Versal Deformation of Singularities and Its Applications to Strange Attractors

Abstract

The aim of this article is to present a systematic mathematical method for the study of global bifurcation diagrams of families of autonomous ordinary differential equations. We employ the normal form and versai family theory developed in USHIKI[22], which is an improved version of classical normal form theory for singularities of vector fields. The classical theory for normal forms is known and has been employed in many authors to study the bifurcation of vector fields. See POINCARE[16][17][18], BIRKHOFF[6], ARNOLD[2][3][4][5], TAKENS[21] and BROER[8] for the classical theory and its modern version. See ARNOLD[4][5], BOGDANOV[7], LANGFORD[13], LANGFORD-IOOSS[14], GUCKENHEIMER[9], HOLMES[10][11], and ARNEODO-COULLET-SPIEGEL-TRESSER[1] for some applications of the normal forms theory to bifurcation problems.KeywordsVector FieldNormal FormLorenz SystemBifurcation ProblemVersal FamilyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.