Bifurcations of Equilibria

Part of the Encyclopaedia of Mathematical Sciences book series (volume 5)


The theory of bifurcations of dynamical systems describes sudden qualitative changes in the phase portraits of differential equations that occur when parameters are changed continuously and smoothly. Thus, upon loss of stability, a limit cycle may arise from a singular point, and the loss of stability by a limit cycle may give rise to chaos. Such changes are termed bifurcations.


Vector Field Singular Point Phase Portrait Bifurcation Diagram Center Manifold 
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  1. 2.
    We emphasize the difference between a local family and a family of germs of vector fields: the field of a local family is defined in a neighborhood of x0 which is independent of ε,ε £ sufficiently near to ε0. The fields belonging to a family of germs do not have this property. 2a We use this bad notation in spite of the fact that the motion along the “stable” manifold is very unstable and hence this manifold should be called “unstable”.Google Scholar
  2. 3.
    If there is no saddle suspension, then for uniformity we shall say that there is a suspension of the “trivial saddle” 0Google Scholar
  3. 4.
    Let A and B be two disjoint classes of germs of vector fields at a singular point 0. We say that the class B is adjacent to the class A (written B → A) if for each germ v in the class B, there exists a continuous deformation taking that germ into the class A. More exactly, there exists a continuous family of germs \( \left\{ {\upsilon _t |t \in [0,1]} \right\} \) such that v 0 = v and v t is a germ in the clas A for all t ∈ (0, 1]Google Scholar
  4. 5.
    Partial results were obtained in the references (Arnol’d (1972); Gavrilov (1980); Khorozov (1979); Guckenheimer (1984); Guckenheimer and Holmes (1983)), and by V.I. Shvetsov in his diploma thesis, Moscow State University, 1983, 15 pp.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

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