# Bifurcations and Catastrophes by Michel Demazure By Michel Demazure

Based on a lecture path, this article provides a rigorous creation to nonlinear research, dynamical platforms and bifurcation concept together with disaster idea. at any place applicable it emphasizes a geometric or coordinate-free method permitting a transparent concentrate on the basic mathematical constructions. It brings out beneficial properties universal to various branches of the topic whereas giving considerable references for extra complex or technical developments.

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Thus b E O”,(u). Similarly, if b E @O(v), ,&en b E W(u). QED. If M is connected, then for every pair of points u and v of P, there is an element a E G suchthat u - ua. 1 that if M is co&ecfed,,the holonomy groups CD(u), u E P, are all conjugate to each other. in G and hence isomorphic with each other. Lie group. ‘~ j T HEOREM 4 . 2 . Let P(i, G) be a principuljbre bundle whose base ‘manifold M is connected and puracompact. Let iD(n) @O(U), u Q P, be the holonomy group and the restricted holonomy group of a connection I?

To the orthogonal group O(n), provided that M is paracompact. ‘: ‘. 6. connectec& Lie group-It. ;d@t product of any of its maximal com@act subgr\$ps H and a’ I Euclidean space (cf. Iwasawa ,[ 11). By the same reasoning as 1 above, the structure group G’qn be reduced to H, / I I. 7. Let L(M) be the bundle of linear frames over a manifold A4 cf dimension n. Let ( , ) be the natural inner product in R” for which e, = (1, 0, . , , O), . . , e, = (0,‘: . ‘,‘O, 1) are orthonormal and which is invariam by O(n) by the very definition of O(n).

Ac%‘e tia)r assume that Y, is defined and continuous for all t, -QD’ < t,< 00. : . n. GSXX~ETRY . G X R w ~OI~OWS. The’value of X at (a, t) A G ‘X R is by definition, eq&d to (I’&, (d/dt)&e Ta(G’) X’ T;(R), ‘wh& z is the natural coordinate system iri R. It is clear-that the infegral curve of x starting from (e, 0) is of the form (By, t) and’;rt, is the desired curve in G. & &t’ut is defined for all t, 0 ‘S; Z S lr Let vb 4 exp tX be a: lo& l+a*meter group of bkal transfbrmaths ,tif G. x7 R generated by X.