Two Results for the Omega Limit Sets of Dynamical Systems

The paper’s Theorem 1 and Theorem 2 are stated and proved correctly. The proof of Theorem 1 constructs p(t)=∇V(φ(t,x0))·f(φ(t,x0)), shows p(t) never vanishes and hence V∘φ is monotone, and then treats the two possibilities V→+∞ or V→L<∞ to conclude ω(x0)⊂A and thus ω(x0)⊂M; the argument carefully avoids needing V to be bounded on Ω or |∇V·f| to be bounded away from 0 near ω-sets outside A . For Theorem 2, the paper first proves S≠∅ and then shows ω(x0)∩S≠∅ via a liminf-distance-to-S argument; finally, defining A={x: φ(t,x)∈S for some t≥0} and invoking Theorem 1 yields ω(x0)⊂M . By contrast, the model’s Theorem 1 proof has two gaps: (i) it asserts ω(y)⊂Ω\A from y∉A, which need not hold if A is not closed; (ii) it argues that V must stay bounded on Ω merely because Ω is compact, but V is only assumed C^1 on Σ⊇Ω\A, so V can blow up near A. These steps are precisely what the paper’s case-splitting avoids, hence the model proof is invalid as written for Theorem 1. The model’s Theorem 2(i) argument (compact ω-set, nonvanishing ∇V·f on it implies uniform gap from 0 and thus contradiction) is fine, but its Theorem 2(ii) conclusion depends on its flawed Theorem 1.