Local normal forms for dynamical systems

2020 Mathematics Subject Classification: Primary: 34C20,32S65,34M25 Secondary: 32Sxx37C2537C8537C20 [MSN][ZBL]

$\def\l{\lambda}$ A local dynamical system is a dynamical system (flow of a vector field, cascade of iterates of a self-map, or sometimes more involved construction) defined in an unspecifiedly small neighborhood of a fixed (rest) point. Application of local invertible self-map ("change of the variables") transforms a local dynamical system to an equivalent system. The local classification problem is to describe the equivalence classes of local dynamical systems, providing, if possible, the simplest or most convenient representative in each class.

The advantage of considering the local system rests in the hope that the classification will be determined by (semi)algebraic conditions imposed on the first few Taylor coefficients of the field (map). This hope is partly justified, see Algebraic decidability of local classification problems.

Local dynamical systems and their equivalence[edit]

By a local dynamical system one usually understands one of the following:

• a (smooth, analytic, formal) vector field $v$ defined^[1] 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, $f(0)=0\}$^[2].

The "dynamics" consists in 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 of the flow maps^[3] $$\exp \RR v=\{\exp tv\in\operatorname{Diff}(\RR^n,0)\,|\, t\in\RR,\ \exp[(t+s)v]=(\exp tv)\circ (\exp sv),\ \tfrac{\rd}{\rd t}|_{t=0}\exp tv=v\} $$ with $v$ as the infinitesimal generator^[4].

Equivalence of local dynamical systems[edit]

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: $$ f\sim f'\iff\exists h:\ f\circ h=h\circ f', \qquad\text{resp.,}\qquad v\sim v'\iff\exists h:\ \rd h\cdot v=v'\circ h. \tag{Cj} $$ 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 $t\in\RR$.

This definition (and the whole ensuing theory) depends in the most crucial way on the regularity condition imposed on the conjugacy $h$. Even if the two local systems themselves are very regular (say, real analytic), the reasonable classification may be sometimes possible only if $h$ is of lower regularity. The following classes are usually considered:

• For real local dynamical systems on $(\R^n,0)$:

1. Real analytic conjugacy with $h,h^{-1}\in \operatorname{Diff}^\omega(\R^n,0)$;
2. Smooth conjugacy by $h\in\operatorname{Diff}^k(\R^n,0)$ with $k$ continuous first derivatives, $k=1,\dots,\infty$; the $C^\infty$-case is probably the most important;
3. Formal conjugacy defined by a tuple $h=(h_1,\dots,h_n)$ of the formal series $\R[[x_1,\dots,x_n]]$ without the free terms and with the nondegenerate Jacobian matrix $\det(\frac{\partial h_i}{\partial x_j})\ne 0$;
4. Topological conjugacy^[5] by $h\in\operatorname{Homeo}(\R^n,0)=\operatorname{Diff}^0(\R^n,0)$.

• For holomorphic dynamical systems on $(\C^n,0)$:

1. Holomorphic conjugacy by a biholomorphism $h\:(\C^n,0)\to(\C^n,0)$, $\det\rd h(0)\ne 0$;
2. Formal conjugacy by a tuple $h=(h_1,\dots,h_n)$ of the formal series $\C[[x_1,\dots,x_n]]$;
3. Topological classification by $h\in\operatorname{Homeo}(\C^n,0)$.

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: Real 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, real 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 $\bigl(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\bigr)$;
• Parabolic singularity, a holomorphic self-map $(\C^1,0)\to(\C^1,0)$, tangent to the identity with finite order: $z\mapsto z+cz^p+\cdots$, $p<+\infty$, $c\ne 0$.

The classification problem for a given singularity type requires 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.

A particular case of classification problems is the study of linearizability. 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 $f(x)=Mx$.

(Local) topological (orbital) equivalence of vector fields[edit]

This is the most coarse classification, which is nevertheless widely used to designate the "same topology of phase portraits". Recall that a smooth vector field $v\in\mathscr X(\R^n,0)$ near an isolated singular point $0$ defines a foliation $\mathscr F$ of the punctured neighborhood $(\R^n,0)\smallsetminus\{0\}$ by (connected, oriented) pieces of phase trajectories of $v$, partition of $(\R^n,0)\smallsetminus\{0\}=\bigsqcup \gamma_a$ into the disjoint union of connected curves $\gamma_a$ tangent to $v$. This partition is often called the "phase portrait" of $v$.

