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Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/8158

Title: ON THE SADDLE LOOP BIFURCATION
Authors: DUMORTIER, Freddy
ROUSSARIE, R
Issue Date: 1990
Publisher: SPRINGER VERLAG
Citation: LECTURE NOTES IN MATHEMATICS, 1455. p. 44-73
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.
Notes: UNIV BOURGOGNE,DEPT MATH,UFR SCI & TECH,TOPOL LAB,CNRS,UA 755,F-21004 DIJON,FRANCE.DUMORTIER, F, LIMBURGS UNIV CENTRUM,UNIV CAMPUS,B-3610 DIEPENBEEK,BELGIUM.
URI: http://hdl.handle.net/1942/8158
DOI: 10.1007/BFb0085390
ISI #: A1990FH64000003
ISSN: 0075-8434
Type: Journal Contribution
Appears in Collections: Research publications

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