توصيفگر ها :
سيستم هاي چند جمله اي مسطح , يكتايي و هذلولوي بودن سيكل حدي , چند سيكل , انشعاب , نماي فاز در ديسك پوانكاره , تابع دولاك , پايداري نقطه پوچ توان و دامنه جذاب
چكيده انگليسي :
Bifurcation diagram and stability for a ane parameter family of planar vector elds Mohsen Ahmadi 2016 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Hamidreza Zohouri Zangeneh hamidz@cc iut ac ir 2010 MSC Keywords Planar polynomial system Uniqueness and hyperbolicity of a limit Cycles Polycycles Bifurcation Phase portrait on a Poincare disk Dulac function Stability Nilpotent point Basin of attraction Abstract This thesis is an extension and generalization of the work s done based on paper 14 by J D Garc a Salda a A Gasull and H Giacomini We consider the one parameter family of planar quintic system x y 3 x3 y x my 5 introduced by A Bacciotti in 1985 It is known that it has at most one limit cycle and that it can exist only when the parameterm is in 0 36 0 6 By using the Bendixson Dulac theorem me give a new uni ed proof of all the previous results We shirink this interval to 0 547 0 6 and we prove the hyperbolicity of the limit cycle Furthermore we considerthe question of the existence of polycycles The main interest and di culty for studying this family is that itis not a semi complete family of rotated vectoor elds When the system has a limit cycle we also determineexplicit lower bounds of basin of attraction of the origin Finally We answer an open question about the changeof stability of the origin for an extension of the above systems A Bacciotti during a conference about the stability of analytic dynamical systems held in Florence in 1985 proposed to study the stability of the origin of the following quintic system x y 3 x3 1 y x y 5 m R Two years later Galeotti and Gori in 12 published an extensive study of 1 They proved that system 1 hasno limit cycles when m 0 36 6 otherwise it has at most one Their proofs are mainly basedon the study of the stability of the limit cycles which is controlled by the sign of its characteristic exponent together with a transformation of the system using a special type of adapted polar coordinates Their proof of