Two vector fields $v$ and $v'$, generating the foliations $\mathscr F,\mathscr F'$, are said to be (orbitally) topologically equivalent, if there exists a germ $h:(\R^n,0)\to(\R^n,0)$ of the orientation preserving homeomorphism, which sends each leaf $\gamma_a$ of $\mathscr F$ into a leaf $\gamma'_{a'}$ of $\mathscr F'$ while preserving the orientation.

Example. The topological equivalence is essentially complete only for germs of analytic vector fields on the plane, where the equivalence classes are characterized by the number and relative position of sectors of different types. For vector fields with a nondegenerate linear part, $v(x)=Ax+\cdots$, $\det A\ne 0$, the topological type is mainly determined by the eigenvalues $\lambda,\mu$ of the linear part, as follows.

In line with these four Poincaré types (saddle, node, focus, center), which in fact constitute four different topological types (saddle, unstable and stable node, center), a degenerate type of a saddle node is usually mentioned, which is represented by the vector field $\dot x=x^2$, $\dot y=y$.

Comparison of classifications (an overview)[edit]

Unlike the left-right classification of germs of maps, the classification by conjugacy changes drastically with the regularity. Probably the easiest is the topological classification of local dynamical systems: a generic local dynamical system can be topologically linearized if it is hyperbolic. The formal classification depends on more delicate properties of eigenvalues (arithmetical identities between them). In the hyperbolic case formal and $C^\infty$-smooth classifications usually coincide, while in general the situation can be more complicated in the relatively low smoothness^[6].

The analytic classification depends on more subtle arithmetic nature of the eigenvalues and in the resonant cases the analytic normal form cannot be finite-parametric.

Example

A parabolic self-map $f\in\operatorname{Diff}(\CC^1,0)$, $f(z)=z+a_2z^2+a_3z^3+\cdots$ (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 Écalle-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, [I93], [IY].

Other local dynamical systems[edit]

When speaking of local dynamics, one can also consder a slightly more general situation of arbitrary finitely generated group action on $\operatorname{Diff}(\RR^n,0)$ or $\operatorname{Diff}(\CC^n,0)$. Instead of just one self-map (vector field), one can consider $r$-tuples of such objects $\{f_1,\dots,f_r\}\subseteq\operatorname{Diff}(\R^n,0)$, $r=2,3,\dots$ (eventually with $r>n$), resp., $\{v_1,\dots,v_r\}$. The "dynamics" generated by such tuples is the subgroup in $\operatorname{Diff}$, generated by these objects as generators: in the discrete time case this is the image of the free group on $r$ letters, in the continuous time case the subgroup is generated by the flow maps $\exp t v_i$, $t\in\R$, $i=1,\dots,r$.

This dynamics may be very complicate in general, especially in the continuous time case, thus some additional restrictions are imposed. For instance, the vector fields are assumed to be commuting, $[v_i,v_j]=0$, $i,j=1,\dots,r$. In this case the flows also commute and the dynamics reduces to a smooth action of the multidimensional real time $\R^r$ on $\operatorname{Diff}(\R^n,0)$: $(t_1,\dots,t_r)\mapsto \exp (t_1v_1+\cdots+t_r v_r)=\exp (t_1v_1)\circ \exp (t_2v_2)\circ\cdots\circ \exp(t_r v_r)$.

Classification of such "multidimensional-time" local dynamical system is formally defined by the same relation: two tuples $\mathbf f=(f_1,\dots,f_r)$ and $\mathbf '=(f_1',\dots,f_r')$ are called equivalent (with the same caveat about the regularity), if there exists a conjugacy $h\in\operatorname{Diff}(\R^n,0)$, which conjugates simultaneously all generators of the two systems: $$ \mathbf f\sim\mathbf h'\iff\exists h:\ h\circ f_i=f_i'\circ h,\qquad \forall i=1,\dots,r. $$

