View source for Local normal forms for dynamical systems

A ''local'' [[dynamical system]] is either 
* a (smooth, analytic, formal) vector field $v$ defined<ref>In the formal case instead of the germ we consider a tuple of formal Taylor series in the variables $x=(x_1,\dots,x_n)$.</ref> 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, $f(0)=0\}$<ref>In the formal and analytic cases one can replace the real field $\RR$ by the field of complex numbers $\CC$.</ref>. 

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<ref>As before, the "real time" $t\in\RR$ can be replaced by the "complex time" $t\in\CC$ given the appropriate context.</ref>
$$\exp \RR v=\{\exp tv\in\operatorname{Diff}(\RR^n,0)\,|\, t\in\RR,\ \exp[(t+s)v]=(\exp tv)\circ (\exp sv),\ \tfrac{\rd}{\rd t}|_{t=0}\exp tv=v\}
$$
with $v$ as the infinitesimal generator<ref>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.</ref>.

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 system]]s: Self-maps tangent to linear automorphisms without modulus one eigenvalues, or vector fields whose linear part has no eigenvalues on the imaginary axis;
* [[Saddle node|Saddle-node]]s, 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 [[Normal form (for singularities)|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]], {{Cite|I93}}, {{Cite|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 <ref>This means that the geometrical reasons for the divergence are "observable" only in the complex domain.</ref>. In other cases the formal divergence affects also the smooth classification even in the case of relatively low smoothness<ref>Yu. Ilʹyashenko, S. Yakovenko, ''Nonlinear Stokes phenomena in smooth classification problems''.  Nonlinear Stokes phenomena,   235--287, Adv. Soviet Math., '''14''', Amer. Math. Soc., Providence, RI,  1993, {{MR|1206045}}. </ref>.                  
         
  
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==Linearization==
$\def\l{\lambda}$
A germ of a vector field with the Taylor expansion $v(x)=Ax+O(\|x\|^2)$ (resp., of a self-map with the Taylor expansion $f(x)=Mx+O(\|x\|^2)$ is ''linearizable'' (formally, smoothly or analytically), if it is conjugated to the linear vector field $v'(x)=Ax$, (resp., to the linear automorphism $f(x)=Mx$. The linear objects are [[#Linear_operators_.28self-maps.29|equivalent]] to their [[Jordan normal form]]s.

Linearizability very strongly depends on the arithmetical properties of eigenvalues $\l_1,\dots,\l_n$ of the operator $A$ (resp., $\mu_1,\dots,\mu_n$ of $M$).

;Definition.
A tuple $\l=(\l_1,\dots,\l_n)\in\CC^n$ is said to be in ''additive'' [[resonance]], if there exists an integer vector $\alpha=(\alpha_1,\dots,\alpha_n)\in\ZZ_+^n$ and index $j\in\{1,\dots,n\}$ such that
$$
\l_j-\left<\alpha,\l\right>=0,\quad|\alpha|\ge 2,\qquad\text{where }\left<\alpha,\l\right>=\sum_{i=1}^n\alpha_i\lambda_i,\ |\alpha|=\sum_{i=1}^n \alpha_i.
$$
A tuple $\mu=(\mu_1,\dots,\mu_n)\in\CC^n_{\ne 0}$ is said to be in a ''multiplicative'' [[resonance]], if there exists an integer vector $\alpha=(\alpha_1,\dots,\alpha_n)\in\ZZ_+^n$ and index $j\in\{1,\dots,n\}$ such that
$$
\mu_j-\mu^\alpha=0,\ |\alpha|\ge 2,\qquad\text{where }\mu^\alpha=\mu_1^{\alpha_1}\cdots\mu_n^{\alpha_n}.
$$
The corresponding ''resonant vector monomial'' is the vector function $v_{j\alpha}:\R^n\to\R^n$ whose only component which is not identically zero, is the monomial $x^\alpha$ at the position $j$, $$v_{j\alpha}(x)=(0,\dots,0,\underset{j}{x^\alpha},0,\dots,0),\qquad j=1,\dots,n,\quad |\alpha|\ge 2.$$ The number $|\alpha|\ge2$ is called the ''order'' of resonance.

