Minimal invariant regions and minimal globally attracting regions for toric differential inclusions

The paper proves that for sufficiently large δ, the region MF,δ constructed from a 2D fan F is both the minimal invariant region (Theorem 5.1) and the minimal globally attracting region for strict solutions (Theorem 6.1). It does so via a detailed construction (Steps 1–5), a boundary-pointing argument for invariance that invokes Nagumo/Blanchini (Proposition 5.11), and a careful global-attraction proof using a convex Lyapunov-like function and subgradient estimates (Section 6) . The candidate solution reaches the same conclusions and tracks the paper’s structure closely (large-δ geometry; invariance; minimality; global attraction; minimal global attraction), but with different proof tactics in two places. First, it asserts a stronger “tangent-cone equality” along interior points of straight connecting segments, whereas the paper only needs and proves inward-pointing containment; the model’s equality claim at those points is not generally correct even though containment—and hence invariance—still holds . Second, for global attraction, the model sketches local distance-like functionals for r ∈ {0,1}, while the paper gives a rigorous convex subgradient argument ensuring uniform decay on compacts for strict solutions . With these caveats, the model’s proof outline is substantively correct and reaches the same results; the paper’s argument is complete and rigorous.