Resonant normal forms as constrained linear systems

TL;DR

This paper demonstrates that nonlinear systems in Poincare'-Dulac normal form can be viewed as constrained linear systems, revealing geometric insights and enabling integration methods for resonant normal forms.

Contribution

It establishes a novel perspective linking resonant normal forms to constrained linear systems, facilitating their integration and geometric understanding.

Findings

Resonant normal forms can be represented as constrained linear systems.

The approach simplifies integration of resonant normal forms.

Provides geometric insights into classical integration methods.

Abstract

We show that a nonlinear dynamical system in Poincare'-Dulac normal form (in $\R^n$) can be seen as a constrained linear system; the constraints are given by the resonance conditions satisfied by the spectrum of (the linear part of) the system and identify a naturally invariant manifold for the flow of the ``parent'' linear system. The parent system is finite dimensional if the spectrum satisfies only a finite number of resonance conditions, as implied e.g. by the Poincare' condition. In this case our result can be used to integrate resonant normal forms, and sheds light on the geometry behind the classical integration method of Horn, Lyapounov and Dulac.