2012.00837
ORDER REDUCTION OF NONLINEAR QUASI-PERIODIC SYSTEMS SUBJECTED TO EXTERNAL EXCITATIONS
Sandesh G. Bhat, Susheelkumar Cherangara Subramanian, Sangram Redkar
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
Both the paper and the model solve the same invariance equation in Lyapunov–Perron coordinates and construct a quasi-periodic invariant manifold (graph) by harmonic balance, obtaining order-by-order homological equations whose solvability hinges on small-divisor non-resonance conditions. The paper’s “reducibility conditions” at quadratic, linear/mixed, and higher orders ((39), (43), (52), (55)) match the model’s denominators at degrees 2, 1, and 3+, respectively, and its linear resonance condition (45) matches the model’s degree-0 denominators with external forcing included. The reduced-order dynamics obtained by restricting to the manifold (67) coincide with substituting z_s = H_r(z_r,t) into the z_r-equations, as the model states. Both accounts are formal (no convergence or analytic regularity is proved), so they align in scope and method.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The manuscript correctly derives order-by-order homological equations for a quasi-periodic invariant manifold in L–P coordinates and identifies precise small-divisor conditions ensuring formal solvability. The contribution is practical and consistent with established methods, but assumptions should be stated more explicitly, and a brief discussion of (non-)convergence would strengthen the presentation.