Resolve channel closure at the resonant end of the zero locus

Resolve the decomposition of third-harmonic closure into its contributing harmonic-feedback channels near the resonant end of the low-frequency zero locus, where the small-rotation expansion no longer applies.

Background

The paper analyzes a third-harmonic zero in a damped two-degree-of-freedom nonlinear absorber. At the baseline working point, the self-generated fifth harmonic provides the complex source direction needed to close the third-harmonic balance. However, near the resonant end of the continued zero locus, the response becomes strongly nonlinear and the fifth-harmonic propagator changes substantially, invalidating the small-rotation approximation used for channel attribution.

The authors explicitly state that they have not resolved the closure into channels in this regime. Thus, the relative contributions of the third-harmonic feedback, fifth-harmonic return, and other nonlinear pathways remain undetermined there.

References

At the resonant end, near \Omega\simeq1.05, |g_5| falls by a factor of 337 while the response amplitude rises and the orbit becomes strongly nonlinear; the zero is still there, but the small-rotation expansion behind Eq.~eq:em-sufficient no longer applies and we have not resolved the closure into channels there.

— Fifth-Harmonic Feedback Controls the Existence of a Third-Harmonic Zero  (2609.01508 - Sarkar, 1 Sep 2026) in Main text, paragraph beginning “That amplification is a property of the working point”

The scaling suggests, but does not demonstrate, an analogous role for an (m{+}2)\Omega channel at other odd harmonics.

— Fifth-Harmonic Feedback Controls the Existence of a Third-Harmonic Zero  (2609.01508 - Sarkar, 1 Sep 2026) in End Matter, paragraph beginning “Finite-amplitude closure and fold”