UDC 517.925

ON THE EXISTENCE OF INVARIANT TORI IN A NEIGHBORHOOD OF THE ZERO SOLUTION OF A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

Yu. N. Bibikov, V. A. Pliss

Consider a system of differential equations of the following form:

\[ \frac{dx_i}{dt}=\lambda_i i x_i+i f_i(x_1,\ldots,x_n,y_1,\ldots,y_n), \]

\[ \frac{dy_i}{dt}=-\lambda_i i y_i-i f_i(y_1,\ldots,y_n,x_1,\ldots,x_n) \quad (i=1,\ldots,n), \tag{1} \]

where \(i^2=-1\), and the \(f_i\) are power series with real coefficients in the variables \(x_1,\ldots,x_n,y_1,\ldots,y_n\), beginning with terms of order not less than the second and converging for sufficiently small values of \(|x_i|\), \(|y_k|\) \((k=1,\ldots,n)\). Concerning the numbers \(\lambda_1,\lambda_2,\ldots,\lambda_n\), we shall assume that they are real and incommensurable, i.e., for any integers \(k_1,k_2,\ldots,k_n\) not all equal to zero, the inequality

\[ \sum_{j=1}^{n} k_j\lambda_j\ne 0 \tag{2} \]

holds.

Various systems of differential equations encountered in applications can be reduced to the form (1). Consider, for example, a system of second-order differential equations depending in an even manner on the first derivatives,

\[ \frac{d^2\xi_i}{dt^2}+\lambda_i^2\xi_i+g_i(\xi_j,\dot{\xi}_j^{\,2})=0 \quad (i,j=1,\ldots,n), \tag{3} \]

where the real numbers \(\lambda_i\) satisfy condition (2), and the \(g_i\) are series with real coefficients in powers of their arguments, beginning with terms of order not less than the second with respect to \(\xi_j,\dot{\xi}_j\). Setting in system (3)

\[ x_k=\xi_k-\frac{i}{\lambda_k}\dot{\xi}_k,\quad y_k=\xi_k+\frac{i}{\lambda_k}\dot{\xi}_k \quad (k=1,\ldots,n) \tag{4} \]

and

\[ \frac{1}{\lambda_k}\, g_k\left(\frac{1}{2}(x_j+y_j),-\frac{\lambda_j^2}{4}(x_j-y_j)^2\right) =f_k(x_j,y_j) \quad (k,j=1,\ldots,n), \]

we obtain system (1). The substitution (4) shows that if \(\xi_i\) are real, then \(x_i\) and \(y_i\) are complex conjugates.

From the very form of system (1) it follows that if, at some instant of time on a solution of this system, the quantities \(x_i\) and \(y_i\) are complex conjugates, i.e. \(x_i=\overline{y_i}\) \((i=1,\ldots,n)\), then throughout the interval on which this solution is defined the equalities \(x_i=\overline{y_i}\) \((i=1,\ldots,n)\) hold. In what follows we shall consider only solutions with complex-conjugate initial values.

In the present work we shall show that in any neighborhood of the origin of coordinates there are \(n\)-dimensional invariant toroidal surfaces. Moreover, it will be shown that the measure of the set of points lying on these surfaces is, as an infinitesimal quantity, equivalent to the measure of the neighborhood under consideration. Thus, the “majority” of solutions of system (1) remain on invariant surfaces and do not leave an arbitrarily small neighborhood of the origin of coordinates.

We shall prove this fact on the basis of the method of successive approximations proposed by A. N. Kolmogorov [1, 2], using which V. I. Arnol'd [3] obtained an analogous result for canonical systems.

§ 1. First we shall show, following I. G. Malkin [4], that system (1) can be reduced by means of an analytic transformation to a special form. More precisely, the following assertion is valid.

Lemma 1.1. Whatever the natural number \(m_0\), there exists a transformation

\[ \left\{ \begin{aligned} x_i&=u_i+\varphi_i(u_j,v_j),\\ y_i&=v_i+\varphi_i(v_j,u_j) \end{aligned} \right. \qquad (i,j=1,\ldots,n), \tag{1.1} \]

where the functions \(\varphi_i\) are polynomials

\[ \varphi_i(u_j,v_j)=\sum_{2}^{2m_0} b_i^{k_j,l_j}u_j^{k_j}v_j^{l_j} \]

with real coefficients \(b_i^{k_j,l_j}\), which brings system (1) to the form

\[ \left\{ \begin{aligned} \dot u_i&=i\lambda_i u_i+iu_iH_i(z_j)+i\Phi_i(u_j,v_j),\\ \dot v_i&=-i\lambda_i v_i-iv_iH_i(z_j)-i\Phi_i(v_j,u_j) \end{aligned} \right. \qquad (i,j=1,\ldots,n), \tag{1.2} \]

where

\[ z_j=u_jv_j,\qquad H_i(z_j)=\sum_{1}^{m_0-1} a_i^{\sigma_j}z_j^{\sigma_j}, \tag{1.3} \]

\[ \Phi_i(u_j,v_j)=\sum_{2m_0+1}^{\infty} M_i^{k_j,l_j}u_j^{k_j}v_j^{l_j} \tag{1.4} \]

and all coefficients \(a_i^{\sigma_j}\) and \(M_i^{k_j,l_j}\) \((i,j=1,\ldots,n)\) are real numbers.

Proof. Substituting (1.1) into the first equations of system (1) and taking (1.2) into account, we obtain the following system of equations for \(\varphi_i(u_j,v_j)\):

\[ \sum_{l=1}^{n}\frac{\partial\varphi_i}{\partial u_l}\lambda_lu_l -\sum_{l=1}^{n}\frac{\partial\varphi_i}{\partial v_l}\lambda_lv_l -\lambda_i\varphi_i+u_iH_i(z_j)= \]

\[ =f_i\bigl(u_j+\varphi_j(u_j,v_j),\,v_j+\varphi_j(v_j,u_j)\bigr) -\Phi_i(u_j,v_j)+ \]

\[ -\sum_{l=1}^{n}\frac{\partial\varphi_i}{\partial u_l} \bigl(u_lH_l+\Phi_l(u_j,v_j)\bigr)- \]

\[ -\sum_{l=1}^{n}\frac{\partial \varphi_i}{\partial v_l}\bigl(v_l H_l+\Phi_l(v_j,u_j)\bigr). \tag{1.5} \]

