Bifurcation diagram and global phase portraits of a family of quadratic vector fields in class I

Release time:2020-06-03

DOI number:10.1007/s12346-020-00402-4

Journal:Qualitative Theory of Dynamical Systems

Key Words:Global phase portrait · Saddle connection · Bifurcation · Generalized normal sector · Rotated vector field

Abstract:We study a family of quadratic vector fields in Class I x'= y, y'= −x −αy +μx^2 − y^2, where (α,μ) ∈ R^2. To study the equilibria at infinity on the Poincaré disk of this system completely, we follow the method of generalized normal sectors of Tang and Zhang (Nonlinearity 17:1407–1426, 2004) and give further two new criterions, which allows us to obtain not only the qualitative properties of the equilibria but also asymptotic expressions of these orbits connecting the equilibria at infinity of this system. Further, the complete bifurcation diagram including saddle connection bifurcation curves of this system is given. Moreover, by qualitative properties of the equilibria, the nonexistence of limit cycle and rotated properties about α and μ, all global phase portraits on the Poincaré disk of this system are also obtained and the number is 19.

Co-author:Haibo Chen

First Author:Jia Man

Indexed by:Journal paper

Correspondence Author:Chen Hebai

Document Code:64

Document Type:J

Volume:19

Page Number:64-1-22

Translation or Not:no

Date of Publication:2020-06-03

Included Journals:SCI