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

• Published in: Applied mathematics and computation Vol. 291; pp. 303 - 322
• Main Authors: Walawska, Irmina, Wilczak, Daniel
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.12.2016
• Subjects: Chaos; Initial value problem; Uniform hyperbolicity; Validated numerics; Variational equations; Chaos; 65L05; Variational equations; 65G20; Initial value problem; Uniform hyperbolicity; Validated numerics
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2016.07.005

•An algorithm for validated computation of monodromy matrices for ODEs is provided.•Smaller truncation error allows larger time steps making the computation faster.•The existence of a chaotic and hyperbolic set for the Rossler system is proved via computer-assisted proofs techniques. 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 C1-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.