A vector field (resp., self-map) is ''resonant'', if the eigenvalues of its linear part exhibit one or more additive (resp., multiplicative) resonances. Otherwise the local dynamical system is called ''non-resonant''. 

;Examples.
* A self-map $M:\CC^1\to\CC^1$, $x\mapsto \mu x$ is (multiplicatively) resonant if and only if $\mu$ is a root of unity, $\mu^d=1$ for some $d\in\NN$. The singleton $\{\mu\}\in\CC^1_{\ne 0}$ satisfies infinitely many resonant identities of the form $\mu=\mu^{\nu d+1}$, $\nu=1,2,\dots$, of orders $d+1,2d+1,\dots$. 
* A tuple $(\l_1,\l_2)$ is additively resonant in two different cases. If $(\l_1:\l_2)=(1:d)$ or $(d:1)$, with $d\in\NN$, then there exists only one resonance between them, $\l_2=d\cdot\l_1$ or $\l_1=d\cdot \l_2$ respectively. The corresponding germ of vector field is usually referred to as the [[resonant node]]. If the ratio $\l_1/\l_2=-\beta_2/\beta_1$, $\gcd(\beta_1,\beta_2)=1$,  is a nonpositive rational number, then the corresponding identity $\left<\beta,\l\right>=0$ implies ''infinitely many'' additive resonance identities of the form 
$$
\l_j=\l_j+\nu\left<\beta,\l\right>,\qquad \nu=1,2,\dots
$$
of orders $\nu|\beta|$. In particular, if one of the numbers vanishes, say, $\l_1=0$, the resonant identities are all of the form $\l_j=\l_j+\nu\l_1$ for all $\nu$ and $j=1,2$. If $|\beta|>1$, the corresponding singularity is called a ''resonant [[saddle]]'', otherwise the standard name is the [[saddle node|saddle-node]].
===Poincaré-Dulac formal normal form===
The central result on the formal classification of local dynamical systems is the Poincaré-Dulac theorem.

;Theorem {{Cite|A83}} (Poincaré-Dulac)
Any vector field (resp., self-map) is formally equivalent to a formal vector field (resp., self-map) which contains only resonant monomials. 
$$
v\underset{\text{form.}}{\sim} v'=Ax+\sum_{(j,\alpha)\text{ res. for $A$}} c_{j\alpha}v_{j\alpha},\qquad\text{res.,}\qquad f\underset{\text{form.}}{\sim} f'=Mx+\sum_{(j,\alpha)\text{ res. for $M$}} c_{j\alpha}v_{j\alpha},\qquad c_{j\alpha}\in\CC.
$$
In particular, a non-resonant vector field (self-map) is formally linearizable.

It is important to notice that if the eigenvalues satisfy a ''unique'' identity $\left<\alpha,\l\right>=0$, then the normal forms are integrable in quadratures: the equation for the (unique) resonant monomial $u(x)=x^{\alpha}$ separates, $\frac{\rd}{\rd t}u=u\,F(u)$, where $F$ is a formal series in one variable $u$; this equation can be integrated. The remaining equations all take the form $\frac{\rd x_i}{x_i\rd t}=\l_i(1+G_i(u))$ with formal series $G_i$ and separated variables. For multi-resonant tuples this is no more the case.

=== Analytic linearization ===
Convergence of the series bringing a local dynamical system to its Poincaré-Dulac normal form is primarily depending on the relative position of the eigenvalues and the imaginary axis (resp., the unit circle). 

The case when all eigenvalues $\l_1,\dots,\l_n$ of the linear part $A=\rd v(0)$ are to one side of the imaginary axis<ref>By a linear change of the independent variable $t\mapsto \sigma t$ one can bring to such form any vector field such that the convex hull of eigenvalues $\l_1,\dots,\l_n$ does not contain zero.</ref> (resp., all eigenvalues $\mu_i$ of $M=\rd f(0)$ are all inside the unit circle or all outside of it) is referred to as the ''[[Poincaré domain]]''. For instance, a self-map $f:(\CC^1,0)\to(\CC^1,0)$ with the multiplicator $\mu=\rd f(0)\in\CC_{\ne 0}$ belongs to the Poincare domain if $|\mu|\ne 1$; a vector field on the plane is in the Poincare domain if the ratio of the eigenvalues $\frac{\l_1}{\l_2}$ is not zero or negative. The only possible additive resonance in the Poincare domain is $(\l_1:\l_2)=(1:d)$.

