One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems
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.
KeywordsUnstable Manifold Invariant Curve Nondegeneracy Condition Bifurcation Condition Fold Bifurcation
Unable to display preview. Download preview PDF.
- 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
- Arnol’d, V.I., Varchenko, A.N. and Gusein-Zade, S.M. (1985), Singularities of Differentiable Maps I, Birkhäuser, Boston, MA.Google Scholar
- 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
- Neimark, Ju.I. (1959), `On some cases of periodic motions depending on parameters’, Dokl. Akad. Nauk SSSR 129, 736–739. In Russian.Google Scholar
- 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