EXISTENCE OF PERIODIC ORBITS FOR PIECEWISE-SMOOTH VECTOR FIELDS WITH SLIDING REGION VIA CONLEY THEORY.

The paper proves: (i) the forward Filippov trajectories of a PSVF with sliding/crossing and tangencies of types A1–B2 on a closed 3-manifold glue to a continuous semiflow (Theorem 3.16), and (ii) if an isolating neighborhood admits a Poincaré section and the Conley index exhibits the stated parity pattern, then there is a periodic orbit (Theorems 2.12 and 3.21). The candidate solution gives a higher-level but logically consistent proof sketch: (a) constructs the Filippov sliding field and argues continuity/semigroup by uniqueness across crossing, sliding, and allowed tangencies; (b) derives the periodic orbit from a mapping–torus/long exact sequence description of the Conley index and a discrete Conley index–Lefschetz argument. The candidate omits some hypotheses explicitly stated in the paper (compact attraction on an ANR, dimension ≤ 3, and exclusion of simultaneous sliding and escape), but these either hold automatically in the paper’s setting or are clearly intended. Hence both are correct; the proofs differ in style and level of detail.