;Theorem.
In the Poincare domain the series bringing the local dynamical system to its Poincare-Dulac normal form, always converges.  

The complementary case, where eigenvalues of the linear part cannot be separated by a line from the origin (resp., by a circle from $1$), is referred to as the ''[[Siegel domain]]''. One-dimensional self-maps are in the Siegel domain, if $|\mu|=1$ (resonant if $\mu$ is a root of unity, otherwise non-resonant). Two-dimensional vector fields are in the Siegel domain, if the ratio of eigenvalues $\l_1/\l_2$ is zero or negative number (resonance occurs if this number is zero or negative rational, otherwise the field is non-resonant).   

Convergence of the formal series linearizing analytic germs in the Siegel domain depends on certain quantitative conditions on the arithmetic nature of the (non-resonant tuples of) eigenvalues. Very roughly, if the (nonvanishing) values of the differences $\delta_k=\inf_{j,\ |\alpha|=k}|\l_j-\left<\alpha,\l\right>|$ (resp., $\delta_k=\inf_{j,|\alpha|\le k}|\mu_j-\mu^\alpha|$), which may decrease to zero as $k\to+\infty$, decrease not too fast (the so called ''Diophantine case''), then the formal conjugacy is convergent. On the contrary, if the ''small denominators'' $\delta_k$ decrease anomalously fast (the so called ''Liouvillean case''), the normalizing series in general diverge. 

The sufficient decay rate of the small denominators $\delta_k\to0$ was first discovered by C. L. Siegel<ref>C. L. Siegel, J. K. Moser,  ''Lectures on celestial mechanics'', Die Grundlehren der mathematischen Wissenschaften, Band 187. Springer-Verlag, New York-Heidelberg,  1971, {{MR|0502448}}</ref> and later improved significantly by A. D. Brjuno {{Cite|Br}}. The sufficient Brjuno condition for self-maps $(\CC^1,0)\to(\CC^1,0)$ was shown to be sharp by J.-C. Yoccoz<ref>J.-C. Yoccoz, '' Théorème de Siegel, nombres de Bruno et polynômes  quadratiques. Petits diviseurs en dimension 1''. Astérisque  No. '''231'''  (1995), 3–88, {{MR|1367353}}.</ref>, see [[Diophantine conditions in dynamics]].

The Diophantine conditions for convergence/divergence to be imposed on the multiplicator $\rd f(0)\in\CC_{\ne 0}=\mu=\exp 2\pi i \theta$, $\theta\in\RR\smallsetminus\QQ$, are most easily formulated in terms of the expansion of rotation angle $\theta$ in the [[continued fraction]], more precisely, in terms of the growth rate of partial denominators,
$$
\theta = q_0 + \cfrac{1}{q_1
          + \cfrac{1}{q_2
          + \cfrac{1}{q_3 + \cdots}}}, \qquad q_0,q_1,\dots \in\NN.
$$
The Siegel condition requires that the denominators' growth is bounded asymptotically by the uniform estimate $\log q_{n+1}=O(\log q_n)$ as $n\to\infty$. The Brjuno condition is equivalent to the summability of the series 
$$
\sum_{n=0}^\infty \frac{\log q_{n+1}}{q_n}<+\infty.\tag{Br}
$$
The necessary condition for convergence, due to Cremer (1938), claims that if 
$$
\sup_{n\ge 0}\frac{\log q_{n+1}}{q_n}=\infty,\tag{Cr}
$$
then there exists a non-linearizable analytic self-map with the multiplicator $\mu=\exp 2\pi i\theta$. For any number violating the Brjuno condition J.-C. Yoccoz constructed in 1987  an example of a quadratic self-map which is non-linearizable.
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===Smooth linearization===
If the local dynamical system $v(x)=Ax+\cdots$ (resp., $f(x)=Mx+\cdots$) is real and exhibits no additive (resp., multiplicative) resonances until sufficiently high order $N\le+\infty$, then this system admits a $C^n$-smooth linearization of smoothness order $n$ which grows to infinity together with $N$. The key assumption used in the proof of this theorem is the hyperbolicity: the non-resonant linear part $A$ (resp., $M$) cannot have eigenvalues on the imaginary axis, $\operatorname{Re}\l_i\ne 0$<ref>Indeed, if $\l$ is an imaginary eigenvalue, then $\l'=\bar\l$ is also an imaginary eigenvalue, which implies that either $\l=0$, or $\l+\l'=0$, in both cases implying infinitely many resonances.</ref> for all $i=1,\dots,n$ (resp., on the unit circle, $|\mu_i|\ne 1$ for all $i=1,\dots,n$<ref>Violation of this condition produces infinitely many resonances via the identity $\mu\mu'=1$, where $\mu'=\bar\mu$ is another eigenvalue.</ref>). This result is known as the Sternberg<ref>S. Sternberg,  ''On the structure of local homeomorphisms of euclidean $n$-space,'' II. Amer. J. Math.  '''80'''  (1958) 623–631, {{MR|0096854}} </ref>-Chen<ref>Chen, Kuo-Tsai, ''Equivalence and decomposition of vector fields about an elementary  critical point'', Amer. J. Math.  '''85'''  (1963) 693–722, {{MR|0160010}}. </ref> theorem, see {{Cite|H|Ch. IX, Sect. 12-14}}. 

