Dynamical analysis of a cubic Liénard system with global parameters (II). (12th May 2016)

• Record Type: Journal Article
• Authors: Chen, Hebai; Chen, Xingwu
• Is Part Of: Nonlinearity. Volume 29:Number 6(2016:Jun.)
• Page Start: 1798
• Page End: 1826
• Publication Date: 2016-05-12
• Subjects: homoclinic loop -- Lienard system -- limit cycle -- rotational vector field; 34C07 -- 34C23 -- 34C37 -- 34K18; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/6/1798
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this paper, we continue to study the global dynamics of a cubic Liénard system for global parameters in the case of three equilibria to follow (2015 Nonlinearity 28 3535–62), which deals with the case of two equilibria. We first analyse qualitative properties of all equilibria and judge the existences of limit cycles and homoclinic loops and their numbers. Then we obtain the bifurcation diagram and all phase portraits as our main results. Based on these results, in the case of three equilibria a positive answer to conjecture 3.2 of (1998 Nonlinearity 11 1505–19), which is about the existence of some function whose graph is exactly the surface of double limit cycles, is obtained. Moreover, a parameter region for the nonexistence of figure-eight loops is given theoretically to compensate for previous numerical results and is illustrated numerically.