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Conditional van der Pol stability from a validated enclosure

Statement

For the van der Pol system x˙=y, y˙=μ(1−x2)y−x, fix μ=1. If a future run of Finite-budget phase-fixed periodic-orbit correction and validation certifies a positive-period nonconstant orbit, and pairwise disjoint Floquet cluster polygons have counts summing to 2, with one polygon containing multiplier 1 at count one and the other count-one polygon strictly inside the unit disk, then that certified orbit is orbitally asymptotically stable. This example states a conditional implication only; it supplies no computed interval, orbit, or numerical certificate.

Facts & Assumptions

Given: The parameter value and the hypothetical successful root and cluster certificate records stated above.

[F1]

A successful validated-root corrector record includes a complete Krawczyk partition with one unique-root box and every other box root-free, outward residual/Jacobian bounds, a nonsingular preconditioner, strict inclusion and a separate contraction bound, and a field component excluding zero. Together with its successful validated trajectory/monodromy jet, these certify a unique phase-fixed nonconstant periodic orbit in its box (Finite-budget phase-fixed periodic-orbit correction and validation, A Krawczyk inclusion certifies a unique phase-fixed nonconstant periodic orbit).

[F2]

The Floquet cluster theorem certifies the algebraic multiplier counts for every matrix in the validated family (Finite complex-contour validation of Floquet multiplier clusters, A verified contour count encloses every Floquet multiplier in a cluster).

[F3]

A complete cluster family with a single neutral multiplier and all other clusters strictly inside the unit disk implies orbital asymptotic stability (Strict Floquet cluster enclosures certify orbital asymptotic stability).

Proof

1.1F1given

Under the stated hypothetical corrector output, [F1] supplies the exact periodic orbit and a monodromy matrix in the enclosure family.

2.1F2F3step 1.1given∎

The assumed cluster certificates apply to that matrix by [F2]. The van der Pol field is polynomial and hence C2. The neutral multiplier cannot be on a certified polygon boundary, so it is interior to its count-one polygon. The other filled polygon is compact and lies strictly inside the unit disk, so its maximum modulus r satisfies r<1; take σ=1−r>0. The two counts sum to the state dimension 2, giving the complete cluster hypotheses of [F3], so the orbit is orbitally asymptotically stable. No numerical conclusion is claimed without those certificates.

Depends on

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