Equating in system (1.5) the coefficients of like powers of \(u_j, v_j\), we obtain a system of linear algebraic equations for determining \(b_i^{k_j,l_j}\) and \(a_i^{\sigma_j}\):

\[ \left[\sum_{j=1}^{n}(k_j-l_j)\lambda_j-\lambda_i\right] b_i^{k_j,l_j}+a_i^{\sigma_j} = c_i^{k_j,l_j} \quad (i,j=1,\ldots,n). \tag{1.6} \]

In this system the term \(a_i^{\sigma_j}\) occurs only in those equations for which \(k_j=l_j+\delta_{ij}\) \((j=1,\ldots,n)\) (\(\delta_{ij}\) is the Kronecker symbol), and in this case \(\sigma_j=l_j\); \(c_i^{k_j,l_j}\) are polynomials with real coefficients in those coefficients \(a_i, b_i\) whose sum of upper indices is less than
\[ k=k_1+l_1+\cdots+k_n+l_n. \]

If \(k_j=l_j+\delta_{ij}\), then
\[ \sum_{j=1}^{n}(k_j-l_j)\lambda_j-\lambda_i=0, \]
and, putting \(b_i^{k_j,l_j}=0\), we arrive at the equation
\[ a_i^{\sigma_j}=c_i^{k_j,l_j}. \]

If, however, \(k_j\ne l_j+\delta_{ij}\) for at least one \(j\), then
\[ \sum_{j=1}^{n}(k_j-l_j)\lambda_j-\lambda_i\ne 0, \]
and the term \(a_i^{\sigma_j}\) is absent from equation (1.6).

Thus, the quantities \(b_i^{k_j,l_j}\) are determined uniquely from (1.6). Hence the polynomials \(\varphi_i(u_j,v_j)\) are also determined. If we substitute (1.1) into the second equations of system (1), then we obtain a system of partial differential equations analogous to system (1.5). It is not difficult to verify that the functions \(\varphi_i(v_j,u_j)\) satisfy this system.

The lemma is proved.

In what follows we shall assume that the series (1.4), representing the functions \(\Phi_i\), converge for \(|u_j|\le 1,\ |v_j|\le 1\), and that for such \(u_j,v_j\) the inequalities
\[ |u_i H_i(z_j)+\Phi_i(u_j,v_j)|<1 \quad (i,j=1,\ldots,n) \tag{1.7} \]
hold. This assumption involves no loss of generality, since these conditions can be achieved by making in system (1.2) the change of variables
\[ u_i=\gamma u_i^*,\qquad v_i=\gamma v_i^* \]
with sufficiently small positive \(\gamma\).

We make one further assumption concerning system (1). Throughout the sequel we shall suppose that the Jacobian of the functions \(H_i\) with respect to the variables \(z_j\) is different from zero at \(z_j=0\):
\[ J=\left|\left.\frac{\partial H_i}{\partial z_j}\right|_{z_j=0}\right| \ne 0 \quad (i,j=1,\ldots,n). \tag{1.8} \]

We now choose a sufficiently small positive number \(\Delta\) so that the following conditions hold. Let
\[ Q_i(z_j)=\sum_{1}^{\infty}\alpha_i^{\sigma_j}(z_k)z_j^{\sigma_j} \quad (i,j,k=1,\ldots,n), \]
where \(\alpha_i^{\sigma_1,\ldots,\sigma_n}(z_k)\) are analytic functions of the variables \(z_k\) for \(|z_k|\le \Delta^2\),
\[ \alpha_i^{\sigma_j}=\alpha_i^{\sigma_j} \quad (i=1,\ldots,n) \quad \text{for } \sigma=\sigma_1+\cdots+\sigma_n\le m_0-1, \]

where \(a_{i}^{\sigma j}\) are from formula (1.3), and

\[ \left|a_{i}^{\sigma j}(z_k)\right|<\Delta^{-\sigma-1/2}\quad \text{for all } \sigma\geq 1. \]

Next, let \(\mu_i\) and \(\alpha\) be such that the inequalities

\[ \left|(\lambda_i-\mu_i)+Q_i(z_j)\right|<\alpha \quad (i=1,\ldots,n) \]

define a domain of the form

\[ |z_i-z_i^*|<\beta_i \quad (i=1,\ldots,n), \]

contained in the domain \(|z_i|\leq \Delta^2\). We shall assume \(\Delta\) so small that the numbers \(\beta_i\) satisfy the inequalities

\[ 0.9\,\beta_i^*<\beta_i<1.1\,\beta_i^*, \]

where the numbers \(\beta_i^*\) correspond to the case when the \(Q_i\) are linear functions of the form
\(Q_i=a_i^{1,0,\ldots,0}z_1+\ldots+a_i^{0,\ldots,0,1}z_n\).

Let \(\rho<\Delta\). We choose the constants \(\mu_1,\ldots,\mu_n\) with the following calculation, so that the domain

\[ \left|(\lambda_i-\mu_i)+H_i(z_j)\right| < \frac{\rho^3}{2n\,[3(2m_0+1)]^{\,n+2}} \quad (i=1,\ldots,n) \tag{1.9} \]

belong to the domain \(|z_i|<\dfrac{1}{4}\rho^2\) and

\[ \left|\sum_{j=1}^{n} p_j\mu_j\right| > \frac{\rho^3}{\left(\sum_{j=1}^{n}|p_j|\right)^{n+1}}, \tag{1.10} \]

where \(p_j\) are arbitrary integers.

The existence of numbers \(\mu_i\) satisfying the conditions posed for arbitrarily small \(\rho\) follows from the following lemma (see [3], p. 115).

Lemma 1.2. For given \(\lambda_1,\ldots,\lambda_n\) and \(\alpha>0\), there exists \(\delta>0\) such that for \(\rho<\delta\) there exist \(\mu_1,\ldots,\mu_n\) satisfying conditions (1.10) and

\[ |\lambda_i-\mu_i|<\alpha\rho^2 \quad (i=1,\ldots,n). \]

We shall transform system (1.2) to a linear system with the aid of a special transformation, which is the limit of a certain sequence of transformations. As the zeroth step we take system (1.2), for which we assign it the index \(0\) and rewrite it in the following form:

\[ \dot u_{i0}=i\mu_i u_{i0}+iu_{i0}\bigl[(\lambda_i-\mu_i)+H_{i0}(u_{j0}v_{j0})\bigr]+i\Phi_{i0}(u_{j0},v_{j0}), \]

\[ \dot v_{i0}=-i\mu_i v_{i0}-iv_{i0}\bigl[(\lambda_i-\mu_i)+H_{i0}(u_{j0}v_{j0})\bigr]-i\Phi_{i0}(v_{j0},u_{j0}) \tag{1.11} \]

\[ (i,j=1,\ldots,n). \]

In what follows we shall consider system (1.11) for \(|u_{i0}|<\dfrac{1}{2}\rho\), \(|v_{i0}|<\dfrac{1}{2}\rho\), \(\rho<\Delta\), and with respect to the numbers \(\mu_i\) we shall assume that they satisfy the conditions indicated above.

