Computing normal forms and formal invariants of dynamical systems by means of word series

TL;DR

This paper introduces a novel method using extended word series with complex numbers to simplify dynamical systems and compute invariants, streamlining the process of normal form reduction and invariant calculation.

Contribution

It develops a new algebraic framework involving a group and Lie algebra of complex numbers for normal form and invariant computation in dynamical systems.

Findings

Provides a universal algebraic approach for normal forms

Enables computation of formal invariants in Hamiltonian systems

Simplifies manipulations using complex numbers instead of vector fields

Abstract

We show how to use extended word series in the reduction of continuous and discrete dynamical systems to normal form and in the computation of formal invariants of motion in Hamiltonian systems. The manipulations required involve complex numbers rather than vector fields or diffeomorphisms. More precisely we construct a group G and a Lie algebra g in such a way that the elements of G and g are families of complex numbers; the operations to be performed involve the multiplication F in G and the bracket of g and result in universal coefficients that are then applied to write the normal form or the invariants of motion of the specific problem under consideration.