Normalization Flow in Dynamical Systems

• Normalization Flow is a continuous dynamical system approach that transforms a local vector field to its normal form by systematically eliminating nonresonant nonlinear terms.
• It employs a one-parameter family of invertible coordinate changes and damping operators to achieve exponential decay of nonresonant coefficients while preserving resonant terms.
• Analytic estimates using majorant equations ensure convergence within shrinking domains, linking algebraic normalization to practical analytic transformations as seen in the Siegel–Brjuno theorem.

A normalization flow is a continuous dynamical system constructed to transform an analytic or formal vector field to its normal form, typically in the context of local differential equations near singularities. In the modern mathematical literature, especially following Chernyshev (Chernyshev, 6 Jan 2026), normalization flows are viewed as one-parameter families of invertible coordinate changes that systematically eliminate nonresonant nonlinear terms from the local expansion of a vector field near a fixed point, generalizing the classic Poincaré–Dulac theory through continuous (rather than iterative or formal) averaging.

1. Algebraic Structure and Flow Equation

Let $F$ be the Lie algebra of formal vector fields on $\mathbb{C}^n$, vanishing at the origin. Any $u\in F$ can be written in coordinates $z=(z_1,\ldots,z_n)$ as

$$u(z) = \Lambda z + \sum_{|k| \geq 2} U^m_k z^k e_m,$$

where $$\Lambda = \operatorname{diag}(\lambda_1,\ldots,\lambda_n)$$ captures the linearization, $k$ is a multi-index, and $e_m$ denotes the $m$-th standard basis vector. Decompose the nonlinear part $u(z) - \Lambda z$ into resonant and nonresonant terms according to resonance conditions

$$L_\lambda = \{k \in \mathbb{Z}_+^n : |k| \geq 2,\ \langle \lambda, k \rangle - \lambda_m = 0\text{ for some }m\}.$$

Then $u_0(z)$ denotes the sum over resonant monomials and $u_*(z)$ the nonresonant.

The normalization flow is a one-parameter $t\geq 0$ evolution in $F$ given by

$\partial_t u(t) = -[\xi u(t), \Lambda z + u(t)],$

with initial condition $u(0) = \hat{u}$. The operator $\xi$ acts by damping nonresonant terms,

$$(\xi u)^m(z) = -\sum_{k \notin L_\lambda} e^{-i \arg\langle\lambda,k\rangle} U^m_k z^k,$$

so that the flow generator at each $t$ is the vector field $V_t = \xi u(t)$.

This flow systematically decays the nonresonant components of $u$, while leaving resonant coefficients unchanged, implementing a continuous averaging procedure that converges to the normal form vector field within the formal power series algebra.

2. Continuous Averaging and Formal Normalization

The normalization flow is interpreted via a continuous pull-back of the vector field by a time-dependent family of diffeomorphisms. For an ODE $z'(t) = f(z(t), t)$ with $f$ vanishing at the origin, the vector field $u(z, t)$ evolves as

$$\partial_t u + [f, u] = 0,\quad u(z, 0) = \hat{u}(z).$$

Selecting $f = \xi u$ yields

$\partial_t u = -[\xi u, u].$

Splitting $u = \Lambda z + u_0 + u_*$ and projecting onto nonresonant monomials $z^k e_m$ (with $\langle \lambda, k\rangle \neq 0$), the linearized evolution is

$$\partial_t U^m_k = -|\langle \lambda, k\rangle| U^m_k,$$

with solution $$U^m_k(t) = U^m_k(0) e^{-|\langle \lambda, k\rangle| t}$$. Resonant terms remain stationary, while nonresonant coefficients decay exponentially to zero as $t \to \infty$. Higher-order terms from $v_0(u)$ and $v_*(u)$ are controlled by induction, ensuring triangularity and convergence in the Tikhonov topology.

Consequently, as $t\to\infty$, all nonresonant monomials are eliminated, leaving only the resonant terms — completing Poincaré–Dulac normalization at the (formal) Lie algebraic level.

3. Analyticity and Domain of Convergence

To obtain convergent (analytic) normalization rather than merely formal normalization, analytic estimates are necessary. An explicit majorant (scalar) equation is solved for the supremum norm, reducing the problem to estimating solutions to a Burgers-type differential equation for a majorant function $F(\zeta, t)$, where $\zeta = |z_1| + \cdots + |z_n|$. This yields the implicit formula

$$F = f(\zeta + 4 n t F),\quad f(\zeta) = a \frac{\zeta^2}{\rho - \zeta},\ a = \frac{\| \hat u \|_\rho}{\rho},$$

and, for each $t\geq 0$, establishes analyticity of $u(t)$ on the polydisk $|z_j| < d(t)$, where

$d(t) \geq \frac{\rho}{2(1+2 a t)}.$

Thus, for normalization time $\delta = t$, the radius of analyticity is bounded below by $O(1/(1+A\delta))$ for some $A>0$.

This explicit control on the shrinking domain ensures the practical convergence of the normalizing transformation for finite times, and, under further arithmetic conditions, even in the infinite time limit.

4. Siegel–Brjuno Theorem via Blockwise Normalization Flow

When the eigenvalues $\lambda$ satisfy the Brjuno arithmetical condition (i.e., $\sum_{j=1}^\infty 2^{-j} \ln a_j<\infty$ for the sequence $a_j = \max(1, \Omega_{2^j+1})$, with $$\Omega_s = \max\{1/|\langle\lambda,k\rangle| : 0 < |k| \leq s,\ k \notin L_\lambda\}$$), analytic normalization exists.

The normalization flow operates on successive homogeneous blocks of order $r=2^m+1$ up to $2r-2$, via operators $\xi_r$ local to such degrees. Each block normalization yields analytic coordinate changes $\nu_m$ shrinking the polydisk radius by at most $e^{-\epsilon_m}$, with $\epsilon_m$ controlled by the small divisors. Infinite composition of these conjugacies converges to an analytic change of coordinates, conjugating the vector field to its (possibly nonlinear) normal form in a neighborhood of the fixed point, thereby providing a new constructive proof of the Siegel–Brjuno theorem (Chernyshev, 6 Jan 2026).

5. Significance and Connections

Normalization flows generalize the traditional iterative (or Lie-theoretic) approach to normal form theory, providing:

• A continuous, explicit deformation process that interpolates between the original vector field and its normal form.
• An operator-theoretic framework that accommodates both the formal and analytic categories, with explicit convergence estimates.
• A concrete construction applicable to analytic vector fields and multidimensional, possibly resonant, singularities.

In this context, normalization flows unify perspectives from the geometry of normal forms, Lie group actions (continuous conjugacies), and the PDE/ODE approach to normalizing transformations. The majorant and analytic estimates supply control over the convergence domain, directly linking algebraic convergence to geometric (analytic) domains.

6. Related Theoretical Frameworks

Continuous normalization flows extend the earlier apparatus of homological equations used in classical normal form theory. In Hamiltonian and complex dynamical systems, analogous flows have facilitated proofs of analytic linearization, Birkhoff normal form, and KAM-type theorems, provided sufficient control over small divisors.

The explicit flow construction also enables finer analysis of the Brjuno–Siegel condition and yields more flexible methods for controlling the loss of analyticity under normalization, compared to those based solely on iterative formal conjugation.

References: (Chernyshev, 6 Jan 2026)