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.

Remark

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 $\operatorname{Diff}(\CC^n,0)$. For instance, one can consider commuting tuples of vector fields $\{v_1,\dots,v_k\}$ on $(\RR^n,0)$, $[v_i,v_j]=0$. This "dynamical system with $k$-dimensional time" corresponds to the action of the abelian group $\R^k$ on $(\R^n,0)$ and is important in the theory of integrable systems. Another, in a sense, completely opposite case, corresponds to the simultaneous classification of several self-maps $\{f_1,\dots,f_k\}\subseteq\operatorname{Diff}(\C^1,0)$. This problem defines (generically) the action of the free group $F_k$ on $k$ generators. Classification (to the extent is possible) of such tuples is important for understanding the complex topology of holomorphic foliations. In most such cases one does not have any finite lists of normal forms, thus the classification problems have to address some more general properties of the corresponding actions.

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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, [I93], [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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Resonances [A83, Chapter V], [IY, Sect. 4]

$\def\l{\lambda}$ 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 equivalent to their Jordan normal forms.

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.

Poincaré-Dulac formal normal form [IY, Sect. 4], [A83, Ch. V]

The central result on the formal classification of local dynamical systems is the Poincaré-Dulac theorem.

Theorem (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 (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 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, 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 [H, Ch. IX]

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, $\operatorname{Re}\l_i\ne 0$ for all $i=1,\dots,n$ (resp., on the unit circle, $|\mu_i|\ne 1$ for all $i=1,\dots,n$). This result is known as the Sternberg-Chen 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 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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Nonlinear normal forms [IY, Sect. 4, 5]

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.

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 system of the differential equations

$$ \left\{ \begin{aligned} \dot x&=y, \\ \dot y&=\phi(x)+y\psi(x), \end{aligned}\right. \qquad \phi,\psi\in\C[[x]], \tag{Cs} $$ with the formal series $\phi,\psi$ in one variable. In contrast with the previous problems, these series are not uniquely defined and can be changed by suitable conjugacies. F. Loray proved in 2006 that a cuspidal singularity can always be brought to an analytic formal form (Cs) with converging series $\phi,\psi$ by an analytic (convergent) conjugacy. However, the proof does not allow for term-by-term computation of either the normal form itself or of the normalizing transformation.

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

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References and basic literature

[sort]
[I93]	Yu. Ilyashenko, Nonlinear Stokes phenomena, Nonlinear Stokes phenomena, 1--55, Adv. Soviet Math., 14, Amer. Math. Soc., Providence, RI, 1993, MR1206041
[IY]	Yu. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations. Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008 MR2363178
[A83]	Arnold V. I., Geometrical methods in the theory of ordinary differential equations. Grundlehren der Mathematischen Wissenschaften, 250. Springer-Verlag, New York-Berlin, 1983, MR0695786
[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.
[H]	P. Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney 1964, MR0171038
[AI88]	V. I. Arnold, Yu. I. Ilyashenko, Ordinary differential equations, Encyclopaedia Math. Sci., 1, Dynamical systems, I, 1--148, Springer, Berlin, 1988, MR0970794
[AAIS]	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, MR1287421