Finally, besides vector fields, one can also consider a problem of local classification of Pfaffian forms. A Pfaffian (differential 1-)form $\xi$ on the real plane $(\R^2,0)$ defines an integrable distribution of lines (eventually with a singularity at the origin) $\{\xi=0\}$ of null spaces which is tangent to a suitable vector field $v_\xi$. If $\xi=A(x,y)\rd x+B(x,y)\rd y$ with, say, analytic germs $A,B$ having an isolated common root at the origin, then the vector field $v_\xi$ takes the form $\dot x=-B(x,y)$, $\dot y=A(x,y)$, which is also analytic. However, the distribution of null spaces is preserved if the 1-form $\xi$ is replaced by a form $u\cdot\xi$, where $u$ is the germ of a non-vanishing function. Classification of null distributions of Pfaffian 1-forms is often referred to as the orbital classification of the respective vector fields: two vector fields are orbitally equivalent if the foliations by integral curves, generated by these fields, are conjugate by a local diffeomorphism (smooth or analytic) of $(\R^n,0)$.

In higher dimension, however, the integrability of the 1-form $\xi$ has to be explicitly postulated; the classification problem for such forms (modulo a scalar multiple, as before) is equivalent to classification of codimension one foliations near a singular point.

Analytic, formal and smooth equivalence[edit]

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:

Resonances[edit]

Linearizability of local dynamical systems very strongly depends on the arithmetical properties of eigenvalues $\l_1,\dots,\l_n$ of the operator $A=\rd v(0)$ (resp., $\mu_1,\dots,\mu_n$ of $M=\rd f(0)$).

A tuple^[1] $\l=(\l_1,\dots,\l_n)\in\CC^n$ is said to be in additive resonance^[2][3], 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. The characteristic property of the resonant vector monomials is their commutation with the linear part: $$ [V,v_{j,\alpha}]=0,\qquad V(x)=Ax,\quad \forall (j,\alpha)\text{ resonant}. $$

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.

Poincaré-Dulac formal normal form[edit]

The central result on the formal classification of local dynamical systems is the Poincaré-Dulac theorem [IY, Sect. 4], [A83, Ch. V]. It claims that 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^[4].

By definition of the resonance monomials, $v$ is in the Poincare-Dulac formal normal form if it commutes with the linear part $V(x)=Ax$, $[v,V]=0$, ditto for the maps. This allows to extend the notion of a normal form for $C^\infty$-smooth fields (self-maps).

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[edit]

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 where all eigenvalues $\l_1,\dots,\l_n$ of the linear part $A=\rd v(0)$ are to one side of the imaginary axis^[5] (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 the "nodal case" $(\l_1:\l_2)=(1:d)$. The corresponding normal form is polynomial, $$ \dot x=\l x,\quad \dot y=d \l y+cx^d,\qquad \l,c\in\R,\ \N\owns d\geqslant 2. $$ Note that this normal form is integrable in quadratures.

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 small denominators, 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^[6] and later improved significantly by A. D. Brjuno [Br]. The sufficient Brjuno condition for self-maps $(\CC^1,0)\to(\CC^1,0)$ was shown to be sharp by J.-C. Yoccoz^[7], 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.

Smooth linearization[edit]

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, $\operatorname{Re}\l_i\ne 0$^[1] for all $i=1,\dots,n$ (resp., on the unit circle, $|\mu_i|\ne 1$ for all $i=1,\dots,n$^[2]). This result is known as the Sternberg^[3]-Chen^[4] theorem, see [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^[5][6] in terms of the hyperbolicity measure, the ratio $$\frac{\max_i|\l_i|}{\min_i|\operatorname{Re}\l_i|},\qquad\text{resp.,}\qquad\frac{\max_i|\mu_i|}{\min_i\bigl||\mu_i|-1\bigr|}.$$

Topological linearization and topological reduction on the center manifold[edit]

The (real) topological classification of hyperbolic local dynamical systems is especially simple. If the linear part $A=\rm dv(0)$ has no eigenvalues on the imaginary axis, then $v$ is topologically equivalent to the "standard saddle" vector field $s(x)$, $$ s(x)=(s_1,\dots,s_n(x)):\quad s_i(x)=x_i,\ i=1,\dots, k,\ s_i(x)=-x_i,\ i=k+1,\dots,n,\qquad\text{for some }k, \ 0\le k\le n. $$ This statement is known as the Grobman-Hartman theorem (for vector fields). For self-maps the hyperbolicity condition requires that the linear part $M=\rd f(0)$ has no eigenvalues on the unit circle. Such a map is topologically equivalent to one of the "standard saddle maps" of the form $$ S(x)=(S_1(x),\dots,S_n(x)):\quad S_i(x)=\pm \tfrac12 x_i,\ i=1,\dots, k,\ S_i(x)=\pm 2 x_i,\ i=k+1,\dots,n, $$ with some $k$ and a certain combination of signs (some of them equivalent to each other).

