Uniqueness and Stability of Limit Cycles in Planar Piecewise Linear Differential Systems Without Sliding Region

The paper proves that a two-zone planar piecewise linear system without sliding has at most one limit cycle, which is hyperbolic and stable iff ξ<0, unstable iff ξ>0 (Theorem 1) , under the canonical form and an integral characterization of Poincaré half-maps (see (1)–(5), (6)–(9)) . It derives derivative formulas for the half-maps and their asymptotics (Propositions 6 and 8) that underpin uniqueness and stability in Section 5 . The candidate solution reaches the same end conclusions but contains critical gaps and mis-specifications: (i) it treats both half-maps as forward-time first-hits, whereas the standard (and the paper’s) setup uses a forward half-map on one side and a backward half-map on the other; cf. the explicit “Backward Poincaré Half-Map” definition (8) . (ii) It asserts P′(u0)→1 at the boundary of the domain (|u0|→∞) to eliminate the integration constant in ln P′, but the paper’s precise asymptotics show half-maps approach nontrivial slopes at infinity (Proposition 8), so this step is generally false . Without a justified basepoint, the claimed uniform inequality P′(u)≶1 and the monotonicity of the displacement D(u)=P(u)−u do not follow. Consequently, while the conclusion mirrors Theorem 1, the model’s proof is incomplete/incorrect where it needs boundary control and time-orientation of the half-maps.