# $C^r$-Lohner algorithm

## TL;DR

This paper introduces a rigorous computational algorithm for solving ordinary differential equations and their derivatives, enabling the proof of complex dynamical phenomena like invariant tori in specific systems.

## Contribution

The paper develops a $C^r$-Lohner algorithm that computes bounds for solutions and derivatives of ODEs, extending previous methods to arbitrary order derivatives.

## Key findings

- Proves existence of multiple invariant tori in pendulum with periodic forcing.
- Establishes invariant tori in Michelson system.
- Demonstrates the algorithm's effectiveness in complex dynamical systems.

## Abstract

We present a Lohner type algorithm for the computation of rigorous bounds for solutions of ordinary differential equations and its derivatives with respect to initial conditions up to arbitrary order. As an application we prove the existence of multiple invariant tori around some elliptic periodic orbits for the pendulum equation with periodic forcing and for Michelson system.