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Local normal forms for dynamical systems

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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, $f(0)=0\}$[1].

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[2] $$\exp \RR v=\{\exp tv\in\operatorname{Diff}(\RR^n,0)\,|\, t\in\RR,\ \exp[(t+s)v]=(\exp tv)\circ (\exp sv),\ \tfrac{\rd}{\rd t}|_{t=0}\exp tv=v\} $$ with $v$ as the infinitesimal generator[3].

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 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.


  1. In the formal and analytic cases one can replace the real field $\RR$ by the field of complex numbers $\CC$.
  2. As before, the "real time" $t\in\RR$ can be replaced by the "complex time" $t\in\CC$ given the appropriate context.
  3. Note that all iterates (resp., flow maps) are defined only as germs, thus the definition of the orbit $O(a)=\{f^{\circ k}(a)\}$ of a point $a\in(\RR^n,0)$ (forward, backward or bi-infinite) requires additional work.

Analytic, formal and smooth equivalence

How to Cite This Entry:
Local normal forms for dynamical systems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_normal_forms_for_dynamical_systems&oldid=25115