Carleman linearization and normal forms for differential systems with quasi-periodic coefficients

We study the matrix representation of Poincaré normalization using the Carleman linearization technique for non-autonomous differential systems with quasi-periodic coefficients. We provide a rigorous proof of the validity of the matrix representation of the normalization and obtain a recursive algorithm for computing the normalizing transformation and the normal form of the differential systems. The algorithm provides explicit formulas for the coefficients of the normal form and the corresponding transformation.

In this paper, we apply Carleman linearization to the problem of constructing the Poincaré normal form for non-autonomous differential equations with quasi-periodic coefficients, as proposed in Chermnykh (1987).
Poincaré normalization for non-autonomous differential systems with quasi-periodic coefficients is used, for example, in celestial mechanics to construct the GPT-method of general planetary theory (Brumberg 1970;Brumberg and Chapront 1973) and GPTcompatible methods of the general Earth's rotation theory (Brumberg and Ivanova 2011) and the Moon's motion theory (Ivanova 2014). Planetary theories have been historically developed to provide ephemerides of planetary bodies; reviews of planetary theories can be found in Seidelmann (1993) and Kholshevnikov and Kuznetsov (2007).
The first step of GPT-method is to reduce the differential system in the vicinity of unperturbed motion by a proper choice of coordinates to the form where the components of vector f are holomorphic functions with respect to the components of vector X ∈ C ν ; f depends on t by means of quasi-periodic functions; f (0, t) = 0 ; the Jacobi matrix (∂f /∂X)| X=0 is of Jordan form (even diagonal) with purely imaginary eigenvalues.
The second step of the GPT-method is to construct iterative transformations of the differential system (1) with quasi-periodic coefficients to the normal form (Birkhoff 1927). The system (1) is subjected to the normalizing iterative transformations excluding all short-period terms and leading to the secular system with slowly changing variables. As a result, one obtains the solution of the secular system avoiding the appearance of the non-physical secular terms.
The most cumbersome operation of GPT-method is the Poincaré normalization of the differential system (1). The evaluation problem in celestial mechanics is of particular importance owing to the large number of terms in the series. The analytical calculations in GPT-method are performed by the Poisson series processor (Brumberg 1995;Ivanova 2001).
In the present paper we develop a recursive algorithm based on Carleman linearization for computing the series. The algorithm provides explicit formulas for the coefficients of the Poincaré normal form and the normalizing transformation. Therefore, the Carleman linearization technique may be advantageous for constructing normal forms in a literal form.
This paper is organized as follows. In the next section, we describe our notations. In sections 'The Weierstrass matrix' and 'The Carleman matrix' , we study two classes of infinite matrices, corresponding to nonlinear mappings and differential systems. In section 'Transformations' , we study the transformations of infinite matrices. Section 'Normal form of a Carleman matrix' presents the recursive algorithm for constructing the Poincaré normalization. The proofs of the propositions are given in section 'Proofs' . Finally, we give an example and discuss the results.
Notations R-real number field, C-complex number field, K-either real or complex number field, Z-the ring of integer numbers, N 0 -the set of integer non-negative numbers, , -scalar product, A-algebra over K of quasi-periodic functions R → K, defined by K-valued finite trigonometric sums, d dt -differential operator in A.
We introduce the countable sets of variables:

Proposition 1 The variables y[m] and x[m] satisfy the linear equations
g[e α , n] X n , (2) where the coefficients g[m, n] for |m| > 1 may be obtained by the recursion formula We introduce natural ordering for the set N ν 0 . Let k ≺ l for k, l ∈ N ν 0 , if |k| < |l| or |k| = |l| and there exists a number β such that k α = l α for α < β and k β > l β .
Using the ordering of N ν 0 , we introduce infinite-dimensional vectors x = (x[n]), y = (y[n]) and the matrix G = (g[m, n]) for m, n ∈ N ν 0 such that

Definition 1
The infinite matrix G is said to be a Weierstrass matrix with a range ν if the elements g[m, n] ∈ A satisfy the condition given by (3) for |m| > 1.
Let ψ denote the constructed correspondence between mappings and matrices The following proposition describes the structure of Weierstrass matrices.
Definition 2 1. The mapping g ∈ A[ν] is said to be invertible if there exists

