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 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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Analytic, formal and smooth equivalence

Technically, the local classification problem for dynamical systems is no different from the for left-right classification problem for germs of smooth maps. In particular, one would assume looking for conjugacy $h$ in the same regularity class as the objects of classification (formal conjugacy for formal germs, smooth conjugacy for smooth germs, analytic conjugacy for analytic germs). This approach usually works for the left-right equivalence (to the extent where a meaningful classification exists).

However, for local dynamical systems a completely new phenomenon of divergence arises: very often a given analytic local dynamical system admits a relatively simple formal normal form (i.e, is formally equivalent to a simple, say, polynomial or even linear germ), yet the formal series for the conjugacy diverges and the analytic classification turns out to be immensely more delicate.

Example

A holomorphic self-map $f\in\operatorname{Diff}(\CC^1,0)$ tangent to the identity, i.e., of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$ (the series converges, $a_2\ne0$) is formally equivalent to the cubic self-map $f'(z)=z+z^2+az^3$, with a formal invariant $a\in\CC$, yet the analytic classification of such self-maps has a functional invariant, the so called Ecalle-Voronin modulus, which shows that the same class of formal equivalence contains continuum of pairwise analytically non-equivalent self-maps distinguished by a certain auxiliary analytic function. The phenomenon is known today under the name of the Nonlinear Stokes phenomenon, [NLSP], [IY].

The $C^\infty$-smooth classification of smooth local dynamical systems occupies an intermediate position: for some (usually, hyperbolic) cases formally equivalent smooth germs are smoothly equivalent even when the formal normalizing series diverge . In other cases the formal divergence affects also the smooth classification even in the case of relatively low smoothness.

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References and basic literature

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[NLSP]	Yu. Ilyashenko, Nonlinear Stokes phenomena, Nonlinear Stokes phenomena, 1--55, Adv. Soviet Math., 14, Amer. Math. Soc., Providence, RI, 199, MR1206041
[IY]	Yu. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations. Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008 MR2363178
[Br]	A. D. Brjuno, Analytic form of differential equations. I, II, Trans. Moscow Math. Soc. 25 (1971), 131--288 (1973); ibid. 26 (1972), 199--239 (1974) MR0377192.
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