4 problems

Global stability conjecture. The equilibrium $E^{s}$ is globally asymptotically stable for all parameters.

Let $f$ be a Filippov system with boundary equilibrium $x^*={\bf 0}$, and let $F$ be its reduced system. Assume the setting and hypotheses of the paper, including $c_1<0$. Lyapunov…

Two-crossing-periodic-orbits conjecture. System $(\mathrm{DPWL})$ has at most two crossing periodic orbits.

Let $Z_0=(X_0,Y_0)$ be a Filippov system around a homoclinic-like loop $\Gamma_0$ at a fold-regular singularity, and suppose that its first return map has a hyperbolic fixed point…