On computation of normal forms A.D. Bruno Institute of Applied Mathematics, Moscow 125047 Russia (bruno@applmat.msk.su)

Keywords: differential equations, stationary point, resonant terms, Hamiltonian system.

Let X ∈ IR^{n} or IC^{n}. In a neighbourhood of the stationary point X = 0 we consider the ODE system

dX/dt =^{def} Ẋ = AX + Φ(X)  (1)

where Φ is a vector power series, Φ = O(∥X∥^{2}).

Problem: Simplify the system (1) preserving main properties of its solutions.

There are two methods for that:

1. If in (1) the matrix A ≠ 0, then we can perform a formal transformation of coordinates

X = BY + Ξ(Y)  (2)

to bring the system (1) into a normal form

Ẏ = CY + Ψ(Y).  (3)

2. If A = 0 then instead of the system (1) we can consider the truncated system

Ẋ = Φ̂_{j}^{(d)}(X),  (4)

where components of the vector Φ̂_{j}^{(d)}(X) are some quasihomgeneous parts of components of the vector-series Φ(X). Generally speaking, there are several different truncated systems (4). To find them we use the Newton polyhedron of the system (1).

In my book [1] it was shown that in any compicated case, combining both methods, one can reduce the system (1) to several simple systems. Here we shall only consider the first method in details.

Let the matrix C = (c_{ij}) be in a normal form (for instance, in the Jordan one). So its diagonal consists of its eigenvalues (λ_{1}, …, λ_{n}) = Λ, and c_{ij} = 0 if λ_{i} ≠ λ_{j}. Let us write components of the vector Ψ in (3) in the form

ψ_{i}(Y) =^{def} y_{i}g_{i}(Y) =^{def} y_{i} ∑ g_{iQ}Y^{Q}, i = 1, …, n,  (5)

where Q = (q_{1}, …, q_{n}) ∈ Z^{n} and Y^{Q} = y_{1}^{q_{1}} … y_{n}^{q_{n}}.

Definition 1 [2,1]. The system (3), (5) is called the resonant normal form, if expansions (5) contain only such terms g_{iQ}Y^{Q} for which the scalar product

⟨Q, Λ⟩ =^{def} q_{1}λ_{1} + … + q_{n}λ_{n} = 0.  (6)

Theorem 1 [2,1]. For each system (1) there exists a formal transformation (2) which brings (1) into its resonant normal form (3).

Let the equation (6) have exactly k integer solutions Q ∈ Z^{n} which are linearly independent over IR. Then the system (1) has the k – fold resonance.

Definition 2 [3,1]. The nonlinear transformation of coordinates

z_{i} = y_{1}^{α_{i1}}y_{2}^{α_{i2}} … y_{n}^{α_{in}}, i = 1, …, n  (7)

is called the power transformation with the matrix α = (α_{ij}).

It is a linear transformation of logarithms of coordinates log Z = α log Y where log Z = (log z_{1}, …, log z_{n})^{∗}.

Theorem 2 [2,1]. If the system (1) has the k – fold resonance, then there exists such a power transformation (7) which brings the resonant normal form (3) into the system

˙ (log z_{i}) = f_{i}(z_{1}, …, z_{k}), i = 1, …, n.  (8)

Here all α_{ij} are integers and detα = ±1.

Thus, to solve the system (8) one must solve its subsystem of order k:

ż_{i} = z_{i}f_{i}(z_{1}, …, z_{k}), i = 1, …, k.  (9)

The system (9) has not a linear part. So to simplify the system (9) we must find its truncated systems

ż_{i} = z_{i}f̂_{ij}^{(d)}(z_{1}, …, z_{k}), i = 1, …, k,  (10)

using the Newton polyhedron of the system (9). The system (10) is quasihomogeneous, so we can again use a power transformation to simplity it.

There are 3 other kinds of the normal form (shortly n.f.):

2. Belitski n.f.

3. Hamiltonian resonant n.f.

4. Hamiltonian Belitski n.f.

Definition 3 [4,5]. The system

Ẏ = CY + Ψ̃(Y)  (11)

is called the Belitski normal form, if it is the resonant n. f. and

\frac{∂Ψ̃}{∂Y}C̄^{∗}Y − C̄^{∗}Ψ̃(Y) = 0,  (12)

where the matrix C̄^{∗} is conjugate to the matrix C.

If components of the vector Ψ̃ we write as

ψ̃_{i} = y_{i} ∑ g̃_{iQ}Y^{Q}, i = 1, …, n

then these expansions for the Belitski n. f. (11) have only resonant terms g̃_{iQ}Y^{Q} with property (6) but some of them are absent.

Theorem 3 [4,5]. For each system (1) there exists a formal transformation (2) into its Belitski n. f.

The power transformation (7) of Theorem 2 brings the Belitski n. f. (11) into the system

(log ż_{i}) = f̃_{i}(z_{1}, …, z_{k}), i = 1, …, n

with the subsystem

ż_{i} = z_{i}f̃_{i}(z_{1}, …, z_{k}), i = 1, …, k.

