An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs

TL;DR

This paper introduces a new implicit algorithm combining high-order Taylor and Hermite-Obreshkov methods to compute validated bounds for solutions of variational equations in ODEs, improving accuracy for dynamical systems proofs.

Contribution

It presents an improved algorithm over the $C^1$-Lohner method, offering sharper bounds for validated solutions of variational equations in ODEs.

Findings

Successfully proved the existence of an attractor in the Rössler system.

Demonstrated the attractor contains a hyperbolic invariant subset.

Confirmed chaotic dynamics conjugated to a subshift of finite type.

Abstract

We propose a new algorithm for computing validated bounds for the solutions to the first order variational equations associated to ODEs. These validated solutions are the kernel of numerics computer-assisted proofs in dynamical systems literature. The method uses a high-order Taylor method as a predictor step and an implicit method based on the Hermite-Obreshkov interpolation as a corrector step. The proposed algorithm is an improvement of the $C^1$-Lohner algorithm proposed by Zgliczy\'nski and it provides sharper bounds. As an application of the algorithm, we give a computer-assisted proof of the existence of an attractor set in the R\"ossler system, and we show that the attractor contains an invariant and uniformly hyperbolic subset on which the dynamics is chaotic, that is, conjugated to subshift of finite type with positive topological entropy.