Before proceeding to the proof of the main result, let us agree on the following. Below we consider regions defined by the inequalities

\[ r^2<|z_i|=|u_i v_i|<R^2 \qquad (i=1,\ldots,n). \tag{1.12} \]

These regions are considered in the space of the variables \(u_i, v_i\); moreover, by the region (1.12) we shall always mean its intersection with the interior of the square \(\{|u_i|<R,\ |v_i|<R\}\). By \(m_0\) we mean a sufficiently large natural number, and by \(\rho\) a sufficiently small positive constant; \(m_0\) and \(\rho\) are absolute constants, which are assumed in advance to have been chosen so that all inequalities appearing below and depending on them are valid.

§ 2. We shall introduce the sequence of transformations inductively. Suppose that \(s\) transformations of system (1.11) have already been carried out, the result of which is the following transformation of variables (the transformation \(T_s\)):

\[ u_{is}=u_{i0}+\psi_{is}(u_{j0},v_{j0}), \]

\[ v_{is}=v_{i0}+\psi_{is}(v_{j0},u_{j0}). \]

Let the transformation \(T_s\) bring system (1.11) to the form

\[ \dot u_{is}= i\mu_i u_{is}+ i u_{is}\bigl[(\lambda_i-\mu_i)+H_{is}(z_{js})\bigr] +i\Phi_{is}(u_{js},v_{js}), \]

\[ \dot v_{is}= -i\mu_i v_{is}- i v_{is}\bigl[(\lambda_i-\mu_i)+H_{is}(z_{js})\bigr] -i\Phi_{is}(v_{js},u_{js}) \tag{2.1} \]

\[ (i,j=1,\ldots,n). \]

Here and in what follows \(z_{is}=u_{is}v_{is}\).

We shall say that \(X(u_j,v_j)\) is a function of class \(A\) in a region \(D\) of the space \(u_i,v_i\) if it is analytic in \(D\), and the coefficients of its expansion in powers of \(u_j,v_j\) are analytic functions of \(z_j=u_jv_j\) in \(D\), real for real \(z_j\).

Introduce for consideration the sequence of numbers \(\{m_s\}\), \(m_{s+1}=2m_s\) \((s=0,1,\ldots)\), and regions \(\mathfrak M_s\), defined by the inequalities

\[ |(\lambda_i-\mu_i)+H_{is}(z_{js})| < \frac{\rho^3}{2n[3(2m_s+1)]^{n+2}} \qquad (i,j=1,\ldots,n). \]

Obviously, \(\mathfrak M_0\) is the region (1.9).

Suppose that the transformation \(T_s\) and system (2.1) satisfy the following induction hypotheses.

I. \(\psi_{is}(u_{j0},v_{j0})\) are functions of class \(A\) in some region \(G_s\subset \mathfrak M_0\) of the space \(u_{i0},v_{i0}\). The transformation \(T_s\) has an inverse defined in \(\mathfrak M_s\), with \(T_s^{-1}(\mathfrak M_s)\subset G_s\), and in the region \(G_s\) the inequality

\[ |\psi_{is}(u_{j0},v_{j0})| < \sum_{k=0}^{s-1}\rho^{m_k} \qquad (i=1,\ldots,n) \tag{2.2} \]

holds.

II.

\[ H_{is}=\sum_{\sigma=1}^{m_s-1} a_{is}^{\sigma} z_{js}^{\sigma} \]

are functions of class \(A\) in the region \(\Omega_s\), defined by the inequalities

\[ |(\lambda_i-\mu_i)+H_{is-1}(z_{js})| < \frac{\rho^3}{2n[3(2m_{s-1}+1)]^{n+2}} \qquad (i,j=1,\ldots,n), \]

where \(\Omega_s \supset \mathfrak{M}_s\), and the estimates hold
\[ \left|\alpha_{is}^j\right|<\frac{1}{R_s^{2\sigma+1}} \qquad (i,\ j=1,\ldots,n), \tag{2.3} \]
where
\[ R_s=\prod_{k=0}^{s-1}(2m_k+1)^{\frac{3\ln\rho}{2m_k+1}}. \tag{2.4} \]

III.
\[ \Phi_{is}=\sum_{k=2m_s+1}^{\infty} M_{is}^{k_j,l_j} u_{js}^{k_j} v_{js}^{l_j} \]
are functions of class \(A\) in \(\mathfrak{M}_s\), and
\[ \left|M_{is}^{k_j,l_j}\right|<\frac{1}{R_s^k} \qquad (i,\ j=1,\ldots,n). \tag{2.5} \]

Lemma 2.1. If the induction assumptions I—III are fulfilled, then there exists a transformation \(T_{s+1}\), defined by the formulas
\[ \begin{aligned} u_{is+1}&=u_{i0}+\psi_{is+1}(u_{j0},v_{j0}),\\ v_{is+1}&=v_{i0}+\psi_{is+1}(v_{j0},u_{j0}) \end{aligned} \qquad (i,\ j=1,\ldots,n), \]
which brings system (1.1) to the form
\[ \begin{aligned} \dot u_{is+1} &=i\mu_i u_{is+1} +i u_{is+1}\bigl[(\lambda_i-\mu_i)+H_{is+1}(z_{js+1})\bigr] +i\Phi_{is+1}(u_{js+1},v_{js+1}),\\ \dot v_{is+1} &=-i\mu_i v_{is+1} -i v_{is+1}\bigl[(\lambda_i-\mu_i)+H_{is+1}(z_{js+1})\bigr] -i\Phi_{is+1}(v_{js+1},u_{js+1}) \end{aligned} \tag{2.6} \]
\[ (i,\ j=1,\ldots,n), \]
and, moreover, assumptions I—III are valid with \(s\) replaced by \(s+1\).

