2007.08879
On the Matrix Measure as a Tool to Study the Stability of Linear and Nonlinear Dynamical Systems on Time Scales
Giovanni Russo, Fabian Wirth
correctmedium confidence
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s Theorem 7.1 establishes a contraction estimate for nonlinear time-scale systems under a uniform negative bound on the matrix measure of the Jacobian, using a variational equation w(t,r)=∂x/∂r and Coppel’s inequality on time scales. The candidate solution proves the same estimate via a direct mean-value representation in x, a pointwise matrix-measure bound v^Δ≤m(B,t)v, and convexity of the matrix measure. Both arguments are correct; the candidate’s proof is sound but should explicitly assume fx∈R (regressive) as in the paper.
Referee report (LaTeX)
\textbf{Recommendation:} no revision
\textbf{Journal Tier:} strong field
\textbf{Justification:}
The nonlinear contraction theorem on time scales and its supporting linear theory (time-scale matrix measures and Coppel’s inequality) are correct and well-integrated. The candidate’s proof offers a valid alternative derivation leveraging the mean-value representation and convexity of the matrix measure. No substantive issues were found; only minor presentational suggestions are offered.