The number of different normal forms of "standard saddles" is finite, which implies the structural stability of hyperbolic local dynamical systems: a small perturbation of a hyperbolic system does not change its topological type (i.e., the topological equivalence class contains an open neighborhood of the hyperbolic system).

The non-hyperbolic case is also partially covered by the so called Shoshitaishvili reduction principle [Ar83, Sect. 32C], [AR, App. C] ^[1]. For vector fields it takes the following form: any sufficiently smooth vector field is topologically equivalent to a product vector field $v'=(v'_h,v'_c)$ on $(\R^h,0)\times (\R^{n-h},0)$, where the field $v'_h$ on $(\R^h,0)$ is hyperbolic and all eigenvalues of the vector field $v'_c$ are on the imaginary axis.

Nonlinear normal forms[edit]

The Poincare-Dulac normal form is linear in the nonresonant case and integrable in the single-resonance case. For more degenerate cases the number of resonant monomials grows very fast, until the limit case $A=0$ (resp., $M=E$, the identity matrix) all monomials are resonant.

Sometimes even in these very degenerate cases one can single out the "leading" nonlinear terms and use them to simplify the remaining part by suitable conjugacy. The first steps of this classification look rather simple [IY, Sect. 4, 5].

1. A (not identically zero analytic) vector field on the 1D-line $(\R^1,0)$ with vanishing linear part is formally and even analytically equivalent to the polynomial vector field $v(z)=z^{p+1}+az^{2p+1}$, $p=1,2,\dots$, or a rational vector field $v(z)=\frac{z^{p+1}}{1+bz^{p}}$. The natural number $p$ and the complex numbers $a$ (or $b$) are formal invariants (cannot be changed by the formal conjugacy).
2. A holomorphic self-map $f(z)=z+a_{p+1}z^{p+1}+\cdots$ with $a_p\ne 0$ is formally equivalent to the polynomial self-map $z\mapsto z+z^{p+1}+az^{2p+1}$ or to the time one (flow) map of one of the two above vector fields. However, the formal series conjugating $f$ to its formal normal form, almost always diverge, see nonlinear Stokes phenomenon. A similar, although somewhat more involved but still polynomial formal normal form can be written for the self-maps tangent to rational rotations $f(z)=\mu z+\cdots$, $\mu=\exp 2\pi i \theta$, $\theta\in\Q$, with the same remark concerning divergence.
3. A cuspidal singularity is a planar vector field with the linearization matrix $\bigl(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\bigr)$. Since this matrix is nilpotent, all both eigenvalues are zero and all monomials are resonant. The formal normal form in this case corresponds to the Liénard system of the differential equations^[1][2]

$$ \left\{ \begin{aligned} \dot x&=y, \\ \dot y&=\phi(x)+y\psi(x), \end{aligned}\right. \qquad \phi,\psi\in\C[[x]], \tag{Cs} $$ with the formal series $\phi,\psi$ in one variable without linear ($\phi$), resp., free ($\psi$) terms. In contrast with the previous problems, these series are not uniquely defined and can be changed by suitable conjugacies. One can show, either by careful estimations^[3] or by elegant use of global classification of holomorphic bundles over $\C P^1$^[4] that a cuspidal singularity can always be brought to an analytic formal form (Cs) by an analytic conjugacy (and then the series $\phi,\psi$ will automatically converge).

Alas, the difficulties on the way of constructing nonlinear normal forms, mount very fast and no general theory in higher dimensions exists.

