An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs
For researchers in dynamical systems, this algorithm improves the precision of validated numerics for computer-assisted proofs, offering sharper bounds than existing methods.
The paper proposes a new algorithm for computing validated bounds for solutions to first-order variational equations for ODEs, improving the C^1-Lohner algorithm by providing sharper bounds. It demonstrates the algorithm's effectiveness through a computer-assisted proof of an attractor set in the Rössler system, showing it contains a chaotic invariant subset.
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 Zgliczyński 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 Rössler 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.