Let us compare its truncated systems

ż_{i} = z_{i}ˆf̃_{ij}^{(d)}(z_{1}, …, z_{k}), i = 1, …, k.  (13)

with truncations (10) of the system (9), obtained from the resonant n. f. (3). Some of them can conside:

f̂_{ij}^{(d)}(Y) = ˆ f̃_{ij}^{(d)}(Y), i = 1, …, k.

Thus, to compute such truncations it is enough to compute the resonant n. f. that is simpler than a computation of the Belitski n.f.

Example 1. In [6] there was considered an invertible system (1) with n = 4, where the matrix A depends of two small parameters: A = A(μ_{1}, μ_{2}). The unperturbed matrix A = A(0, 0) is the 4 × 4 Jordan block with zero eigenvalue. Then the system (1) is its own resonant n.f. By means of a special program [7] for computation of the Newton polyhedron we have computed the Newton polyhedron of that system (1). It gave 5 truncated systems; 4 of them are integrable and one (the main truncation) is nonintegrablite. We have computed the Belitski n. f. of the system (1) and its Newton polyhedron. It gave 4 truncated systems for the Belitski n. f.; 3 of them are integrable and the fourth is exactly the same main truncation that was obtained for initial system (1).

Now let us consider the Hamiltonian system (1):

ẋ_{i} = \frac{∂h}{∂x_{i+m}}, ẋ_{i+m} = −\frac{∂h}{∂x_{i}}, i = 1, …, m  (14)

where n = 2m and the Hamiltonian function h is a power series

h = ∑ h_{Q}X^{Q}.  (15)

Let Λ be the vector of eigenvalues of the matrix of the linear part of the system (14).

Definition 4 [8]. Hamiltonian system (14) is called the resonant normal form if the expansion (15) contains only terms h_{Q}X^{Q} with the property (6).

Theorem 4 [8, § 12]. For any Hamiltonian system (14), there exists a canonical transformation (2) which brings (14) into the Hamiltonian resonant n. f.

For the Hamiltonian resonant n. f., one can compute the reduced subsystem

ż_{i} = z_{i}h̃_{i}(z_{1}, …, z_{k}), i = 1, …, k.  (16)

On the other side, for the Hamiltonian system (14), we can compute the resonant n. f. (3), using a noncanonical transformation (2) and obtain its reduced sybsystem (10). Sometimes the main truncations of the systems (10) and (16) differ by a linear transformation of variables only. In such cases, to obtain the main truncation we can use the noncanonical normalizing transformation, that demands essentially more simple computation that the canonical one. Moreover, to compute coefficients of the truncated system (10) we can use the explicit formula (35) in [1, Ch. III, § 1]. In [9] such approach was used in a system related to the water–wade problem.

Let C be the matrix of the linear part of the Hamiltonian system (14).

Definition 5 [5]. The Hamiltonian system (14) is called the Hamiltonian Belitski normal form if the scalar product

⟨C̄^{∗}X, ∂h/∂X⟩ = 0.

Theorem 5 [5]. There exists a formal canonical transformation (2), which brings the Hamiltonian system (14) into its Hamiltonian Belitski n. f.

Indeed the normal form and the normalizing transformation are not unique. Using more special normalizing transformation, we obtain more special normal form, but it demands more complicated computation.

Thus, me have 4 normal forms:

(i) Resonant n. f.

(ii) Belitski n. f.

(iii) Hamiltonian resonant n. f.

(iv) Hamiltonian Belitski n. f.

There are inclusions:

(i) ⊃ (iii) ⊃ (iv),
(i) ⊃ (ii) ⊃ (iv).

Apparently in some cases we can solve a problem, computing less special n. f., i.e. using more simple computation. Five methods of computation of normal forms are discussed in [10].

The work was partly supported by the International Science Foundation, Grant M3W300.

References

[1] Bruno, A.D. Local Methods in Nonlinear Differential Equations. Springer -Verlag, Berlin, 1989.

[2] Bruno, A.D. Normal form of differential equations. Soviet Math. Dokl. 5(1964) 1105 – 1108.

[3] Bruno, A.D. The asymptotic behavior of solutions of nonlinear systems of differential equations. Soviet Math. Dokl. 3(1962) 464 – 467.

[4] Belitski, G.R. The normal forms of local mappings. Uspehi Mat. Nauk, 30:1(1975) 223 (Russian).

[5] Belitski, G.R. Normal Forms, Invariants and Local Mappings. Kiev: Naukova Dumka, 1979 (Russian).

[6] Soleev, A. and Aranson, A.B. First approximations of a reversible ODE system. Preprint N 28 of the Keldysh Inst. of Appl. Math., Moscow, 1995 (Russian).

[7] Soleev, A. and Aranson, A.B. Computation of a polyhedron and of normal cones of its faces. Preprint N 36 of the Keldysh Inst. of Appl. Math., Moscow, 1994 (Russian).

[8] Bruno, A.D. Analytical form of differential equations. Trans. Moscow Math. Soc.26(1972) 199–239.

[9] Bruno, A.D. and Soleev, A. Local analysis of a singularity of an invertible ODE system. Preprints Nos. 40, 47, 54 of Keldysh Inst. of Appl. Math., Moscow, 1995 (Russian).

[10] Bruno, A.D. Methods of computation of the normal form. Russian Academy Sci. Doklady. Mathem. (1995) (to appear).