[Paper Review] $\mathcal C^1$-HO - an implicit algorithm for validated enclosures of the solutions to variational equations for ODEs

Irmina Walawska, Daniel Wilczak|arXiv (Cornell University)|Sep 24, 2015

Quantum chaos and dynamical systems1 citations

TL;DR

This paper introduces $Χ^1$-HO, a high-order implicit algorithm for computing validated enclosures of solutions to first-order variational equations of ODEs. By combining a high-order Taylor predictor with an Hermite-Obreshkov-based corrector, it improves upon the $C^1$-Lohner method, yielding sharper bounds and enabling a computer-assisted proof of a chaotic invariant set in the Rossler system with positive topological entropy.

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 Zgliczynski 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 Rossler 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.

Motivation & Objective

• To develop a more accurate algorithm for computing validated bounds of solutions to variational equations of ODEs.
• To improve upon the $C^1$-Lohner method by reducing overestimation in interval enclosures.
• To enable rigorous computer-assisted proofs of chaotic dynamics in ODE systems.
• To provide sharper and more reliable bounds through high-order accurate numerical integration with implicit correction.

Proposed method

• The algorithm uses a high-order Taylor method as a predictor to advance the solution over time steps.
• An implicit corrector based on Hermite-Obreshkov interpolation is applied to refine the predicted solution and reduce local truncation errors.
• The method ensures validated enclosures by rigorously bounding all numerical errors using interval arithmetic.
• The algorithm maintains $C^1$-smoothness of the solution enclosures, essential for rigorous dynamical systems analysis.
• It combines predictor-corrector strategies with high-order accuracy to improve convergence and reduce overestimation.
• The approach is designed to be applicable to computer-assisted proofs in dynamical systems, particularly for detecting chaotic invariant sets.

Experimental results

Research questions

• RQ1 Can a high-order implicit method reduce overestimation in validated enclosures of variational ODE solutions compared to existing $C^1$-Lohner methods?
• RQ2 Does the improved accuracy of the $Χ^1$-HO algorithm enable the rigorous detection of chaotic dynamics in ODE systems?
• RQ3 Can the algorithm be used to validate the existence of a uniformly hyperbolic invariant set in the Rossler system?
• RQ4 What is the topological entropy of the chaotic dynamics restricted to the invariant set in the Rossler system?
• RQ5 How does the algorithm's performance compare to prior methods in terms of bound sharpness and computational cost?

Key findings

• The $Χ^1$-HO algorithm produces sharper validated enclosures of variational solutions than the $C^1$-Lohner method, reducing overestimation.
• The algorithm successfully computes rigorous bounds for the variational equations of the Rossler system.
• A computer-assisted proof confirms the existence of an attractor set in the Rossler system.
• The attractor contains a uniformly hyperbolic invariant subset with chaotic dynamics.
• The dynamics on this subset is conjugated to a subshift of finite type, confirming positive topological entropy.
• The method enables the first rigorous validation of chaotic behavior in the Rossler system using validated numerics.