Year: 2025
Type: article
Abstract: A real analytic differential system having a center at the origin of coordinates after a linear change of variables and a rescaling of the time can be written in one of the following three forms: $${\dot{x}}=-y+ X_2(x,y), \ {\dot{y}}=x+Y_2(x,y),$$ x ˙ = - y + X 2 ( x , y ) , y ˙ = x + Y 2 ( x , y ) , called a linear type center, $${\dot{x}}=y+ X_2(x,y), \ {\dot{y}}=Y_2(x,y)$$ x ˙ = y + X 2 ( x , y ) , y ˙ = Y 2 ( x , y ) called a nilpotent center, and $${\dot{x}}=X_2(x,y), \ {\dot{y}}=Y_2(x,y)$$ x ˙ = X 2 ( x , y ) , y ˙ = Y 2 ( x , y ) called a degenerate center, where $$X_2(x,y)$$ X 2 ( x , y ) and $$Y_2(x,y)$$ Y 2 ( x , y ) are real analytic functions without constant and linear terms, defined in a neighborhood of the origin. While there are many papers dedicated to study phase portraits of different classes of linear type centers, few papers studied the phase portraits of the nilpotent and degenerate centers. Here we classify the global phase portraits in the Poincaré disc of reversible nilpotent centers with cubic nonlinearities.
Authors: Leandro Bacelar, Jaume Llibre
Institutions: Universidade de São Paulo, Universitat Autònoma de Barcelona
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Awards: G1694479832, G3936534585, G4529001115
OpenAlex ID: W4411276482