The Theory of Bifurcations in the Work of Poincaré

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


Poincaré already used “putting into general position” almost as is done today. For example, his classification of the singular points of generic vector fields on the plane (saddle, node, focus) is the prototype for many classifications in catastrophe theory. The theory of bifurcations of periodic solutions, also created by Poincaré, anticipated the general theory of bifurcations, not only with respect to results, but also (and particularly) with respect to methods. Already in his dissertation (1879) (solving a problem of classification of differential equations posed by Darboux, who was the discover of the normal form of a symplectic structure) created the general method of normal forms, which leads to the classification of catastrophes if one applies it to functions instead of to differential equations.


Periodic Solution Normal Form Catastrophe Theory Versal Deformation Dirty Hand 
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  1. 2.
    In Mather’s terminology (1968) “contact finite detenninacy” (however, Poincaré proved more, since he did not use difleomorphisms of z-space).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

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