Normal Forms and Symmetries for Dynamical Systems

Abstract

In this chapter, we consider a smooth dynamical system (DS) on a manifold M, which we take, for ease of notation, as being embedded in R ⁿ, with a zero at a point x ₀ which we take, again for ease of notation, to be at the origin of R ⁿ. We have then

$$ \dot x = f(x) f:M \to TM, f(0) = 0, $$

((1))

or, equivalently, a tangent vector field (VF) on M,

$$ X = \sum\limits_{i = 1}^n {f^i (x)\partial _i \left( {\partial \equiv \frac{\partial } {{\partial x^i }}} \right)} , $$

((2))

and we are interested in the normal forms classification of such DS, or equivalently of such VF.