Characterization of Invariance, Periodic Solutions and Optimization of Dynamic Financial Networks

The paper’s Theorem 1 states exactly that the region of attraction in orthant k is the set of initial conditions V(0) satisfying J[k](C^t(V(0) − V[k]) + V[k] − V̄) ≥ 0 for all t ≥ 0, and argues this by expressing the trajectory in orthant k as V(t) − V̄ = V[k] − V̄ + C^t(V(0) − V[k]) and enforcing orthant membership via J[k](V(t) − V̄) ≥ 0 (the statement and sketch are explicit in the PDF) . The candidate solution proves the same set equality and uses the same closed-form linear evolution within a fixed mode, adding a clean induction for sufficiency and a standard norm argument from 1^T C < 1^T to conclude C^t → 0, which is consistent with the paper’s Schur-stability assertion . There is no substantive conflict; the model’s proof is a slightly more detailed version of the paper’s sketch.