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

作者：Walawska Irmina; Wilczak Daniel^*

来源：Applied Mathematics and Computation, 2016, 291: 303-322.

DOI：10.1016/j.amc.2016.07.005

• Validated numerics
• Initial value problem
• Variational equations
• Uniform hyperbolicity
• Chaos

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.

• 出版日期2016-12-1