By Bernard Candelpergher, Francine Diener, Marc Diener (auth.), Jean-Pierre Françoise, Robert Roussarie (eds.)

ISBN-10: 354046722X

ISBN-13: 9783540467229

ISBN-10: 3540535098

ISBN-13: 9783540535096

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**Extra resources for Bifurcations of Planar Vector Fields: Proceedings of a Meeting held in Luminy, France, Sept. 18–22, 1989**

**Example text**

0 ) = X ~ x - 7 ( # ) y ;,it is trivial to 55 prove t h a t the transition map Du(x) from a0 to #o has the form : Du(x) = x~(U)[1 + q)l(X, #)] (19) where ¢1 is a continuous function and ¢1(0, #) ------O. T h e transition map R~(x) from ~-o to ao is C ~ and can be e x p a n d e d : R~,(y) = a(tt) +/3(#)y[1 + ¢ ( y , #)] (20) where ¢, a,/3 are C ¢¢,/3(#) > 0 and ¢(y, #) = O(y). The Poincar~ map relative to a = a0 is equal to Rt, o Du. From (19), (20) we o b t a i n : P~,(z) = c~(#) +/3(#)x'Y(u)(1 + el(X, #))(1 -F ¢(xZ(#)(1 + ¢1(x, #)), #)) So :Pu(x) = (~(#) +/3(#)x~(U)(1 + ¢2(x, #)) (21) with again ¢~ a continuous function such t h a t 62(0, #) = 0.

Anal. , 71(1979),333-350. [16] P. Henrici, Apphed and compn~atwnal complex analysis, Vol. I, John Wiley and Sons, New York, 1974. [17] M. Herve, Several complex variables, Oxford Univ. Press, 1963. , Dover, New York, 1977. [19] N G. Lloyd, Lzmzt cycles of polynomzal systems, some recent developments, New Directions in Dynamical Systems, LMS Lecture Notes, Series No. 127, Cambridge Umverslty Press, 1988, 192-234. [20] N. G. Lloyd and 5. M. Pearson, Condztzons for a certre and the bzfurcatzon of hm,t cycles zn a 43 class of cubic systems, Preprint, The University College of Wales, ]990.

Thus, we must identify as explicitly as possible the dk under this hypothesis. This will be accomplished using the Structure Lemma. It is perhaps worthwhile to note that the determination of the zeros of dk(~, 0), which solves our bifurcation problem, does not depend on the global convergence of the series expansion for the displacement function. /=1 36 Thus, by composition, we obtain power series representations for the functions which appear in the Structure Lemma: ~(a(~)) = [ 2 ~ J , ~(~(~)) = k=l ~(a(~)) = ~ ~ , ~ , ~ e~, k=l ~ , ~ , ~(a(~)) = ~ ~,~.

### Bifurcations of Planar Vector Fields: Proceedings of a Meeting held in Luminy, France, Sept. 18–22, 1989 by Bernard Candelpergher, Francine Diener, Marc Diener (auth.), Jean-Pierre Françoise, Robert Roussarie (eds.)

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