Local normal forms for dynamical systems

A local dynamical system is either

• a (smooth, analytic, formal) vector field $v$ defined on a neighborhood $(\RR^n,0)$, $v:(\RR^n,0)\owns x\mapsto T_x(\RR^n,0)$, and vanishing at the origin, $v(0)=0$, or
• a (smooth, analytic, formal germ of a) invertible self-map $f\in\operatorname{Diff}(\RR^n,0)=\{$invertible maps of $(\RR^n,0)$ to itself fixing the origin, $f(0)=0\}$.

The "dynamical" idea is the possibility to iterate the self map, producing the cyclic group $$ f^{\circ\ZZ}=\{\underbrace{f\circ \cdots\circ f}_{k\text{ times}}\,|\,k\in\ZZ\}\subseteq\operatorname{Diff}(\RR^n,0), $$ or a one-parametric group $$\exp \RR v=\{\exp tv\in\operatorname{Diff}(\RR^n,0)\,|\, t\in\RR,\ \exp[(t+s)v]=(\exp tv)\circ (\exp sv),\ \tfrac{\rd}{\rd t}|_{t=0}\exp tv=v\} $$ with $v$ as the infinitesimal generator.

Two local dynamical systems of the same type are equivalent, if there exists an invertible self-map $h\in\operatorname{Diff}(\RR^n,0)$ which conjugates them: $$ f\sim f'\iff\exists h:\ f\circ h=h\circ f', \qquad\text{resp.,}\qquad v\sim v'\iff\exists h:\ \rd h\cdot v=v'\circ h. $$ Here $\rd h$ is the differential of $h$, acting on $v$ as a left multiplication by the Jacobian matrix $\bigl(\frac{\partial h}{\partial x}\bigr)$. Obviously, the equivalent systems have equivalent dynamics: if $h$ conjugates $f$ with $f'$, it also conjugates any iterate $f^{\circ k}$ with $f'^{\circ k}$, and conjugacy of vector fields implies that their flows are conjugated by $h$: $h\circ(\exp tv)=(\exp tv')\circ h$ for any $t\in\RR$.

A singularity (or singularity type) of a local dynamical system is a subspace of germs defined by finitely many semialgebraic constraints on the initial Taylor coefficients of the germ. Examples:

• Hyperbolic local dynamical systems: Self-maps tangent to linear automorphisms without modulus one eigenvalues, or vector fields whose linear part has no eigenvalues on the imaginary axis;
• Saddle-nodes, self-maps having only one simple egenvalue $\mu=1$, resp., vector fields, whose linearization matrix has a simple eigenvalue $\lambda=0$;
• Cuspidal germs of vector fields on $(\RR^2,0)$ with the nilpotent linearization matrix $\bigl(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\bigr)$.

The classification problem for a given singularity type is to construct a list (finite or infinite, eventually involving parameters) of normal forms, such that any local dynamical system of the given type is equivalent to one of these normal forms.

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