Proof. Apply to system (2.1) the transformation \(S_s\):
\[ \begin{aligned} u_{is+1}&=u_{is}+\varphi_{is}(u_{js},v_{js}),\\ v_{is+1}&=v_{is}+\varphi_{is}(v_{js},u_{js}) \end{aligned} \qquad (i,\ j=1,\ldots,n), \]
with the intention that it be brought to the form
\[ \begin{aligned} \dot u_{is+1} &=i\mu_i u_{is+1} +i u_{is+1}\bigl[(\lambda_i-\mu_i)+H_{is}(z_{js+1})+G_{is}(z_{js+1})\bigr] +i\Phi_{is+1},\\ \dot v_{is+1} &=-i\mu_i v_{is+1} -i v_{is+1}\bigl[(\lambda_i-\mu_i)+H_{is}(z_{js+1})+G_{is}(z_{js+1})\bigr] -i\Phi_{is+1} \end{aligned} \tag{2.7} \]
\[ (i,\ j=1,\ldots,n), \]
where
\[ G_{is}(z_{js+1})=\sum_{\sigma=m_s}^{2m_s-1} M_{is}^{\sigma_j+\delta_{ij},\sigma_j} z_{js+1}^{\sigma_j}, \]
\[ \Phi_{is+1}(u_{js+1},v_{js+1}) =\sum_{k=4m_s+1}^{\infty} M_{is+1}^{k_j,l_j}u_{js+1}^{k_j}v_{js+1}^{l_j}, \]
and \(M_{is}^{k_j,l_j}(z_{js+1})\) are analytic real functions of \(z_{js+1}\).

Introduce the notation
\[ \Delta X=X(u_{js+1},v_{js+1})-X(u_{js},v_{js})= \]
\[ = X(u_{js}+\varphi_{js},\,v_{js}+\varphi_{js})-X(u_{js},v_{js}). \]

Let, furthermore,

\[ \Delta H_{is}=\Psi_{is}+\chi_{is}, \]

where \(\Psi_{is}\) are linear functions of \(\varphi_{is}(u_{js},v_{js})\) and \(\varphi_{is}(v_{js},u_{js})\). Obviously,

\[ \begin{aligned} \Psi_{is}={}& \sum_{\sigma=1}^{m_s-1}\sum_{j=1}^{n} \Biggl[ \varphi_{js}(u_{ks},v_{ks}) \left( a_{is}^{\sigma j}\sigma_j\frac{1}{u_{js}} + \frac{\partial a_{is}^{\sigma j}}{\partial z_{js}}v_{js} \right) \\ &\qquad\qquad +\varphi_{js}(v_{ks},u_{ks}) \left( a_{is}^{\sigma j}\sigma_j\frac{1}{v_{js}} + \frac{\partial a_{is}^{\sigma j}}{\partial z_{js}}u_{js} \right) \Biggr]z_{1s}^{\sigma_1}\cdots z_{ns}^{\sigma_n} \end{aligned} \tag{2.8} \]

\[ (j,k=1,\ldots,n). \]

We define \(\varphi_{is}(u_{js},v_{js})\) as a solution of the following system of partial differential equations

\[ \sum_{l=1}^{n} \left( \frac{\partial\varphi_{is}}{\partial u_{ls}}u_{ls} - \frac{\partial\varphi_{is}}{\partial v_{ls}}v_{ls} \right) \left[\mu_l+(\lambda_l-\mu_l)+H_{ls}(z_{js})\right] - \tag{2.9} \]

\[ -\varphi_{is}\left[\mu_i+(\lambda_i-\mu_i)+H_{is}(z_{js})\right] = u_{is}\Psi_{is} - \sum_{k=2m_s+1}^{4m_s} M_{is}^{k_i,l_i}u_{js}^{k_i}v_{js}^{l_i} \]

\[ \left(\sum_{j=1}^{n}(k_j-l_j-\delta_{ij})^2>0\right)(i,j=1,\ldots,n). \]

Then the functions \(\Phi_{is+1}\) in (2.7) satisfy the equalities

\[ \begin{aligned} \Phi_{is+1}(u_{js+1},v_{js+1})={}& \sum_{k=4m_s+1}^{\infty} M_{is}^{k_i,l_i}u_{js}^{k_i}v_{js}^{l_i} - u_{is}(\Delta G_{is}+\chi_{is}) \\ &- \varphi_{is}\bigl(\Delta H_{is}+G_{is}(z_{js+1})\bigr) \\ &+ \sum_{l=1}^{n} \left( \frac{\partial\varphi_{is}}{\partial u_{ls}}\Phi_{ls}(u_{js},v_{js}) - \frac{\partial\varphi_{is}}{\partial v_{ls}}\Phi_{ls}(v_{js},u_{js}) \right) \end{aligned} \tag{2.10} \]

\[ (i,j=1,\ldots,n). \]

We shall seek a solution of system (2.9) in the form

\[ \varphi_{is}(u_{js},v_{js})=\sum L_{is}^{k_i,l_i}u_{js}^{k_i}v_{js}^{l_i} \quad (i,j=1,\ldots,n), \tag{2.11} \]

where \(L_{is}^{k_i,l_i}\) are analytic real functions of \(z_{js}\). Then

\[ \frac{\partial L_{is}}{\partial u_{ls}}u_{ls} - \frac{\partial L_{is}}{\partial v_{ls}}v_{ls} =0 \quad (i,l=1,\ldots,n). \tag{2.12} \]

Put

\[ \omega_i= \sum_{j=1}^{n} \left[ (k_j-l_j-\delta_{ij})\mu_j + (k_j-l_j-\delta_{ij})(\lambda_j-\mu_j+H_{js}) \right] \tag{2.13} \]

\[ (i=1,\ldots,n). \]

Taking (2.8) and (2.12) into account, substituting (2.11) into (2.9), and equating the coefficients of like powers of \(u_{js}, v_{js}\), we obtain:

\[ \omega_i L_{is}^{k_j,l_j} = -\,M_{is}^{k_j,l_j} + \sum_{\sigma=1}^{m_s-1}\sum_{j=1}^{n} \left[ a_{is}^{\sigma_j}\sigma_j \left( L_{js}^{k_1-\sigma_1,l_1-\sigma_1,\ldots,k_j-\sigma_j+1,l_j-\sigma_j,\ldots,k_n-\sigma_n,l_n-\sigma_n} + \right.\right. \]

\[ \left.\left. + L_{js}^{l_1-\sigma_1,k_1-\sigma_1,\ldots,l_j-\sigma_j+1,k_j-\sigma_j,\ldots,l_n-\sigma_n,k_n-\sigma_n} \right) + \right. \]

\[ \left. + \frac{\partial a_{is}^{\sigma_j}}{\partial z_{js}} \left( L_{js}^{k_1-\sigma_1,l_1-\sigma_1,\ldots,k_i-\sigma_i-1,l_i-\sigma_i,\ldots,k_j-\sigma_j,l_j-\sigma_j-1,\ldots,k_n-\sigma_n,l_n-\sigma_n} + \right.\right. \]

\[ \left.\left. + L_{js}^{l_1-\sigma_1,k_1-\sigma_1,\ldots,l_i-\sigma_i,k_i-\sigma_i-1,\ldots,l_j-\sigma_j,k_j-\sigma_j-1,\ldots,l_n-\sigma_n,k_n-\sigma_n} \right) \right] \tag{2.14} \]

\[ (i=1,\ldots,n). \]

