Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields [ Back ]

Date:
06.10.14   
Times:
15:30 to 16:30
Place:
UAB - Dept. Matemàtiques (C1/-128)
Speaker:
Ilker Evrim Çolak
University:
Universitat Autònoma de Barcelona

Abstract:

If an analytic system has a center, then after an affine change of variables and a rescaling of the time variable, it can be written in one of the following three forms:

i) $x'=-y+P(x,y),\ y'=x+Q(x,y),$ called a linear type center,

ii) $x'=y+P(x,y), y'=Q(x,y),$ called a nilpotent center,

iii) $x'=P(x,y),\ y'=Q(x,y),$ called a degenerate center,

where P(x,y) and Q(x,y) are real analytic functions without constant and linear terms, defined in a neighborhood of the origin.

In [1] we had provided the global phase portraits on the Poincaré disk of all the Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields. This time we present the bifurcation diagrams for these phase portraits. If time permits we shall also talk about how this work extends to nilpotent centers as well. This is a joint work with Jaume Llibre and Claudia Valls.

[1] I.E. Colak, J. Llibre and C. Valls “Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields”, J. Differential Equations 257 (2014), 1623-1661.