By Michel Demazure

Based on a lecture path on the Ecole Polytechnique (Paris), this article provides a rigorous creation to some of the key principles in nonlinear research, dynamical platforms and bifurcation idea together with disaster idea. anyplace acceptable it emphasizes a geometric or coordinate-free process which permits a transparent concentrate on the basic mathematical buildings. Taking a unified view, it brings out good points universal to various branches of the topic whereas giving abundant references for extra complex or technical developments.

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Also, and this is essential for many applicatons, one can attempt to escape the limitation of normed spaces and take as E and F spaces of Coo functions, spaces of distributions, and so on. Unfortunately, this is much more difficult, and there is no real generalization of the Local Inversion Theorem which retains its power and simplicity. See the discussion ofthese of these questions by J. D. Bost [BO], who gives a good approximation to such weH as certain applications, notably the famous Kolmogorova theorem as well Arnol'd-Moser (KAM) theorem on invariant tori which plays an essential role in the theory of conservative dynamical systems.

But there are more fundamental reasons. Theoretical results have to be able to be applied to objects as simple as a plane curve with double points or a cone, and we have apriori a priori a choice between two approaches: throw out all singular points - the solution chosen here - or broaden the definition to admit any subset deeeo, but also why not of class cr). er). Unfortunately fined by equations (of class Coo, (or fortunately for students) this second solution is totally impracticable. 24). However, the choice of this type of definition will not prevent us from studying certain singularities, in particular those which occur 'inevitably' as limits of regular situations.

0 It is often more natural geometrically to consider the affine tangent space to V, which is the translate a + T Taa V. 3. To ealculate Taa V we may use either of the two descriptions deseriptions given calculate T deseription "by equations" it is often convenient eonvenient to above. If we use the description proeeed cjJ is a Coo function defined on a neighproceed as follows. First note that if ¢> Ethen bourhood of a and v is a veetor vector in E then we have a partial expansion cjJ(a + tv) = = ¢>(a) cjJ(a) ¢>(a + t(v,d¢>(a)) t(v, dcjJ(a)) + o(t).