Verified Computation of High-Order Poincaré Maps

Poincaré maps often prove to be invaluable tools in the study of long-term behaviour and qualitative properties of a given dynamical system. While the analytic theory of these maps is fully explored, finding numerical algorithms that allow the computation of Poincaré maps in concrete problems is far from trivial. For the verification it is desirable to approximate the Poincaré map over as large a domain as possible. Knowledge of the flow of the system is a prerequisite for any computation of Poincaré maps. Taylor model based verified integrators compute final coordinates as high-order polynomials in terms of initial coordinates, with a small remainder error interval which typically is many orders of magnitude smaller than the initial domain. We present a method to obtain a Taylor model representation of the Poincaré map from the original Taylor model flow representation. First a high-order polynomial approximation of the time necessary to reach the Poincaré section is determined as a function of the initial conditions. This is achieved by reducing the problem to a non-verified polynomial inversion. This approximate crossing time is inserted into the Taylor model of the time-dependent flow, leading to an approximate Poincaré map. A verified correction is performed heuristically which provides a rigorous enclosure of the Poincaré map. Key-Words: Poincaré map, verified computation, differential algebra, Taylor model.