UDC 517.925

MATHEMATICS

A. D. BRYUNO

ON THE CONVERGENCE OF TRANSFORMATIONS OF DIFFERENTIAL EQUATIONS TO NORMAL FORM

(Presented by Academician L. S. Pontryagin on July 16, 1965)

Let \(\varphi_1(X), \ldots, \varphi_n(X)\) be power series in \(x_1, \ldots, x_n\) without constant terms, convergent in some neighborhood of the point \(X=0\). Then \(X=0\) will be a singular point of the system of differential equations

\[ dx_i/dt=\varphi_i(X), \qquad i=1,\ldots,n. \tag{1} \]

Suppose further that \(\xi_i(Y)\) and \(\psi_i(Y)\), \((i=1,\ldots,n)\), are power series in \(y_1,\ldots,y_n\) without constant terms, convergent in some neighborhood of the point \(Y=0\), such that the transformation

\[ x_i=\xi_i(Y), \qquad i=1,\ldots,n, \tag{2} \]

fixed at zero, transforms system (1) into the system

\[ dy_i/dt=\psi_i(Y), \qquad i=1,\ldots,n. \tag{3} \]

In his dissertation \((^1)\), Poincaré proved that there exists a biholomorphic transformation at zero of the form (2), reducing system (1) to the form

\[ dy_i/dt=\lambda_i y_i, \qquad i=1,\ldots,n, \tag{4} \]

if the eigenvalues \(\lambda_1,\ldots,\lambda_n\) of the matrix \(\|\partial\varphi_i/\partial x_j|_0\|\) satisfy certain stringent conditions. The proof consists of two stages. First it is proved that there exist formal power series \(\xi_i(Y)\) giving a formal transformation of system (1) to the form (4). The second stage consists in proving the convergence of the series \(\xi_j(Y)\).

Let us write system (3) in the form

\[ dy_i/dt=y_i g_i(Y)=y_i \sum_{Q\in N_i} g_{iQ}Y^Q, \qquad i=1,\ldots,n, \tag{5} \]

where \(Q=(q_1,\ldots,q_n)\), \(Y^Q=y_1^{q_1}\cdots y_n^{q_n}\); \(N_i=\{Q:\) integers \(q_1,\ldots,q_{i-1}, q_{i+1},\ldots,q_n\geq 0,\ q_i\geq -1,\ \sum_1^n q_k\geq 0\}\), \(i=1,\ldots,n\). Denote by \(\Lambda=(\lambda_1,\ldots,\lambda_n)\) the vector of eigenvalues of the matrix \(\|\partial\varphi_i/\partial x_j|_0\|\).

In my paper \((^2)\) the following theorem was proved:

There exists an invertible formal transformation (2) of system (1) into a system (3) such that: \(\alpha)\) the matrix \(\|\partial\psi_i/\partial y_j|_0\|\) is in Jordan normal form; \(\beta)\) when system (3) is written in the form (5), \(g_{iQ}\) differ from 0 only for those \(Q\) for which the scalar product \((Q,\Lambda)=0\). In this case system (3) shall be called the normal form of system (1).

This theorem is the completion of a long series of works on normal form. In parallel, the following question was studied: for which normal forms (3) does the holomorphy of \(\varphi_i\) imply the holomorphy of the transformation (2)? It had long been shown that in some cases the series \(\xi_i\) diverge. For example, the system \(dx_1/dt=x_1^2,\ dx_2/dt=x_2-x_1\) is reduced to normal form

\[ dy_1/dt=y_1^2,\quad dy_2/dt=y_2 \]

by the transformation

\[ x_1=y_1,\quad x_2=y_2+\sum_{k=1}^{\infty}(k-1)!\,y_1^k, \]

which diverges for every \(y_1\ne 0\).

The convergence of transformations to normal form has been proved up to the present time in the following 4 cases (the conditions are imposed on the eigenvalues \(\lambda\) and on the normal form in the notation (5)):

a) Suppose the numbers \(\lambda_1,\ldots,\lambda_n\) are represented by points of the complex plane, and the convex hull of the points \(\lambda\) does not contain zero \((^3)\);

b) \(n=2,\ \lambda_1=-\lambda_2\ne 0\) and \(g_1(k,k)=-g_2(k,k)\), \(k=1,2,\ldots\) (the Poincaré center case \((^4)\));

c) \(n=3,\ \lambda_1=-\lambda_2\ne 0\) are real, \(\lambda_3=i\), and \(g_{1(k,k,0)}=-g_{2(k,k,0)},\ g_{3(k,k,0)}=0,\ k=1,2,\ldots\) (the hyperbolic case of a periodic solution of a Hamiltonian system \((^5)\));

d) for all integral \(Q\ne 0\)

\[ |(Q,\Lambda)|>\left(\sum_1^n |q_i|\right)^{-\nu},\quad \nu\ge n+1\quad (^{6,9}). \]

Condition a) is necessary and sufficient for the equation \((Q,\Lambda)=0\) to have only a finite number of solutions \(Q\in N=N_1\cup\cdots\cup N_n\) and, for the remaining \(Q\in N\),

\[ |(Q,\Lambda)|>\varepsilon\sum_1^n |q_k|. \]

Therefore this condition is, in its way, limiting. The remaining cases of convergence can be generalized.

Condition 1. The normal form (5) of system (1) is such that, if \(G_Q=(g_{1Q},\ldots,g_{nQ})\), then \(G_Q=\mu_Q\Lambda+\nu_Q\bar\Lambda\), where \(\mu_Q\) and \(\nu_Q\) are complex numbers.

Generally speaking, the normal form is not determined uniquely by the given system.

Lemma 1. If at least one normal form of a given system (1) satisfies condition 1, then every normal form of this system satisfies condition 1.