Let us replace in (2.14) the quantities \(L_{is}\) once again by formulas (2.14). As a result, all terms containing the coefficients \(L_{is}\) cancel mutually, and we arrive at the formulas

\[ L_{is}^{k_1,l_1,\ldots,k_n l_n} = -\frac{1}{\omega_i}M_{is}^{k_1,l_1,\ldots,k_n,l_n} + \]

\[ + \frac{1}{\omega_i^2} \sum_{\sigma=1}^{m_s-1}\sum_{j=1}^{n} \left[ a_{is}^{\sigma_j}\sigma_j \left( -\,M_{js}^{\ldots,k_i-\sigma_i-1,l_i-\sigma_i,\ldots,k_j-\sigma_j+1,l_j-\sigma_j,\ldots} + \right.\right. \]

\[ \left.\left. + M_{js}^{\ldots,l_i-\sigma_i,k_i-\sigma_i-1,\ldots,l_j-\sigma_j+1,k_j-\sigma_j,\ldots} \right) + \right. \]

\[ \left. + \frac{\partial a_{is}^{\sigma_j}}{\partial z_{js}} \left( -\,M_{js}^{\ldots,k_i-\sigma_i-1,l_i-\sigma_i,\ldots,k_j-\sigma_j,l_j-\sigma_j-1,\ldots} + M_{js}^{\ldots,l_i-\sigma_i,k_i-\sigma_i-1,\ldots,l_j-\sigma_j,k_j-\sigma_j-1,\ldots} \right) \right]. \tag{2.15} \]

\[ (i=1,\ldots,n). \]

Since in formulas (2.15) \(2m_s+1 \leq k \leq 6m_s\), we have

\[ \varphi_{is}(u_{js},v_{js}) = \sum_{k=2m_s+1}^{6m_s} L_{is}^{k_j,l_j}u_{js}^{k_j}v_{js}^{l_j} \qquad (i,j=1,\ldots,n). \]

We shall show that the \(L_{is}\) are real analytic functions of \(z_{js}\) in the domain \(\mathfrak M_s\). Indeed, by assumption the functions \(M_{is}, a_{is}\), and \(\omega_i\) in formulas (2.15) are such. Therefore it remains only to show that \(\omega_i\neq 0\) in \(\mathfrak M_s\). We shall prove that in \(\mathfrak M_s\)

\[ |\omega_i| > \frac12 \left| \sum_{j=1}^{n}(k_j-l_j-\delta_{ij})\mu_j \right| \qquad (i=1,\ldots,n), \tag{2.16} \]

whence, in view of the incommensurability of the \(\mu_i\), the assertion being proved follows. Since in (2.15) \(|k_j-l_j|<3(2m_s+1)\), in \(\mathfrak M_s\), by the definition of \(\mathfrak M_s\),

\[ \left| \sum_{j=1}^{n}(k_j-l_j-\delta_{ij})(\lambda_j-\mu_j+H_{js}) \right| < \frac{\rho^3}{2\,[3(2m_s+1)]^{n+i}} \qquad (i=1,\ldots,n), \]

and since, in view of (1.10),

\[ \left|\sum_{j=1}^{n}(k_j-l_j-\delta_{ij})\mu_j\right|> \frac{\rho^3}{[3(2m_s+1)]^{n+1}} \qquad (i=1,\ldots,n), \]

(2.16) holds in \(\mathfrak{M}_s\).

Let the inequalities

\[ |(\lambda_i-\mu_i)+H_{is}(z_{js})|< \frac{l}{3}-\frac{\rho^3}{2n[3(2m_s+1)]^{n+2}} \quad \left( \begin{array}{l} i=1,\ldots,n,\\ l=1,2,3 \end{array} \right) \tag{2.17} \]

be equivalent to the following:

\[ |z_{is}-z_{is}^*|<\beta_{is}^{(l)} \quad \left( \begin{array}{l} i=1,\ldots,n,\\ l=1,2,3 \end{array} \right). \]

Denote by \(\mathfrak{M}_s^{2/3}\) and \(\mathfrak{M}_s^{1/3}\) the domains (2.17) with \(l=2\) and \(l=1\), respectively. The domain (2.17) with \(l=3\) is \(\mathfrak{M}_s\).

Since, by the induction hypothesis, \(T_s^{-1}(\mathfrak{M}_s)\subset G_s\subset \mathfrak{M}_0\), it follows that

\[ |u_{i0}|<\frac{1}{2}\rho,\qquad |v_{i0}|<\frac{1}{2}\rho,\qquad \rho<\Delta. \]

Moreover, by virtue of (2.2),

\[ |\psi_{is}(u_{j0},v_{j0})|<\sum_{k=0}^{s-1}\rho^{m_k} \qquad (i=1,\ldots,n), \]

therefore in \(\mathfrak{M}_s\)

\[ |u_{is}|<|u_{i0}|+|\psi_{is}|<\rho<\Delta,\qquad |v_{is}|<\rho<\Delta\quad (i=1,\ldots,n). \]

From (2.4) and Lemma 1 of the Appendix with \(\alpha=3\), it follows that

\[ R_s>\rho^{1/2}. \tag{2.18} \]

From (2.18) and (2.3) we then obtain

\[ |a_{is}^{\sigma_j}|<\frac{1}{R_s^{2\sigma+1}}<\rho^{-\sigma-1/2}<\Delta^{-\sigma-1/2}. \]

Consequently, on the basis of the definition of the quantity \(\Delta\) in § 1, we may assert that

\[ \beta_{is}^{(l_1)}<\beta_{is}^{(l_2)} \qquad (i=1,\ldots,n),\quad \text{if } l_1<l_2, \]

and there exists \(L>0\) such that

\[ |\beta_{is}^{(l_1)}-\beta_{is}^{(l_2)}|> \frac{L\rho^3}{(2m_s+1)^{n+2}} \quad \left( \begin{array}{l} l_1\ne l_2,\\ i=1,\ldots,n \end{array} \right). \tag{2.19} \]

