The existence of quasi-periodic invariant tori and double Hopf bifurcation of van der Pol’s oscillator with delayed feedback

Xuemei Li, Bochao Yu

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

Both the paper and the candidate solution proceed via the same pipeline: 4D center-manifold reduction at a nonresonant double Hopf point, parameter-dependent normal form, amplitude–phase (polar) decomposition, existence of 2-tori and a secondary Hopf (torus–Hopf) in the truncated amplitude subsystem, and KAM persistence of the 2- and 3-tori for most parameter values under a twist and nonresonance scheme. The candidate’s twist determinant reduces exactly to the paper’s condition (32) once the special structure p12=2p11 and p21=2p22 (and similarly for qjk) is used. Minor differences are mainly notational or in the choice of KAM references; no substantive logical conflict was found.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

Technically careful work deriving a parameter-dependent normal form at a nonresonant double Hopf for a delayed van der Pol oscillator and proving KAM persistence of 2- and 3-tori. The main steps accord with standard RFDE/normal-form/KAM technology, and explicit coefficient formulas are valuable. Minor clarifications in Section 3.2 on the exact nondegeneracy used for 3-tori and on reconciling the full angle equations with the simplified truncated system would improve readability.