. A 374, 208–213 (2009)

    Article  MathSciNet  Google Scholar 

  16. Gershenfeld, N., Grinstein, G.: Entrainment and communication with dissipative pseudorandom dynamics. Phys. Rev. Lett. 74, 5024–5027 (1993)

    Article  Google Scholar 

  17. Grebogi, C., Ott, E., Yorke, J.A.: Metamorphoses of basin boundaries in nonlinear dynamical systems. Phys. Rev. Lett. 56, 1011–1014 (1986)

    Article  MathSciNet  Google Scholar 

  18. Nusse, H.E., Yorke, J.A.: Bifurcations of basins of attraction from the view point of prime ends. Topol. Appl. 154, 2567–2579 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Alexander, J.C., Kan, I., Yorke, J.A., You, Z.: Riddled basins. Int. J. Bifurc. Chaos 2, 795–813 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Aguirre, J., Viana, R.L., Sanjuán, M.A.F.: Fractal structures in nonlinear dynamics. Rev. Mod. Phys. 81, 333–386 (2009)

    Article  Google Scholar 

  21. Sommerer, J.C., Ott, E.: A physical system with qualitatively uncertain dynamics. Nature 365, 138–140 (1993)

    Article  Google Scholar 

  22. Ott, E., Sommerer, J.C., Alexander, J.C., Kan, I., Yorke, J.A.: Scaling behavior of chaotic systems with riddled basins. Phys. Rev. Lett. 71, 4134–4137 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ashwin, P., Terry, J.R.: On riddling and weak attractors. Physica D 142, 87–100 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ashwin, P.: Riddled basins and coupled dynamical systems. In: Lect. Notes Phys., vol. 671, pp. 181–207. Springer, Berlin (2005)

    Google Scholar 

  25. Ding, M., Grebogi, C., Ott, E., Yorke, J.A.: Transition to chaotic scattering. Phys. Rev. A 42, 7025–7040 (1990)

    Article  MathSciNet  Google Scholar 

  26. Woltering, M., Markus, M.: Riddled-like basins of transient chaos. Phys. Rev. Lett. 84, 630–633 (2000)

    Article  Google Scholar 

  27. Souza, S.L.T., Caldas, I.L., Viana, R.L., Balthazar, J.M.: Sudden changes in chaotic attractors and transient basins in a model for rattling in gearboxes. Chaos Solitons Fractals 21, 763–772 (2004)

    Article  MATH  Google Scholar 

  28. Grebogi, C., McDonald, S.W., Ott, E., Yorke, J.A.: Final state sensitivity: an obstruction to predictability. Phys. Lett. A 99, 415–418 (1983)

    Article  MathSciNet  Google Scholar 

  29. Nandi, A., Dulla, D., Bhattacharjee, J.K., Ramaswamy, R.: The phase-modulated logistic map. Chaos 15, 023107 (2005)

    Article  MathSciNet  Google Scholar 

  30. Xie, J., Ding, W.: Hopf-Hopf bifurcation and T2 Torus of a vibro-impact system. Int. J. Non-Linear Mech. 40, 531–543 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang, Y., Kong, G., Yu, J.: Two codimension-3 bifurcations and non-typical routes to chaos of a shaker system. Acta Phys. Sin. 57, 6182–6187 (2008)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yongxiang Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Y., Luo, G. A special type of codimension two bifurcation and unusual dynamics in a phase-modulated system with switched strategy. Nonlinear Dyn 67, 2727–2734 (2012). https://doi.org/10.1007/s11071-011-0184-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-0184-2

Keywords

Navigation