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Extra info for Bifurcations of Planar Vector Fields: Proceedings of a Meeting held in Luminy, France, Sept. 18–22, 1989
Becomes: y 2 ( x ) =/7 + xg(~')(1 + (~) with ~ ( ~ ' ) = ~,a-~(~), ¢ ( x , / 7 ) = ¢ ( x , a - ~ ( ~ ) ) Clearly, "~ is differentiable at 0 and (~(x,~) verifies (~(0,~) - (~(x,0) =-=-0. We can choose domains [0,#0], [0,x0] to apply the "Case A " : f;~ is (C °, Id)-conjugate to + z ~(°) = t7 + x~. If we return to the parameter # : ]~. is (C °, Id)-conjugate to a(#) + x ~. Now, because a has the necessary properties, we can apply the "Case B" to a ( # ) + x ~" and F~. Finally we have that f . is (C °, Id)-conjugate to F..
4. (i) The number of periodic trajectories of $~(o) at which a continuous local family of limit cycles emerges in the system g~(¢) is bounded by the number of zeros (counted with multiplicity) of an arbitrary linear combmatwn of the functions B1, t73 and Bs. (ii) IfA~(e)A3(e) -- 0, then there is at most one periodic trajectory orgy(o) at which a local family of limit cycles emerges in the system ~(~). Moreover, if O < Go < co, there is an analytic curve e ~ ~(~) such that a continuous local family of limit cycles emerges at Go in the system g~(~).
16] P. Henrici, Apphed and compn~atwnal complex analysis, Vol. I, John Wiley and Sons, New York, 1974.  M. Herve, Several complex variables, Oxford Univ. Press, 1963. , Dover, New York, 1977.  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.  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.