By Claude Godbillon (auth.)

These notes are an elaboration of the 1st a part of a path on foliations which i've got given at Strasbourg in 1976 and at Tunis in 1977. they're involved normally with dynamical sys tems in dimensions one and , particularly on the way to their purposes to foliated manifolds. a big bankruptcy, despite the fact that, is lacking, which might were facing structural balance. The ebook of the French version was once re alized by-the efforts of the secretariat and the printing place of work of the dept of arithmetic of Strasbourg. i'm deeply thankful to all those that contributed, specifically to Mme. Lambert for typing the manuscript, and to Messrs. Bodo and Christ for its copy. Strasbourg, January 1979. desk of Contents I. VECTOR FIELDS ON MANIFOLDS 1. Integration of vector fields. 1 2. normal conception of orbits. thirteen three. Irlvariant and minimaI units. 18 four. restrict units. 21 five. course fields. 27 A. Vector fields and isotopies. 34 II. THE neighborhood BEHAVIOUR OF VECTOR FIELDS 39 1. balance and conjugation. 39 2. Linear differential equations. forty four three. Linear differential equations with consistent coefficients. forty seven four. Linear differential equations with periodic coefficients. 50 five. edition box of a vector box. fifty two 6. Behaviour close to a novel aspect. fifty seven 7. Behaviour close to a periodic orbit. fifty nine A. Conjugation of contractions in R. sixty seven III. PLANAR VECTOR FIELDS seventy five 1. restrict units within the airplane. seventy five 2. Periodic orbits. eighty two three. Singular issues. ninety four. The Poincare index.

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Let y denote a periodie orbit of period T of a 60 veetor field X of elass e r , and let y be a point of~, and N a submanifold of eodimension I of M whieh is transverse to X and satisfies N n ~ =ly~. The maximal flow generated by X is denoted by ~ = (ept). •. (t,O'Y2'· of y in whieh N is defined by YI= ··,ym) =0 o. Then the equation may be solved for t near 1", yielding anumerieal funetion a on an open neighbourhood U of y in N whieh has the same elass of differentiability as X and satisfies the following properties: i) a(y) = 1"; ii) t(a(z),z)(N forallz(U.

I) Letting Z(t,t o ) denote the resolvent of (E) and choosing for X the fundamental matrix solution z(t,O) we obtain D= Z(~,O) . Hence (E) has a non-vanishing and periodicsolution of period ~ if and only if 52 one of its eharaeteristie numbers equals 1: for v= eCO) this eondition is indeed equivalent to Dv = Z(T, 0) v = v. ii) If (E) is considered as an equation of period 2T then X(t+2T) X(t)D 2 . Henee in this ease the matrix B may be chosen to be real. iii) The similarity elass of the matrix D is invariant under linear and periodie ehanges of variables (of period T).

4. A smooth map t~X(t) of J into the space of m X m square matri- ees is then ealled a fundamental matrix solution of (E) if the eolumns of X form m independent solutions of (E) on J. Such a map is a solution of the linear matrix differential equation X' =A(t)X. Henee we deduee: i) if X and Y are two fundamental matrix solutions of (E) we have X(t)X(to)-l=Y(t)Y(to)-l ii) e of (E) for all t and to in J; for a fundamental matrix solution X of (E) and a solution for all t and to in J. For these reasons the matrix funetion Z(t,to ) = X(t)X(t o ) -1 whieh is independent of the choice of the fundamental matrix solution 46 X, is called the resolvent of (E).