Daniel Wilczak, Piotr Zgliczynski
The goal of this paper is to show how to produce a piece of rigorous bifurcation diagram of periodic orbits for an ODE. We study the Rossler system, one of the textbook examples of ODEs generating nontrivial dynamics, for the parameter range containing two period doubling bifurcations.
Irmina Walawska, Daniel Wilczak
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.
Daniel Wilczak, Piotr Zgliczyński
The Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and with parameter $ν=0.1212$ is considered. We give a computer-assisted proof the existence of symbolic dynamics and countable infinity of periodic orbits with arbitrary large periods.
Daniel Wilczak, Piotr Zgliczynski
We present a topological method for the efficient computer assisted verification of the existence of the homoclinic tangency which unfolds generically in a one-parameter family of planar maps. The method has been applied to the Henon map and the forced damped pendulum ODE.