Corollary 1 Invertible Weierstrass matrices W ν (A) form a group with multiplication, which is isomorphic to the group of invertible mappings A[ν] with composition.
Corollary 2 Let f , g ∈ A[ν] and g be invertible. We introduce the mapping Then, the following diagram is commutative

The Carleman matrix
Here, we construct a class of infinite matrices representing ordinary differential equations.
, represented by a formal power series:

Proposition 4 The variables x[n]
(2) satisfy the following differential equations: where the coefficients f [m, n] for |m| > 1 may be obtained from (4) by the formula Using the ordering of N ν 0 , we introduce the infinite-dimensional vector x = (x[n]) and matrix F = (f [m, n]) for m, n ∈ N ν 0 such that: where d ′ dt denotes differentiation of components.

Definition 3 The infinite matrix F is said to be a Carleman matrix of range ν
if the elements f [m, n] ∈ A for |m| > 1 satisfy the condition (5).
Let ϕ denote the constructed correspondence between vector fields and matrices It is easy to see that ϕ is an isomorphism of linear spaces. The following proposition describes the structure of the Carleman matrix.

Transformations
Now, we consider the substitution defined by the Weierstrass matrix G into differential equation (6) defined by the Carleman matrix F. We prove that the result of the substitution (7) gives the differential equation defined by the Carleman matrix (Proposition 6) and may be interpreted in terms of vector fields (Proposition 7).
Then, the following diagram is commutative:

Normal form of a Carleman matrix
In this section, we provide a definition of the normal form of a Carleman matrix. We also introduce a method for reducing the corresponding differential equations to the normal form.
Let C ν T (A) denote the linear space over K of upper triangular Carleman matrices with main diagonal elements from K. Let W ν T (A) denote the semi-group of upper triangular Weierstrass matrices with main diagonal elements from K.
Let A be the algebra over K of K-valued finite trigonometric sums Consider the matrix H ∈ C ν T (A) with elements Let ∈ K ν denote the first ν main diagonal elements of H : α = h[e α , e α ], α = 1, . . . , ν .
Definition 4 (m, n, k)-resonance holds if the following condition is satisfied: where m, n ∈ N ν 0 , k ∈ Z σ . The harmonic h[m, n, k] exp�iωt, k� is said to be resonant if (m, n, k)-resonance holds. A Carleman matrix H is reduced to the normal form if all of its non-zero harmonics are resonant.

Theorem For any Carleman matrix
Proof We obtain the components of G and H in the following order: We restrict ourselves to the case g[n, n] = 1 for n ∈ N ν 0 .
It follows from the equation FG − d ′ dt G = GH that we can make the non-resonant harmonic h[e α , n, k]exp�iωt, k� vanish by a proper choice of each g[e α , n, k], namely If (e α , n, k)-resonance holds, we cannot eliminate the corresponding resonant harmonic via a choice of g[e α , n, k]. In this case, g[e α , n, k] may be assigned an arbitrary value. Then, one obtains the resonant harmonic in H as follows By Proposition 6, one obtains the components of G and H below the ν-row by (3) and (5)

Remark 1
The leading ν rows of G and H determine the normalizing transformation X = g(t, Y) and the normal form of the differential equation d dt X = h(t, X), respectively, where G = ψ(g) and H = ϕ(h).

Remark 2
The elements of the inverse matrix G −1 = (g * [m, n]) may be obtained together with the elements of G in the order (8). By equation G −1 G = E, one obtains that:

Proofs
In this section, we provide proofs of the propositions introduced above.

Proof of Proposition 1 Using the properties of homogeneous polynomials, we obtain:
Proof of Proposition 2 Part 1. It is an immediate consequence of the definition provided in (3).
For later considerations, it will be useful to prove a property of Weierstrass matrices.

Proof of Proposition 4 One obtains that:
This mathematics proves the proposition. Remark 4 Another definition for the Carleman matrix may be introduced. Let α = 1, . . . , ν and m ∈ N ν 0 . Then:
Step 2. Let H = G −1 d ′ dt G. We prove that H ∈ C ν (A). Let the superscript point denote differentiation d dt .

It follows that:
This mathematics proves (14).