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A near-unit multiplier enclosure does not decide stability

Statement

Suppose a validated transverse multiplier enclosure is the real interval $[0.99, 1.01]$ . It intersects the unit circle at $1$ and contains both a strictly stable scalar multiplier, for example $0.995$ , and a strictly unstable one, for example $1.005$ . A cluster count may establish how many multipliers the interval represents, but the strict Floquet separation hypothesis of Strict Floquet cluster enclosures certify orbital asymptotic stability is not established; the stability verdict is inconclusive.

Facts & Assumptions

Given: The interval enclosure $[0.99, 1.01]$ for a non-phase multiplier and the unit-circle separation rule.

[F1]

A validated shooting jet encloses a family of exact monodromy matrices and their multiplier data (Certified residual, Jacobian, trajectory, and monodromy enclosures for periodic shooting).

[F2]

The cluster procedure certifies algebraic root counts inside a polygon, while orbital stability requires strict non-phase separation from the unit circle (Finite complex-contour validation of Floquet multiplier clusters, A verified contour count encloses every Floquet multiplier in a cluster, Strict Floquet cluster enclosures certify orbital asymptotic stability).

Proof

1.1givenalgebra

Both $0.995$ and $1.005$ lie in $[0.99, 1.01]$ , but the first has modulus less than one and the second has modulus greater than one. Thus the interval alone admits opposite stability classifications, and it also intersects the unit circle at $1$ .

2.1F1F2step 1.1∎

A contour count can certify the number of roots enclosed without sharpening their location relative to the circle. Since the enclosure does not place every non-phase multiplier strictly inside the unit disk or strictly outside it, the hypotheses required for a stability verdict in [F2] are not met. The sound output is inconclusive.

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Sources

S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica 19 (2010) (standard reference, not scraped)
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Chapter 1, §1.5 (standard reference, not scraped)