The European Physical Journal Special Topics

, Volume 165, Issue 1, pp 45–59 | Cite as

Chaotic orbits prediction and atypical bifurcations in a class of piece-wise linear noninvertible maps



In this paper we apply one of the main results from the theory of noninvertible maps to predict the oscillations amplitude of a chaotic attractor, using only the lines where the piece-wise linear (PWL) map is not differentiable, their first iterates (called critical lines or curves), and their finite-rank iterates. This approach is also valid for smooth differentiable noninvertible maps, where the critical lines are defined as the first iterates of the set of points for which the Jacobian determinant of the map cancels. Moreover, a novel bifurcation encountered in the case of PWL map is presented: chaotic orbit arising from the collision of a typical bifurcation for smooth maps (the Neimark-Hopf, or center bifurcation) with the bifurcation typical only for piece-wise smooth and non differentiable maps (the border-collision bifurcation). Finally, it is shown that this novel bifurcation can give birth to two (or more) chaotic attractors, and the conditions allowing to predict the number of chaotic attractors which could coexist are proposed and discussed in the context of possible applications.


European Physical Journal Special Topic Chaotic Attractor Critical Line Chaotic Orbit Discontinuity Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. M.I. Feigin, J. Appl. Math. Mech. 34, 861 (1970)Google Scholar
  2. M.I. Feigin, J. Appl. Math. Mech. 38, 810 (1974)Google Scholar
  3. M.I. Feigin, J. Appl. Math. Mech. 59, 853 (1995)Google Scholar
  4. S. Banerjee, C. Grebogi, Phys. Rev. E 59, 4052 (1999)Google Scholar
  5. M. di Bernardo, A. Nordmark, G. Olivar, Physica D: Nonlinear Phenomena (in press)Google Scholar
  6. M. di Bernardo, F. Garafalo, L. Ianelli, F. Vasca, Int. J. Control 75, 1243 (2002)Google Scholar
  7. M. di Bernardo, M.I. Feigin, S.J. Hogan, M.E. Homer, Chaos, Solitons Fractals 10, 1881 (1999)Google Scholar
  8. R.I. Leine, Physica D: Nonlinear Phenom. 223, 121 (2006)Google Scholar
  9. H.E. Nusse, J.A. Yorke, Physica D 57, 39 (1992)Google Scholar
  10. H.E. Nusse, J.A. Yorke, Phys. Rev. E 49, 1073 (1994)Google Scholar
  11. H.E. Nusse, J.A. Yorke, Int. J. Bifur. Chaos 5, 189 (1995)Google Scholar
  12. Zh. T. Zhusubaliyev, E. Mosekilde, Phys. Lett. A (2007) (in press)Google Scholar
  13. C. Rosa, M.J. Correia, P.C. Rech, Chaos, Solitons Fractals (2007) (in press)Google Scholar
  14. I. Taralova-Roux, Vol. 2 (Lille, France, 1996), p. 631Google Scholar
  15. D. Fournier-Prunaret, O. Feely, I. Taralova-Roux, Nonlin. Anal. 47, 5343 (2001)Google Scholar
  16. O. Feely, D. Fournier-Prunaret, I. Taralova-Roux, D. Fitzgerald, Int. J. Bifurc. Chaos 10, 307 (2000)Google Scholar
  17. I. Taralova-Roux, O. Feely, Recent Advances in Circuits and Systems (World Scientific, 1998), p. 136Google Scholar
  18. I. Taralova-Roux, O. Feely International Symposium on Non-linear Theory and its Applications NOLTA’98, Vol. 2 (Crans-Montana, Switzerland, 1998), p. 755Google Scholar
  19. I. Taralova-Roux , O. Feely, Computers and Computational Engineering in Control (WSES Press, 1999), p. 314Google Scholar
  20. I. Taralova, D. Fournier-Prunaret, IEEE Transactions on Circuits and Systems-I, Fundamental Theory and Applications, Vol. 49 (2002), p. 1592Google Scholar
  21. C. Mira, G. Millerioux, S. Rouabhi, Eur. J. Oper. Res. 139, 461 (2002)Google Scholar
  22. O. Feely, J. Franklin Inst. 331, 903 (1994)Google Scholar
  23. C. Mira, Chaotic Dyn. (World Scientific, Singapore, 1987)Google Scholar
  24. C. Mira, Chaos, Solitons Fract. 11, 251 (2000)Google Scholar
  25. C. Mira, L. Gardini, A. Barugola, J.C. Cathala, Scientific Series on Nonlinear Science, Serie A 20 (1996)Google Scholar
  26. O. Feely, Int. J. Circ. Theor. Appl. 35, 515 (2007)Google Scholar

Copyright information

© EDP Sciences and Springer 2008

Authors and Affiliations

  1. 1.IRCCyN UMR 6597Nantes Cedex 3France

Personalised recommendations