Dynamical analysis of a cubic Liénard system with global parameters (II)

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 , 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.

Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line

“…The Liénard second order differential equation (1) can be written as the equivalent planar differential system When F (x) and g(x) are polynomial functions in the variable x, the Liénard differential system (2) is called the generalized polynomial Liénard differential system has been studied extensively, see [14,38,41] for center conditions, [13,16,21,22,27,37] for the number of limit cycles, [1,40] for the amplitude of limit cycles, [15,26] for integrability conditions, [38,39] for isochronous conditions, and [2,3,17] for global phase portraits and bifurcation diagrams.…”

Global Analysis of a Liénard System with Quadratic Damping

“…In 2016, Llibre [10] studied the centers of the analytic differential systems and analyzed the focus-center problem. H. Chen and X. Chen [11][12][13] investigated the dynamical behaviour of a cubic Liénard system with global parameters, analyzing the qualitative properties of all the equilibria and judging the existence of limit cycles and homoclinic loops for the whole parameter plane. They gave positive answers to Wang Kooij's [14] two conjectures and further properties of those bifurcation curves such as monotonicity and smoothness.…”

Global phase portraits of the key pitchfork bifurcation

“…例如, 二次系统就有超过 2,000 多种不同拓扑结构的全局相图. 据此, 大部分已有文献都是对某些特殊系统的全局结构进行分析, 如 二次系统 [1][2][3][4][5][6][7][8] 、三次系统 [9][10][11][12][13][14][15][16] 、四次系统 [17][18][19][20] 、Liénard 系统 [21][22][23][24] 、Hamiltonian 系统 [25][26][27] 和 Lotka-Volterra 系统 [28] 等. 此外, 文献 [29] 详细介绍了如何利用平面多项式相图软件 P4 (planar polynomial phase portraits) 绘制全局相图.…”