STAYING THE COURSE: LOCATING EQUILIBRIA OF DYNAMICAL SYSTEMS ON RIEMANNIAN MANIFOLDS DEFINED BY POINT-CLOUDS

The paper correctly identifies that, away from equilibria and under a pointwise invertibility assumption on ∇X, the nullspace of ∇Y is one-dimensional and defines a line field (Proposition 2.6; see also the coordinate formula A(Y)=(X^T X)^{-1/2} Q A(X)), aligning with the model’s kernel characterization via (∇X)^{-1}X and the existence of (local) generalized isoclines obtained by integrating a locally oriented unit section of that line field . However, two central claims in the paper are unsupported or stated without necessary hypotheses: (i) the Euclidean Proposition 2.3 asserts that near any isolated equilibrium, every unit direction V is realized by points with X(x)=λV, but this fails for degenerate equilibria; the model correctly inserts the needed nondegeneracy (invertible DX at the equilibrium) and supplies a degree-theoretic proof of surjectivity, remedying the gap . (ii) The paper’s statement that generalized isoclines “end at an equilibrium” is asserted narratively (e.g., in conclusions and examples) but no rigorous continuation/compactness proof is given; the model provides the standard ODE argument on compact manifolds to justify it . Thus, on the main mathematical points, the model’s solution is correct and complete under explicit hypotheses, while the paper’s statements are either missing assumptions or lack proofs.