What is the graph of a dynamical system?

The paper’s Theorem 1 states that if the global attractor G is connected, then the chain graph is connected, and gives a four-case proof in Section 7.3. However, two key steps are left unjustified: (i) in Case M the argument jumps from the existence of points in C arbitrarily close to N to the existence of a weak edge from some node of M to some node of N, without constructing the required ε-chains or citing a lemma that would imply this (see “Since G is connected … Hence, there is a weak edge …” in the proof of Theorem 1) ; and (ii) in the Remaining Case the sets M+ and N+ are asserted to be closed without providing the needed upper semicontinuity statements for α- and ω-limit sets (also in ). By contrast, the candidate solution supplies a complete, stand-alone proof: it gives an explicit ε-chain lemma showing how an ω-limit set intersecting both node blocks creates a directed edge, proves upper semicontinuity of x↦ω(x) on compact G, and then uses these to produce a separation of G by disjoint open sets—contradicting connectedness. The model also cleanly reduces continuous time to the time-1 map, in line with the paper’s intent around Proposition 4, whose full details are likewise deferred in the paper . Hence the theorem is correct, but the paper’s proof is incomplete, whereas the model’s proof is complete and correct.