From (2.3) and (2.19), using Cauchy’s estimate for the derivatives of analytic functions, we obtain in the domain \(\mathfrak{M}_s^{2/3}\):

\[ \left|\frac{\partial a_{is}^{\sigma}}{\partial z_{js}}\right| < \frac{(2m_s+1)^{n+2}}{L\rho^3 R_s^{2\sigma+1}} < \frac{\rho^{-3}(2m_s+1)^{n+3}}{R_s^{2\sigma+1}}, \tag{2.20} \]

\[ \left|\frac{\partial^2 a_{is}^{\sigma}} {\partial z_{j_1s}\partial z_{j_2s}}\right| < \frac{\rho^{-6}(2m_s+1)^{2n+5}}{R_s^{2\sigma+1}} \qquad (i,j,j_1,j_2=1,\ldots,n). \tag{2.21} \]

(Here and below \(a_{is}^{\sigma}\) is understood to mean the coefficient, largest in absolute value, among the \(a_{is}^{\sigma_j}\) with \(\sum_{j=1}^{n}\sigma_j=\sigma\).)

If we set

\[ \frac{\partial H_{is}}{\partial z_{js}}=\sum_{\sigma=0}^{m_s-2} a_{jis}^{\sigma} z_{js}^{\sigma}, \]

then

\[ |a_{jis}^{\sigma}|<(\sigma+1)|a_{is}^{\sigma+1}|+\left|\frac{\partial a_{is}^{\sigma}}{\partial z_{js}}\right| \qquad (i,j=1,\ldots,n). \]

Then, taking into account (2.3) and (2.20),

\[ |a_{jis}^{\sigma}|< \frac{\sigma+1}{R_s^{2\sigma+3}} +\frac{\rho^{-3}(2m_s+1)^{n+3}}{R_s^{2\sigma+1}} = \frac{(\sigma+1)R_s^{-2}+\rho^{-3}(2m_s+1)^{n+3}}{R_s^{2\sigma+1}}. \]

Hence, on the basis of (2.18),

\[ |a_{jis}^{\sigma}|< \frac{\rho^{-3}(2m_1+1)^{n+4}}{R_s^{2\sigma+1}} \qquad (i,j=1,\ldots,n). \tag{2.22} \]

By analogous arguments, using (2.21), we obtain in the domain \(\mathfrak M_s^{2/3}\) the estimate

\[ |a_{j_1j_2is}^{\sigma}|< \frac{\rho^{-6}(2m_s+1)^{2n+6}}{R_s^{2\sigma+1}} \qquad (i,j_1,j_2=1,\ldots,n), \tag{2.23} \]

where \(a_{j_1,j_2,i,s}^{\sigma}\) are the coefficients in the expansions of

\[ \frac{\partial^2 H_{is}}{\partial z_{j_1s}\partial z_{j_2s}}. \]

In view of (1.10) and (2.16) we have

\[ |\omega_i|>\frac{\rho^3}{2k^{n+1}} \qquad (i=1,\ldots,n). \tag{2.24} \]

Then from (2.3), (2.5), (2.15), (2.20), and (2.24) it follows that

\[ |L_{is}^k|< \frac{2\rho^{-3}k^{n+1}}{R_s^k} + \frac{4\rho^{-6}k^{2n+2}}{R_s^{k+1}} \sum_{\sigma=1}^{m_s-1}\sum_{j=1}^n \left[2\sigma_j+2\rho^{-3}(2m_s+1)^{n+3}\right] \]

\[ (i=1,\ldots,n), \]

whence

\[ |L_{is}^k|< \frac{\rho^{-10}k^{3n+5}}{R_s^k} \qquad (i=1,\ldots,n), \tag{2.25} \]

since \(k\ge 2m_s+1\).

Next, if we set

\[ \frac{\partial \varphi_{is}}{\partial u_{js}} = \sum_{k=2m_s}^{\infty} L_{j,i,s}^{k\,j_i,l_i}\,u_{js}^{k_i}v_{js}^{l_i} \qquad (i,j=1,\ldots,n), \]

then, arguing as in the derivation of estimate (2.22), and using Cauchy’s estimate, (2.19), and (2.25), we obtain in the domain \(\mathfrak M_s^{1/3}\)

\[ |L_{j,i,s}^{k}|< \frac{\rho^{-12}k^{4n+7}}{R_s^k}. \tag{2.26} \]

The coefficients of the expansion of

\[ \frac{\partial \varphi_{is}}{\partial v_{js}} \]

are estimated analogously.

Denote by \(S_k(\varphi)\) the sum of the absolute values of all coefficients in the expansion of \(\varphi\) in powers of its arguments with sum of indices equal to \(k\). We note that the number of such coefficients for functions of \(2n\) variables is less than \(k^{2n}\) \((k \ge 2)\). Then, by virtue of (2.25), (2.26), and (2.22),

\[ S_k(\varphi_{is}) < \frac{\rho^{-10} k^{5n+5}}{R_s^k}, \tag{2.27} \]

\[ S_k\left(\frac{\partial \varphi_{is}}{\partial u_{js}},\ \frac{\partial \varphi_{is}}{\partial v_{js}}\right) < \frac{\rho^{-12} k^{6n+7}}{R_s^k} \tag{2.28} \]

and

\[ S_\sigma\left(\frac{\partial H_{is}}{\partial z_{js}}\right) < \frac{\rho^{-3}(2m_s+1)^{3n+4}}{R_s^{2\sigma+1}} \qquad (i,\ j=1,\ldots,n). \tag{2.29} \]

Put

\[ \Theta_s=(2m_s+1)^{\frac{\ln \rho}{2m_s+1}}, \]

\[ r_s=\Theta_s R_s,\qquad \rho_s=\Theta_s r_s,\qquad R_{s+1}=\Theta_s\rho_s=\Theta_s^3 R_s. \]

Since \(R_0=1\), the \(R_s\) so defined coincide with the \(R_s\) defined by formula (2.4). By virtue of (2.18),

\[ R_s>r_s>\rho_s>R_{s+1}>\rho^{1/2}. \tag{2.30} \]

