# Verified Computation of High-Order Poincaré Maps

### Abstract

Poincare 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
Poincare maps in concrete problems is far from trivial. For the verification it is
desirable to approximate the Poincare map over as large a domain as possible. Knowledge of
the flow of the system is a prerequisite for any computation of Poincare 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 Poincare map from the
original Taylor model flow representation. First a high-order polynomial approximation
of the time necessary to reach the Poincare 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 Poincare map. A verified correction is
performed heuristically which provides a rigorous enclosure of the Poincare map given,
demonstrating achieved accuracy with verification.

J. Grote, K. Makino, M. Berz,
*Transactions on Systems* **11,4** (2005) 1986-1992

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