By Bernard Candelpergher, Francine Diener, Marc Diener (auth.), Jean-Pierre Françoise, Robert Roussarie (eds.)
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Extra resources for Bifurcations of Planar Vector Fields: Proceedings of a Meeting held in Luminy, France, Sept. 18–22, 1989
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Bautin, On the number of hmzt cycles whzch appear wz~h the vamatwn of coe]ficzents from an equdzbmum posztzon of focus or center type, Amer. Math. Soc. 'IYansl , 100(1954), 1-19.  P. F. Byrd and M. D. Friedman, Handbook of elhptac zntegrals for engzneers and physzczsts, Springer-Verlag, Berlin, 1954.  C. Chicone and M. Jaeobs~ Bzfurcatwn of critical pemods for plane vector fields, Trans. Amer. Math. , 312(1989), 433-486  C. Chicone and M. 5aeobs, Bzfurcatwn of hm~t cycles from quadratic ~sochrones, to appear in 5.
Then X . and Y. are (C °, Id)-equivalent. Because of the theorems 1 and 2 we obtain that 3' 6]0, I[U]I, cx~[ is a modulus of stability for (C °,C~)-equivalence, with 1 _< r _< ec. By this we mean that the (C °, C~)-equivalence classes are completely determined by 7. A C 1 assumption is sufficient to formulate the genericity of the family. In the proofs of the theorems we will use the C a assumption for technical reasons. In particular we use it to establish a Cl-linearisation of the Poincar6-map of the saddle loop at the value # = 0: Theorem 3 Let X be a C ~ vector field with a saddle loop P at a saddle point s with ratio of hyperboticity 7 7~ 1.