We shall show that in \(\mathfrak{M}_s^{1/3}\)

\[ S_k(\varphi_{is})<\frac{1}{r_s^k},\qquad S_k\left(\frac{\partial \varphi_{is}}{\partial u_{js}},\ \frac{\partial \varphi_{is}}{\partial v_{js}}\right) <\frac{1}{r_s^k} \qquad (i,\ j=1,\ldots,n). \tag{2.31} \]

For this, in view of (2.27) and (2.28), it is sufficient to verify the validity of the inequality

\[ \rho^{-12} k^{6n+7}<\Theta_s^{-k}, \]

or

\[ -12\ln\rho+(6n+7)\ln k < -k\ln\rho\,\frac{\ln(2m_s+1)}{2m_s+1}. \]

It follows that the estimates (2.31) are valid, since for \(k \ge 2m_s+1\)

\[ 12<\frac{k\ln(2m_s+1)}{2(2m_s+1)},\qquad 6n+7< -\frac{\ln\rho\cdot k\ln(2m_s+1)}{2(2m_s+1)\ln k}. \]

It was shown earlier that in the domain \(\mathfrak{M}_s\), \(|u_{is}|<\rho\), \(|v_{is}|<\rho\); all the more is this true in the domain \(\mathfrak{M}_s^{1/3}\). Hence, by virtue of (2.31),

\[ |\varphi_{is}|< \sum_{k=2m_s+1}^{\infty} S_k(\varphi_{is})\rho^k < \sum_{k=2m_s+1}^{\infty}\left(\frac{\rho}{r_s}\right)^k \qquad (i=1,\ldots,n), \]

which, in view of (2.30), gives

\[ |\varphi_{is}|<\rho^{m_s} \qquad (i=1,\ldots,n) \tag{2.32} \]

in the domain \(\mathfrak{M}_s^{1/3}\). Similarly, in \(\mathfrak{M}_s^{1/3}\),

\[ \left|\frac{\partial \varphi_{is}}{\partial u_{js}}\right|<\rho^{m_s},\qquad \left|\frac{\partial \varphi_{is}}{\partial v_{js}}\right|<\rho^{m_s} \qquad (i,\ j=1,\ldots,n). \tag{2.33} \]

Set \(\psi_{is+1}=\psi_{is}+\varphi_{is}\). In other words, define the transformation \(T_{s+1}\) as the product \(T_{s+1}=T_s S_s\). In view of (2.33), the Jacobian of the transformation \(S_s\) is nonzero.

Let \(G_{s+1}=T_s^{-1}\mathfrak{M}_s^{1/3}\). Since \(T_s^{-1}\mathfrak{M}_s\subset G_s\), all the more \(G_{s+1}\subset G_s\subset \mathfrak{M}_0\). In \(\mathfrak{M}_s^{1/3}\) the estimates (2.32) hold. Therefore, from (2.2) it follows that in \(G_{s+1}\)

\[ |\psi_{is+1}|<|\psi_{is}|+|\varphi_{is}|<\sum_{k=0}^{s}\rho^{m_k} \qquad (i=1,\ldots,n). \tag{2.34} \]

Thus the validity of induction assumption I has been established (with the exception of the relation \(T_{s+1}^{-1}\mathfrak{M}_{s+1}\subset G_{s+1}\), which will be proved below, when the set \(\mathfrak{M}_{s+1}\) has been defined).

For what follows we need an estimate of the function \(\Delta H_{is}\) in \(\mathfrak{M}_s^{1/3}\). From (2.29) we have

\[ \left|\frac{\partial H_{is}}{\partial z_{js}}\right| < \sum_{\sigma=0}^{m_s-2} \frac{\rho^{-3}(2m_s+1)^{3n+4}}{R_s^{2\sigma+1}}\rho^{2\sigma} \qquad (i,j=1,\ldots,n), \]

whence

\[ \left|\frac{\partial H_{is}}{\partial z_{js}}\right| < \rho^{-4}(2m_s+1)^{3n+4} \qquad (i,j=1,\ldots,n). \]

From (2.32), using Lagrange’s formula, we obtain

\[ |\Delta H_{is}|< n\rho^{-4}(2m_s+1)^{3n+4}\rho^{m_s} < (2m_s+1)^{3n+4}\rho^{m_s-5} \tag{2.35} \]

\[ (i=1,\ldots,n). \]

Since

\[ |(\lambda_i-\mu_i)+H_{is}(z_{js+1})| < |(\lambda_i-\mu_i)+H_{is}(z_{js})|+|\Delta H_{is}| \]

\[ (i,j=1,\ldots,n), \]

it follows that in the domain \(\mathfrak{M}_s^{1/3}\)

\[ |(\lambda_i-\mu_i)+H_{is}(z_{js+1})| < \frac{1}{3}\, \frac{\rho^3}{2n[3(2m_s+1)]^{\,n+2}} + (2m_s+1)^{3n+4}\rho^{m_s-5} \]

\[ (i,j=1,\ldots,n). \]

Hence it follows that

\[ |(\lambda_i-\mu_i)+H_{is}(z_{js+1})| < \frac{\rho^3}{2n[3(2m_s+1)]^{\,n+2}} \qquad (i,j=1,\ldots,n). \]

Now consider the function

\[ G_{is}(z_{js+1}) = \sum_{\sigma=m_s}^{2m_s-1} M_{is}^{\sigma_i+\delta_{ij},\,\sigma_j} z_{js+1}^{\sigma_j}. \]

If \(u_{is}, v_{is}\) and \(u_{is+1}, v_{is+1}\) are connected by the transformation \(S_s\), then from what has been said, on the basis of (2.5) and (2.34), we conclude

\[ |M_{is}^{\sigma_i+\delta_{ij},\,\sigma_j}| < \frac{1}{R_s^{2\sigma+1}}, \qquad |u_{is+1}| < \frac{1}{2}\rho+\sum_{k=0}^{s}\rho^{m_k}<\rho, \qquad |v_{is+1}|<\rho \]

\[ (i,j=1,\ldots,n). \]

Arguing as in the derivation of (2.32), we obtain:
\[ |G_{is}(z_{js+1})|<\rho^{m_s}\qquad (i,\ j=1,\ldots,n) \tag{2.36} \]
in the domain \(F_s=S_s\mathfrak{M}_s^{1/3}=T_{s+1}G_{s+1}\).

