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A Krawczyk inclusion certifies a unique phase-fixed nonconstant periodic orbit

Statement

Suppose Finite-budget phase-fixed periodic-orbit correction and validation returns validated-root for a rational shooting box $Z = X \times Θ$ , with a complete Krawczyk partition of $Z$ containing exactly one unique-root box $Y$ and only root-free boxes otherwise, outward residual and Jacobian enclosures on $Y$ , a separately verified contraction bound, and a component of $f (X_{Y})$ excluding zero, where $X_{Y}$ is the physical-state projection of $Y$ . Then there is exactly one zero $(x_{*}, T_{*}) \in Z$ of the phase-bordered shooting residual, and it lies in $Y$ . It generates a nonconstant periodic solution with $ψ (x_{*}) = 0$ , period $T_{*} > 0$ , and the returned trajectory and monodromy boxes contain its exact trajectory and monodromy. No uniqueness outside $Z$ is asserted.

Facts & Assumptions

Given: The successful finite corrector record, its rational box and all interval enclosures and strict inequalities.

[F1]

Strict Krawczyk inclusion proves existence, and the separate interval norm bound $q < 1$ proves uniqueness in the box; strict inclusion alone does not prove uniqueness (Krawczyk inclusion proves existence, disjointness excludes roots, and contraction proves uniqueness).

[F2]

Provided the exact flow exists uniquely on $[0, θ]$ , a zero of the phase-bordered residual with positive period and nonzero vector field is exactly a phase-fixed nonconstant periodic solution (Zeros of the phase-bordered BVP are phase-fixed periodic orbits).

[F3]

The successful corrector returns the exact chart $T (u) = a + D u$ , a complete Krawczyk partition of $[0, 1]^{d + 1}$ and its mapped physical partition of $Z$ , with one normalized unique-root box and root-free boxes otherwise; it also returns physical residual/Jacobian, trajectory, monodromy and nonstationarity enclosures on its unique-root image box $Y$ (Finite-budget phase-fixed periodic-orbit correction and validation).

[F4]

A successful validated Euler range-oracle run establishes existence and uniqueness through scaled time $1$ and encloses the exact trajectory and variational solution (The validated Euler tube encloses the exact periodic flow and monodromy matrix).

[F5]

The positive-side rational affine chart transfers zeros, root-free/existence/uniqueness conclusions and complete partitions bijectively; its normalized derivative is $D F (T (u)) D$ (Rational boxes admit dyadic enclosure boundaries and exact affine root charts).

Proof

1.1F1F3F5given

By [F3], the actual Krawczyk run applies to $\hat{F} = F \circ T$ on the dyadic unit box, with normalized Jacobian enclosure for $D F (T (u)) D$ . Its unique normalized box $A_{Y}$ satisfies [F1], so strict inclusion gives a zero and the separate contraction bound gives at most one there. Every other normalized terminal box is root-free. By [F5], $T$ sends this complete partition onto $Z$ and bijects the zeros, so $Y = T (A_{Y})$ contains exactly one zero of the physical residual and every other physical image box is root-free. This is exactly one zero in all of $Z$ .

2.1F2F3F4step 1.1

Since $Θ \subset (0, \infty)$ , this root has positive period. The field range component excludes zero on its physical-state projection $X_{Y}$ , so $f (x_{*}) \neq 0$ . The full-box jet returned by [F3] contains a successful Euler run on $Y$ , so [F4] establishes existence and uniqueness of the exact flow through physical time $T_{*}$ . With this hypothesis verified, apply [F2] to obtain the phase-fixed nonconstant periodic solution.

3.1F1F3F4step 1.1∎

The successful run's range-oracle and interval inclusions contain each exact trajectory and variational solution by [F4]; [F3] makes these the trajectory and monodromy boxes of the root from step 1.1. Uniqueness is restricted to $Z$ , as established in step 1.1.

Depends on

Used by

Cited to discharge well-definedness by Finite-budget phase-fixed periodic-orbit correction and validation.

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Sources

S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica 19 (2010) (standard reference, not scraped)
R. B. Kearfott, Interval Computations: Introduction, Uses, and Resources, Euromath Bulletin 2(1) (1996) (standard reference, not scraped)