The order $N(n)$ as a function of the required smoothness $n$ grows no faster than linearly: it is sufficient to verify absence of resonances till order $N\le C\cdot n$, where the constant $C$ depends on the relative position of eigenvalues and the imaginary axis (resp., the unit circle) and can be expressed<ref>V. S. Samovol,  ''Equivalence of systems of differential equations in the neighborhood of  a singular point''  (Russian),  Trudy Moskov. Mat. Obshch.  44  (1982), 213–234, {{MR|0656287}}</ref><ref>G. R. Belitsky, ''Equivalence and normal forms of germs of smooth mappings'', Russian Math. Surveys '''33''' (1978), no. 1, 107--177, {{MR|0490708}}</ref> in terms of the hyperbolicity measure, the ratio $$\frac{\max_i|\l_i|}{\min_i|\operatorname{Re}\l_i|},\qquad\text{resp.,}\qquad\frac{\max_i|\mu_i|}{\min_i\bigl||\mu_i|-1\bigr|}.$$ 

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== References and basic literature ==
{| class="sortable" 
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| valign="top"|{{Ref|I93}} ||valign="top"|Yu. Ilyashenko, ''Nonlinear Stokes phenomena'',  Nonlinear Stokes phenomena,   1--55, Adv.  Soviet Math., 14, Amer. Math. Soc., Providence, RI,  1993, {{MR|1206041}} 
|-
|valign="top"|{{Ref|IY}}||valign="top"|  Yu. Ilyashenko, S.   Yakovenko, ''Lectures on analytic differential  equations''. Graduate   Studies in Mathematics, '''86'''. American  Mathematical Society,   Providence, RI,  2008 {{MR|2363178}}

|-
|valign="top"|{{Ref|A83}}||valign="top"|   Arnold  V. I.,  ''Geometrical methods in the theory of ordinary   differential    equations''. Grundlehren der Mathematischen   Wissenschaften, '''250'''.   Springer-Verlag, New York-Berlin,  1983,   {{MR|0695786}}
|-
|valign="top"| {{Ref|Br}} ||valign="top"| 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) {{MR|0377192}}.
|-
|valign="top"| {{Ref|H}} ||valign="top"| P. Hartman,  ''Ordinary differential equations'', John Wiley & Sons, Inc., New York-London-Sydney  1964, {{MR|0171038}}  
|-

|valign="top"| {{Ref|AI88}} ||valign="top"| V. I. Arnold, Yu. I.  Ilyashenko, ''Ordinary differential equations'', Encyclopaedia Math. Sci., '''1''',  Dynamical systems, I, 1--148, Springer, Berlin,  1988, {{MR|0970794}} 
|-

|valign="top"| {{Ref|AAIS}} ||valign="top"|V. I.Arnold, V. S. Afrajmovich, Yu. S.  Ilʹyashenko, L. P.   Shilnikov, ''Bifurcation theory and catastrophe theory'', Encyclopaedia Math. Sci., '''5''', Dynamical systems, V,  Springer, Berlin, 1994, {{MR|1287421}}
|}