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In [Chen et al., 2020], the third author and other coauthors studied global dynamics of the following system: ẋ = y − b 1 x − b 2 2 (| x + 1 | − | x − 1 |) , ẏ = − b 3 x + b 2 2 (| x + 1 | − | x − 1 | ) − b 4 , in the parameter region { (b 1 , b 2 , b 3 , b 4) ∈ ℝ 4 : b 2 < 0 , b 4 ≠ 0 }. To study completely the piecewise linear system, we consider the parameter region { (b 1 , b 2 , b 3 , b 4) ∈ ℝ 4 : b 2 > 0 , b 4 ≠ 0 } in this paper. Firstly, we study the local dynamics, such as the bifurcations of equilibria (including the equilibrium at infinity). Secondly, the number and stability of limit cycles are studied completely. Then, we analyze the existence of upper and lower saddle connections and homoclinic loops. Moreover, we show that there are no heteroclinic loops in this parameter region. Finally, we give the bifurcation diagram and all global phase portraits on the Poincaré disc are given as well as some numerical examples. [ABSTRACT FROM AUTHOR]

Chen [2016a, 2016b] studied global dynamics of the Filippov systems ẍ + (α + β x 2) ẋ + x ± sgn (x) = 0 , respectively. To study the global dynamics of ẍ + (α + β x 2) ẋ ± x ± sgn (x) = 0 completely, since the dynamics of ẍ + (α + β x 2) ẋ − x − sgn (x) = 0 is very simple, we are only interested in the global dynamics of ẍ + (α + β x 2) ẋ − x + sgn (x) = 0 in this paper. Firstly, we use Briot–Bouquet transformations and normal sector methods to discuss these degenerate equilibria at infinity. Secondly, we discuss the number of limit cycles completely. Then, the sufficient and necessary conditions of existence of the heteroclinic loop are found. To estimate the upper bound of the heteroclinic loop bifurcation function on parameter space, a result on the amplitude of a unique limit cycle of a discontinuous Liénard system is given. Finally, the complete bifurcation diagram and all global phase portraits are presented. The global dynamic property of system ẍ + (α + β x 2) ẋ − x + sgn (x) = 0 is totally different from systems ẍ + (α + β x 2) ẋ + x ± sgn (x) = 0. [ABSTRACT FROM AUTHOR]