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چكيده انگليسي :

Generalized Hopf Bifurcation for PlanarVector Fields via the Inverse Integrating Factor Reyhane Hosseiny r hoseyni@math iut ac ir 27 8 2015 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Rasoul Asheghi r asheghi@cc iut ac ir 2010 MSC 37G15 37G10 34C07 Keywords inverse integrating factor generalized Hopf bifurcation Poincar map limit cycle enilpotent focus Abstract In this thesis we study the maximum number of limit cycles that can bifurcate from a focussingular point p0 of an analytic autonomous di erential system in the real plane under an analyticperturbation We consider p0 being a focus singular point of the following three types non degenerate degenerate without characteristic directions and nilpotent In a neighborhood of p0 the di erential system can always be brought by means of a change to generalized polar coordinates r to an equation over a cylinder in which the singular point p0corresponds to a limit cycle 0 This equation over the cylinder always has an inverse integratingfactor which is smooth and non at in r in a neighborhood of 0 We de ne the notion of vanishingmultiplicity of the inverse integrating factor over 0 This vanishing multi plicity determines themaximum number of limit cycles that bifurcate from the singular point p0 in the non degenerate caseand a lower bound for the cyclicity otherwise Moreover we prove the existence of an inverse inte grating factor in a neighborhood of many types of singular points namely for the three types of focusconsidered in the previous paragraph and for any isolated singular point with at least one non zeroeigenvalue The relation between limit cycles of planar di erential systems and the inverse integrating factor was rst shown in an article of Giacomini Llibre and Viano appeared in 1996 From that moment on many research articles are devoted to the study of the properties of the inverse integrating factor andits relation with limit cycles and their bifurcations This thesis is a summary of all the results aboutthis topic We include a list of references together with the corresponding related results aiming atbeing as much exhaustive as possible The thesis is nonetheless self contained in such a way that allthe main results on the inverse integrating factor are stated and a complete overview of the subject isgiven Each section contains a di erent issue to which the inverse integrating factor plays a role theintegrability problem the center problem vanishing set of an inverse integrating factor bifurcation oflimit cycles from either a period annulus or from a monodromic limit set and some generalizations we address the problem of existence of inverse integrating factors for an analytic planar vector eldin a neighborhood of its nonwandering sets It is proved that there always exists a smooth inverseintegrating factor in a neighborhood of a limit cycle obtaining a necessary and su cient condition forthe existence of an analytic one This condition is expressed in terms of the Ecalle Voronin modulusof the associated Poincare map The existence of inverse integrating factors in a neighborhood of anelementary singularity is also established and we give the rst known examples of analytic vector elds in R2 not admitting a C inverse integrating factor in any neighborhood of either a limit cycleor a weak focus Moreover it is shown that a C 1 inverse integrating factor of a C 1 planar vector eldmust vanish identically on the polycycles which are limit sets of its ow We show that the inverse integrating factor de nes an ordinary di erential equation for the transitionmap along the orbit When the regular orbit is a limit cycle we can determine its associated Poincare

حسيني قرچه قيائي، ريحانه

انشعاب هاپف تعميم يافته براي ميدان هاي برداري مسطح با عامل انتگرال ساز وارون

نگاشت پوانكاره , سيكل حدي , كانون پوچ توان