Condition 2. There exists an \(\varepsilon>0\) such that, for every \(Q\in N\), either \((Q,\Lambda)=0\), or \(|(Q,\Lambda)|>\varepsilon\).

Theorem 1. If, for system (1), some normal form satisfies condition 1, \(\Lambda\) satisfies condition 2, and the \(\varphi_i(X)\) are holomorphic at \(X=0\), then there exists a transformation (2), holomorphic at \(Y=0\), which brings system (1) to normal form.

In cases b) and c), conditions 1 and 2 are fulfilled:

b) condition 1 is fulfilled, since \(G_{(k,k)}=g_{1(k,k)}\lambda_1^{-1}\Lambda,\ k=1,2,\ldots\); condition 2 is fulfilled, since \(|(Q,\Lambda)|=|\lambda_1|\,|q_1-q_2|\ge |\lambda_1|\), if \(q_1\ne q_2\).

c) condition 1 is fulfilled, since

\[ G_{(k,k,0)}=\frac12 g_{1(k,k,0)}\lambda_1^{-1}(\Lambda+\bar\Lambda),\quad k=1,2,\ldots; \]

condition 2 is fulfilled, since the equation \((Q,\Lambda)=\lambda_1(q_1-q_2)+iq_3=0\) determines, in the real space \((q_1,q_2,q_3)\), the line \(q_1=q_2,\ q_3=0\). The distance from this line to the points of the integer lattice not lying on it is bounded below.

Condition 3. Let \(\omega(p)=\min |(Q,\Lambda)|\) over all such \(Q\) that \(\sum |q_i|=p,\ (Q,\Lambda)\ne 0,\ q_i\) are integers, and let \(\omega_k=\omega(p_k)\) be the successive minima of \(\omega(p)\) as \(p\) increases from \(1\) to \(\infty\), i.e. \(p_1=1,\ p_k<p_{k+1}\), and \(\omega_k>\omega_{k+1}\), and for every \(p<p_{k+1}\) one necessarily has \(\omega(p)\ge \omega(p_k)\). Then

\[ \sum_{k=1}^{\infty}\frac{\ln \omega^{-1}(p_k)}{p_k}<\infty . \]

It is essential that those \(Q\) for which \((Q,\Lambda)=0\) do not participate in the formation of the successive minima. A somewhat different notion of successive minima of linear forms was used by Voronoi for the purposes of generalizing the continued-fraction algorithm \((^7)\).

If \(n=2\), \(\lambda=\lambda_1^{+1}\lambda_2\) is real and \(r_k\) is the denominator of the \(k\)-th convergent of the continued fraction of the number \(\lambda\), then condition 3 is equivalent to the condition

\[ \sum_1^\infty r_k^{-1}\ln r_{k+1}<\infty . \]

Lemma 2. For every \(n\) there exists a \(c_n>1\) such that \(p_k>c_n^k\cdot 2^{-1/2}\), \(k=1,2,\ldots\).

Theorem 2. If the system (1) has some normal form (4), \(\Lambda\) satisfies condition 3, and the \(\varphi_i(X)\) are holomorphic at \(X=0\), then there exists a transformation (2), holomorphic at \(Y=0\), which brings system (1) to normal form.

In case г), condition 3 is satisfied, the equation \((Q,\Lambda)=0\) has no solutions in integers \(Q\ne 0\), and the normal form is (4). Siegel’s condition in terms of \(\omega(p)\) is \(\omega(p_k)>p_k^{-\nu}\); then

\[ \sum_1^\infty p_k^{-1}\ln \omega^{-1}(p_k) < \nu \sum_1^\infty p_k^{-1}\ln p_k . \]

The convergence of the latter series follows from Lemma 2.

It remains to note that the normal form (5), satisfying condition 1, is integrable. Let the integral vectors \(R_1,\ldots,R_d\) form a basis in the lattice that is the intersection of the integral lattice with the solutions of the equation \((Q,\Lambda)=0\). Then, first,

\[ Y^{R_i}=\operatorname{const}\equiv c_i \qquad (i=1,\ldots,d) \]

are first integrals and, second,

\[ g_i(Y)\equiv \sum_{(Q,\Lambda)=0} g_{iQ}Y^Q = \widetilde g_i\bigl(Y^{R_1},\ldots,Y^{R_d}\bigr),\quad i=1,\ldots,n, \]

are functions only of the first integrals. Finally,

\[ d\ln y_i=\widetilde g_i(c_1,\ldots,c_d)\,dt,\qquad i=1,\ldots,n \]

(see also § 2 of my paper \({}^{8}\)).

Received
13 VII 1965

REFERENCES

\({}^{1}\) H. Poincaré, Thèse, Paris, 1879.
\({}^{2}\) A. D. Bruno, DAN, 157, No. 6, 1276 (1964).
\({}^{3}\) H. Dulac, Bull. Soc. Math. France, 40, 324 (1912).
\({}^{4}\) H. Poincaré, J. Math. pure et appl., 3 ser., 7 (1881).
\({}^{5}\) J. Moser, Comm. Pure and Appl. Math., 9, 673 (1956); K. Siegel, Collected Translations. Mathematics, 5, 2, 157 (1961).
\({}^{6}\) C. L. Siegel, Nachr. Akad. Wiss. Göttingen, math.-phys. Kl. II-a, 21 (1952); K. Siegel, Collected Translations. Mathematics, 5, 2, 149 (1961).
\({}^{7}\) G. F. Voronoi, Collected Works, 1, Kiev, 1952.
\({}^{8}\) A. D. Bruno, Izv. Akad. Nauk SSSR, Ser. Mat., 29, No. 2, 329 (1965).
\({}^{9}\) V. A. Pliss, Differential Equations, 1, No. 2 (1965).