Normal forms of dynamical systems in Python

This document describes how to use a Python package to compute normal forms of autonomous differential equations

\[\frac{dx}{dt} = f(x)\]

where \(f:\mathbb{R}^n\rightarrow\mathbb{R}^n\).

The first section of this documents presents some background and a few examples of normal forms, and the remaining sections present the Python classes that the main normal_forms.normal_form.normal_form class uses.

The table of contents of this document is

• Normal forms
  • Background
  • A 1-D example: saddle-node bifurcation
  • A 2-D example: Duffing’s equation
• Taylor’s Series
  • A \(\mathbb{R}\rightarrow\mathbb{R}\) example
  • A \(\mathbb{R}\rightarrow\mathbb{R}^2\) example
  • A \(\mathbb{R}^2\rightarrow\mathbb{R}\) example
• Lie operators for vector fields