5 years ago

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

184 of parameters, we

184 of parameters, we define G .factors through F by the existence of a change of coordinates (S,X,A, ~) such that G(z, A, a) = S(z, A, a) F(X(...), A(...), ~(a)). where (S,X,A) is an unfolding of the identity in ~D and ~(0) = 0. In that situation all the bifurcation diagrams of G = 0 can be found in F = 0 modulo the change of coordinates. The main question is to find an universal unfolding of f, ie. an unfolding with the smallest number of parameters needed for each other possible unfolding to factor through it. Singularity theory uses algebraic ideas to solve these questions. E has a natural struc- ture of E,-module. Let f 6 E be fixed, we shall need the following subsets of E : T(,f, U) = { S,f + ,f.X I S e ~, S(0) = 0, X ~ E, X(0) = 0, X~(0) = 0 } + Ex < a2/x > T(,f, unf) = { ..q,f+,f=XIS 6 .&4, X ~ E } +Ex

4.2 The recognition problem 185 This problem is concerned with determining when a bifurcation function is equivalent to a given one. We are interested in knowing when a germ is equivalent to an initial segment of its own Taylor series and so interested in criteria for deciding whether f + p is equivalent to f for f,p E E. We define the set of higher order terms of f by ~(f)={pEE] g+p62~f, Vg62)f}. P(f) is a submodule of E which depends only on the 9- equivalence class of f and has the related property of being intrinsic. A submodule M C E is said to be intrinsic if Y f, g 6 E, g 6 M and h 62)g =~ h 6 M. For any linear subspace L of E we define the intrinsic part of L, denoted ItrL, to be the maxima/intrinsic submodule contained in L. The fundamental result is then Theorem 4 (Gaffney) If the codiraension of f is finite, 7~(f) D ItrT(f, U). The proofs of the recognition and classification theorems use this result to calculate the higher order terms that can be discarded in a Taylor series and then uses explicit changes of coordinates to bring the low order terms into the required normal form. The following lemma gives some useful criteria to find the intrinsic part of a submodule. Lemma 5 a) , If c I , J C I C J , K C J C K. 4.3 Unfolding theory An unfolding F of f E E is universal if each other unfolding of f factors through F and if F has the minimal number of parameters among the unfoldings with that property. The universal unfolding of f is calculated, if it exists, by means of the tangent space T(f, unf). The fundamental theorem is the following. Theorem 6 Let F be a k-parameter unfolding off E E. Then F is a universal unfolding of f iff = T(f, un])+ 1% It clearly follows from this theorem that if we have a basis (Vl... vk) of (E/T(f, unf)) then the unfolding G = f + k 5"~i=, ai vl is an universal unfolding of f. 4.4 Classification We are now giving the results organized in function of n. We stopped at the topological codimension 1, where the first bifurcation into non reversible subharmonic solutions occur (case IX). The topological codimension is the number of parameters needed to get all the different 1infolding diagrams under continuous (instead of smooth) changes of coordinates. The remaining parameters are called moduli and are invariant of the smooth changes of

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