On the saddle loop bifurcation

Abstract

It is shown that the set of C-infinity (generic) saddle loop bifurcations has a unique modulus of stability gamma epsilon]0, 1[union cup]1, infinity[ for (C(o), C(r))-equivalence, with 1 less-than-or-equal-to r less-than-or-equal-to infinity. We mean for an equivalence (x,mu) --> (h(x,mu), psi(mu)) with h continuous and psi of class C(r). The modulus gamma is the ratio of hyperbolicity at the saddle point of the connection. Already asking psi to be a lipeomorphism forces two saddle loop bifurcations to have the same modulus, while two such bifurcations with the same modulus are (C(o), +/- Identity)-equivalent. A side result states that the Poincare map of the connection is C1-conjugate to the mapping x --> x-gamma. In the last part of the paper is shown how to finish the proof that the Bogdanov-Takens bifurcation has exactly two models for (C(o), C-infinity)-equivalence.