One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems

  • Yuri A. Kuznetsov
Part of the Applied Mathematical Sciences book series (AMS, volume 112)


In this chapter, which is organized very much like Chapter 3, we present bifurcation conditions defining the simplest bifurcations of fixed points in n-dimensional discrete-time dynamical systems: the fold, the flip, and the Neimark-Sacker bifurcations. Then we study these bifurcations in the lowest possible dimension in which they can occur: the fold and flip bifurcations for scalar systems and the Neimark-Sacker bifurcation for planar systems. In Chapter 5 it will be shown how to apply these results to n-dimensional systems when n is larger than one or two, respectively.


Unstable Manifold Invariant Curve Nondegeneracy Condition Bifurcation Condition Fold Bifurcation 
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.

Bibliographical notes

  1. Whitley, D. (1983), `Discrete dynamical systems in dimensions one and two’, Bull. London Math. Soc. 15, 177–217.MathSciNetzbMATHCrossRefGoogle Scholar
  2. van Strien, S. (1991), Interval dynamics, in E. van Groesen and E. de Jager, eds, `Structures in Dynamics’, Vol. 2 of Studies in Mathematical Physics, North-Holland, Amsterdam, pp. 111–160.Google Scholar
  3. Arnol’d, V.I., Varchenko, A.N. and Gusein-Zade, S.M. (1985), Singularities of Differentiable Maps I, Birkhäuser, Boston, MA.Google Scholar
  4. Newhouse, S., Palis, J. and Takens, F. 1983), `Bifurcations and stability of families of diffeomorphisms’, Inst. Hautes Etudes Sci. Publ. Math. 57, 5–71.Google Scholar
  5. Neimark, Ju.I. (1959), `On some cases of periodic motions depending on parameters’, Dokl. Akad. Nauk SSSR 129, 736–739. In Russian.Google Scholar
  6. Sacker, R. (1965), `A new approach to the perturbation theory of invariant surfaces’, Comm. Pure Appl. Math. 18, 717–732.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Ruelle, D. and Takens, F. (1971), `On the nature of turbulence’, Comm. Math. Phys. 20, 167–192.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Marsden, J. and McCracken, M. (1976), Hopf Bifurcation and Its Applications, Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  9. Chenciner, A. (1979), Bifurcations de difféomorphismes de R2 au voisinage d’un point fixe elliptique, in G. Iooss, R. Heileman and R. Stora, eds, `Chaotic Behavior of Deterministic Systems (Les Houches, 1981)’, North-Holland, Amsterdam, pp. 273–348.Google Scholar
  10. Wan, Y.-H. (1978b), `Computations of the stability condition for the Hopf bifurcation of diffeomorphisms on R2’, SIAM J. Appl. Math. 34, 167–175.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Marsden, J. and McCracken, M. (1976), Hopf Bifurcation and Its Applications, Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Yuri A. Kuznetsov
    • 1
    • 2
  1. 1.Department of MathematicsUtrecht UniversityUtrechtThe Netherlands
  2. 2.Institute of Mathematical Problems of BiologyRussian Academy of SciencesPushchino, Moscow RegionRussia

Personalised recommendations