Solving nonlinear ordinary differential equations using the invariant manifolds and Koopman eigenfunctions

The paper’s Theorem 2 states exactly the claim under review: for invariant-manifold generators Mi with d/dt Mi = Mi Ni, the product ϕ = ∏i Mi^{pi} is a Koopman eigenfunction if and only if the weighted sum ∑i pi Ni is a constant λ; the proof differentiates ϕ and obtains dϕ/dt = (∑i pi Ni) ϕ, which yields the eigenfunction relation when the sum is constant . The candidate solution reproduces the same product/chain-rule derivation and explicitly gives both directions (including the converse) while carefully stating domain conditions (branches, zeros) that the paper glosses over. Minor gaps in the paper’s exposition include not explicitly writing the “only if” direction and not stating domain/branch assumptions for complex exponents, but these do not alter correctness of the main claim; the model’s write-up supplies these details. Background definitions used (Koopman eigenfunction, invariant-manifold generators M, N) align with the paper’s setup .