Determine the global connectivity of the third-harmonic zero set

Determine the global connectivity of the zero set of the driven-mass third-harmonic response, including whether the traced finite-amplitude branch connects to the weak-drive or zero-frequency endpoints under the stated parameter constraints.

Background

The paper establishes a finite-amplitude branch of exact third-harmonic zeros and derives conditions constraining possible weak-drive limits. The authors exclude certain nearby leading-order crossings and note that reaching the endpoints would require continuation through parameter regimes not observed on the traced component.

The global topology of the complete zero set is nevertheless unresolved. In particular, it is unknown whether the traced component connects to the zero-frequency endpoint, the weak-drive endpoint, or other components through parameter values outside the continuation examined in the paper.

References

Reaching either endpoint would require a sign-changing continuation through \beta_1=0, not observed on the traced component; global connectivity of the zero set is not established, nor is the joint corner (F,\Omega)\to(0,0) or a limit with \beta_1\to\infty.

— Fifth-Harmonic Feedback Controls the Existence of a Third-Harmonic Zero  (2609.01508 - Sarkar, 1 Sep 2026) in End Matter, “Phase reduction and a closure criterion” subsection

Reaching either endpoint would require a sign-changing continuation through \beta_1=0, not observed on the traced component; global connectivity of the zero set is not established, nor is the joint corner (F,\Omega)\to(0,0) or a limit with \beta_1\to\infty.

— Fifth-Harmonic Feedback Controls the Existence of a Third-Harmonic Zero  (2609.01508 - Sarkar, 1 Sep 2026) in End Matter, “Phase reduction and a closure criterion” subsection

This is a local statement; global uniqueness and global structural stability are not established.

— Fifth-Harmonic Feedback Controls the Existence of a Third-Harmonic Zero  (2609.01508 - Sarkar, 1 Sep 2026) in End Matter, “Transversality” subsection