Global dynamics of degenerate linear differential systems with symmetry and two parallel switching lines (Q2143190)

The authors investigate the global dynamics of a degenerate linear differential system with symmetry with respect to the origin, expressed in the following normal form: \[ \begin{cases} \dot x=\alpha_1x+\beta_1y+\gamma_1\textrm{sgn}(x+1)+\gamma_1\textrm{sgn}(x-1), \\ \dot y=\alpha_2x+\beta_2y+\gamma_2\textrm{sgn}(x+1)+\gamma_2\textrm{sgn}(x-1), \end{cases} \] where \[ \alpha_1\beta2 = \alpha2\beta1, \alpha_1^2 + \alpha_2^2 + \beta_1^2 + \beta_2^2 > 0, \gamma_1^2 + γ_2^2 > 0, \] endowed with two paralleled switching lines given by \[ \Sigma^- := \{(x, y) \in\mathbb{R}^2\,:\, x = -1\},\quad\Sigma^+:=\{(x, y)\in\mathbb{R}^2\,:\, x = 1\}. \] After analyzing the qualitative properties of all equilibria including infinity and the number of closed orbits, they obtain all global phase portraits on the Poincaré disc. From these main results, they find necessary and sufficient conditions for the existence of crossing limit cycles, crossing heteroclinic loops and sliding heteroclinic loops, respectively, and prove that the numbers of these three types of closed orbits are all at most 1. Moreover, switching lines maybe pseudo singular lines or boundary singular lines.