Local dynamical systems with additional structure (Hamiltonian)[edit]

2020 Mathematics Subject Classification: Primary: 37J40 Secondary: 37J10 [MSN][ZBL]

In parallel with the "general" dynamical systems, it is important to consider dynamical systems induced by special structure. For instance, one can assume that the local phase space $(\R^n,0)$ is equipped with a Riemannian metric, and consider the class of gradient vector fields, with two such fields being equivalent if there exists a local isometry conjugating these fields (or, what is equivalent in this case, their potentials).

Another, much more important class consists of Hamiltonian systems (in continuous or discrete time) on even-dimensional space. Recall that a symplectic structure on an even-dimensional neighborhood $(\R^{2n},0)$ is the germ of a nondegenerate closed 2-form $\omega\in\varLambda^2(\R^n,0)$. By the Darboux theorem, in suitable local coordinates $(x_1,\dots,x_n,y_1,\dots,y_n)$ such a form looks as $$ \omega =\sum_{i=1}^n \rd y_i\land\rd x_i. $$ A local diffeomorphism $f\in\operatorname{Diff}(\R^{2n},0)$ is called symplectic, or canonical, if it preserves the symplectic structure, $f^*\omega=\omega$. A germ of the vector field $v$ is called canonical, if all its flow maps preserve $\omega$. In this case the Lie derivative $L_v\omega$ vanishes identically, and thus by the homotopy formula the contraction $i_v\omega=\omega(v,\cdot)\in\varLambda^1$ must be a closed, hence exact 1-form: $$ 0=L_v\omega=i_v (\rd\omega)+\rd (i_v\omega)=\rd i_v(\omega)=\rd\big(\omega(v,\cdot)\big)\implies i_v\omega=\rd H. $$ The function $H:(\R^{2n},0)\to(\R,0)$ such that its differential $\rd H$ coincides with the contraction $\omega(v,\cdot)$ is called the Hamiltonian of the vector field $v$ preserving $\omega$. Two Hamiltonians $H,H'$ on the symplectic neighborhood $(\R^{2n},0)$ are called canonically equivalent, if there exists a canonical (symplectic) transformation $f$ such that $H\circ f=H'$. Such transformation necessarily conjugates also the corresponding Hamiltonian vector fields.

The local classification problem for Hamiltonian systems is reduced therefore to the (right) classification of smooth functions by the action of symplectomorphisms, $$ H,H':(\R^{2n},0)\to(\R^1,0),\quad H\sim H'\iff\exists f\in\operatorname{Diff}(\R^{2n},0):\quad f^*\omega=\omega,\ f^*H=H'. $$ The corresponding matrix classification problem was discussed here. For practical reasons, the most important case is that where the Hamiltonian vector field has only imaginary eigenvalues, that is, with the quadratic part linearly equivalent to $$ H_2=\sum_1^n \frac12\omega_i(x_i^2+y_i^2),\qquad \omega_1,\dots,\omega_n\in\R $$ (some of the frequencies may well be zero).

For this classification problem, the notion of resonance has to be modified: of course, any of the $n$ pair of imaginary conjugate eigenvalues $\pm i\omega$ produces infinitely many Poincare-Dulac (additive) resonances, but all these resonant monomials are non-Hamiltonian and hence irrelevant. The resonances which correspond to Hamiltonian monomials, all have the form $$ \left<\omega,\alpha\right>=0,\qquad \omega=(\omega_1,\dots,\omega_n)\in\R^n,\ \alpha=(\alpha_1,\dots,\alpha_n)\in\Z^n_+,\ |\alpha|\ge 2.\tag{HR} $$ The formal normal form of Hamiltonian vector fields (an analog of the Poincare linearization theorem) claims [Ar74] that if the tuple of frequencies is non-resonant, then the Hamiltonian is formally symplectically equivalent to the series in the variables $I_i(x,y)=\tfrac12(x_i^2+y_i^2)$ only, $$ \begin{gathered} H(x,y)=\frac12\sum_1^n \omega_i(x_i^2+y_i^2)+O\bigl(|x|+|y|)\bigr)^3 \implies H\sim\sum_1^n\omega_i I_i+F(I_1,\dots,I_n),\\ I_i=I_i(x,y)=\tfrac12(x_i^2+y_i^2),\ F\in\R[[I_1,\dots,I_n]]. \end{gathered} $$

References and basic literature[edit]