Explicit Numerical Computation of Normal Forms for Poincaré Maps

Introduction

This project provides a basic code sample for computing normal forms and when the the fixed points is elliptic Hamiltonian also twist maps. The sample is provided for a specific model. While the normal form construction does not necessarily require Hamiltonian dynamics, we illustrate the procedure using the classical Hénon-Heiles system, a standard Hamiltonian model in celestial mechanics.

For further details, see the manuscript GJJZ24.

Requirements

The code is written in C and has been successfully compiled with the GNU C Compiler version 12.2.0. The programs are self-contained, requiring no extra libraries, and they depend on appropriate initial guesses to function correctly.

Output files from the Taylor v2.2 package are provided in the 0-model folder. These include structures like MY_JET and MY_FLOAT, and various functions. The folder also contains instructions on generating these output files using taylor if users wish to extend the sample.

Usage

To use the code, follow the folder structure in the specified order, adhering to the instructions in the accompanying README files:

• 0-model: Contains the model and outputs from taylor used in subsequent folders. Everything required for the sample is included here.
• 1-fixed-point: Computes a fixed point of the Poincaré map for the Hénon-Heiles system, fixing an energy level and a spatial section at {x = 0}.
• 2-jet-fixed-point: Computes the high-order derivatives of the Poincaré map at the fixed point computed in 1.
• 3-normal-form: Performs a change of coordinates on the jet obtained in 2 to derive its normal form.
• 4-twist: Computes a twist map based on the output of 3, leveraging the elliptic fixed point from 1 and the Hamiltonian nature of the system.

Warnings

This code is a sample designed to illustrate the process of computing normal forms for Poincaré maps. Specifically, it addresses 2-by-2 systems, solving linear systems and computing eigenpairs.

Note that this sample does not include components such as parameter-dependent normal forms or torus visualizations, as these are outside the scope of the demonstration and might detract from clarity.

Enhancements

You may consider the following improvements:

• Implement parameter-dependence normal form.
• Develop the torus visualization for a radius within the validity range.
• Improve the linear algebra parts by incorporating custom libraries.
• Run the code with extended precision (this requires modifying the output files from taylor in the 0-model folder).
• Use wrappers to run the code, particularly the part of taylor, in a different programming language, e.g., Python.

Authors and Acknowledgments

The authors of the paper and this code project are:

• J. Gimeno
• À. Jorba
• M. Jorba-Cuscó
• M. Zou

Date: December 2024.

License

This project is licensed under the GNU General Public License v3.0. See the LICENSE file for details.

For more information about the GPL v3.0, please visit the official GNU website: GNU General Public License v3.0.