On the shifts of orbits and periodic orbits under perturbation and the change of Poincaré map Jacobian of periodic orbits

The paper derives δx_cyc = −[DP^m − I]^{-1} δP^m for maps (its Eq. 14) and δ_⊥x_cyc = [I − (I − b̂ b̂^T) DXT]^{-1} (I − b̂ b̂^T) δXT for flows (its Eq. 16), under the hyperbolicity condition excluding the unit eigenvalue except along the flow tangent, which ensures invertibility and uniqueness. These statements and their intended scope are explicit in the text (Eq. 14 and surrounding discussion; Eq. 15–16; hyperbolicity definition) . The model’s solution reproduces the same results via the same core idea: differentiate the periodicity condition and solve the linear system; for flows, include a potential period shift δT and then project onto the orthogonal Poincaré plane so that the δT-term cancels, consistent with the paper’s note that period changes can introduce terms but are neutralized by the projection used for Eq. 16 .