On the shifts of stable and unstable manifolds of a hyperbolic cycle under perturbation

The paper explicitly states the projector-sum formulae for the first-order shift of the intersection point of stable and unstable manifolds on a 2-D section (its Eq. (13)) and in 3-D (its Eq. (14)), and gives the higher-codimension generalization (its Eq. (15)), all derived from the normal-projection constraints on the intersection shift, with the observation that the sum of the two rank-1 projectors is invertible (on the section) or invertible on the normal plane in 3-D. These are the same relationships and logic used in the candidate solution, which formalizes the derivation by writing the linearized membership constraints P_u(δX×−δX^u)=0 and P_s(δX×−δX^s)=0, summing to (P_u+P_s)δX×=P_uδX^u+P_sδX^s, and inverting on the normal sum while fixing the tangent-direction gauge. The paper’s exposition is less explicit about restricting the inverse to the normal subspace and about the tangential gauge choice in 3-D and higher codimension, but the intended meaning matches what the model proves. In short, both present the same method; the model adds the missing linear-algebraic details and transversality hypotheses that the paper assumes implicitly. See Eqs. (13)–(15) and the accompanying text on perpendicular components and projector invertibility in the paper ; see also the definitions of perpendicular components in Eqs. (11)–(12) and the remark that only perpendicular components are intrinsic .