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arXiv · 2605.07500

Computer-Assisted Proofs in Dynamical Systems: A Case Study of a Heteroclinic Orbit in the Shimizu--Morioka System

Olivier Hénot·Akitoshi Takayasu
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Abstract

The radii polynomial approach is an a posteriori validation method based on the contraction of a quasi-Newton operator. We apply this strategy to give a computer-assisted proof of a transverse heteroclinic orbit in the Shimizu--Morioka system, validating the equilibria and eigenpairs, the local invariant manifolds via the parameterization method, and the connecting orbit via a boundary-value problem. For each subproblem we present a four-step procedure: $(i)$ zero-finding formulation, $(ii)$ approximate zero, $(iii)$ approximate inverse, and $(iv)$ bound estimates. This highlights the unifying structure behind the a posteriori validation method. Alongside the analysis, we include code snippets implemented in Julia using the RadiiPolynomial library.

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Olivier Hénot, Akitoshi Takayasu. 2026-05-08. Computer-Assisted Proofs in Dynamical Systems: A Case Study of a Heteroclinic Orbit in the Shimizu--Morioka System. https://arxiv.org/abs/2605.07500

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