Define the domain \(\mathfrak{M}_{s+1}\) by the inequalities
\[ |(\lambda_i-\mu_i)+H_{is}(z_{js+1})+G_{is}(z_{js+1})|< \frac{\rho^3}{2n[3(4m_s+1)]^{n+2}} \]
\[ (i,\ j=1,\ldots,n). \]

We shall show that \(\mathfrak{M}_{s+1}\subset F_s\). For this purpose consider the equalities
\[ |(\lambda_i-\mu_i)+H_{is}(z_{js})|= \frac{1}{3}\,\frac{\rho^3}{2n[3(2m_s+1)]^{n+2}} \]
\[ (i,\ j=1,\ldots,n). \tag{2.37} \]

From (2.37) it follows that
\[ |(\lambda_i-\mu_i)+H_{is}(z_{js+1})+G_{is}(z_{js+1})|> \]
\[ >\frac{1}{3}\,\frac{\rho^3}{2n[3(2m_s+1)]^{n+2}} -|\Delta H_{is}|-|G_{is}| \]
\[ (i,\ j=1,\ldots,n). \]

Since on the boundary of the domain \(\mathfrak{M}_s^{1/3}\) the estimates (2.35) and (2.36) hold, it follows that on its image under the transformation \(S_s\)
\[ |(\lambda_i-\mu_i)+H_{is}(z_{js+1})+G_{is}(z_{js+1})|> \]
\[ >\frac{\rho^3}{2n[3(4m_s+1)]^{n+2}}\qquad (i,\ j=1,\ldots,n), \]
whence it follows that \(\mathfrak{M}_{s+1}\subset F_s\). Then, since \(F_s=T_{s+1}G_{s+1}\), we have \(T_{s+1}^{-1}\mathfrak{M}_{s+1}\subset G_{s+1}\). If now we put
\[ H_{is+1}(z_{js+1})=H_{is}(z_{js+1})+G_{is}(z_{js+1}),\qquad m_{s+1}=2m_s\qquad (i,\ j=1,\ldots,n), \]
then we find that the domain \(\mathfrak{M}_{s+1}\) defined above coincides with the one defined in the induction hypotheses. Thus induction hypothesis I is proved.

Next, by the definition of \(\Omega_s\), we obtain, as also for \(F_s\), that \(\mathfrak{M}_{s+1}\subset\Omega_{s+1}\). In addition, the coefficients of the expansions of the functions \(H_{is+1}(z_{js+1})\) are either \(a_{is}\) or \(M_{is}\), and therefore, by assumption, are analytic and real for real \(z_{js+1}\) and satisfy in \(\Omega_{s+1}\) the estimates (2.3) and (2.5). Moreover, (2.3) and (2.5) hold with \(R_s\) replaced by \(R_{s+1}\), since \(R_s>R_{s+1}\). These remarks prove induction hypothesis II.

We now consider the functions \(\Phi_{is+1}(u_{js+1},v_{js+1})\) satisfying relations (2.10). Since the expansions of the functions on the right-hand side of (2.10) contain no terms of dimension lower than \(4m_s+1=2m_{s+1}+1\), the expansions of \(\Phi_{is+1}\) contain no terms of dimension lower than \(2m_{s+1}+1\). Moreover, \(\Phi_{is+1}\) belong to the class \(A\) in \(\mathfrak{M}_{s+1}\), since the functions occurring on the right-hand side of (2.10) have this property.

It remains to estimate the coefficients of the expansions of \(\Phi_{is+1}\). To this end, consider majorants of the functions on the right-hand side of (2.10), replacing their coeffi-

coefficients by constant numbers, using estimates of these coefficients in the domain \(\mathfrak M_s^{1/3}\), which are all the more valid in \(\mathfrak M_{s+1}\). We shall denote the majorant functions by a tilde.

If \(|u_{is}|<\rho_s,\ |v_{is}|<\rho_s\), then, by virtue of (2.31),

\[ |\widetilde{\varphi}_{is}| < \sum_{k=2m_s+1}^{\infty}\left(\frac{\rho_s}{r_s}\right)^k = \frac{(2m_s+1)^{\ln\rho}}{1-\Theta_s}, \tag{2.38} \]

\[ \left|\frac{\partial\widetilde{\varphi}_{is}}{\partial u_{js}}\right| < \frac{(2m_s+1)^{\ln\rho}}{1-\Theta_s}, \qquad \left|\frac{\partial\widetilde{\varphi}_{is}}{\partial v_{js}}\right| < \frac{(2m_s+1)^{\ln\rho}}{1-\Theta_s} \]

\[ (i,\ j=1,\ldots,n). \]

and, by virtue of (2.29),

\[ \left|\frac{\partial \widetilde H_{is}}{\partial z_{js}}\right| < \frac{\rho^{-4}(2m_s+1)^{3n+4}}{1-\Theta_s^4} \qquad (i,\ j=1,\ldots,n). \tag{2.39} \]

Let us show that

\[ R_{s+1}+\frac{(2m_s+1)^{\ln\rho}}{1-\Theta_s}<\rho_s, \tag{2.40} \]

or, what is the same thing,

\[ \frac{(2m_s+1)^{\ln\rho}}{(1-\Theta_s)^2}<\rho_s. \]

Since \(\rho_s>\rho^{1/2}\) by virtue of (2.30), it is sufficient to prove that

\[ \frac{(2m_s+1)^{\frac12\ln\rho}}{1-\Theta_s}<\rho^{1/4}. \]

The validity of the last inequality follows from Lemma 2 of the Appendix for \(\alpha=\dfrac12\).

From (2.40) and (2.38) we obtain that the image of the square \(\{|u_{is}|\leq\rho_s,\ |v_{is}|\leq\rho_s\}\) contains the square \(\{|u_{is+1}|\leq R_{s+1},\ |v_{is+1}|\leq R_{s+1}\}\) inside itself. Therefore the estimates (2.37) and (2.38) are valid if \(|u_{is+1}|\leq R_{s+1},\ |v_{is+1}|\leq R_{s+1}\).

From Lagrange’s formula, (2.38), and (2.39), we obtain

\[ |\widetilde{\varphi}_{is}\Delta \widetilde H_{is}| < 2n\, \frac{\rho^{-4}(2m_s+1)^{2\ln\rho+3n+4}} {(1-\Theta_s)(1-\Theta_s)^2} \qquad (i=1,\ldots,n). \]

It is not difficult to verify that

\[ \frac{2n\,\rho^{-4}(2m_s+1)^{\ln\rho+3n+4}} {(1+\Theta_s)(1+\Theta_s)^2} <1. \]

Moreover, from Lemma 2 of the Appendix, for \(\alpha=\dfrac13\), it follows that

\[ \frac{(2m_s+1)^{\ln\rho}}{(1-\Theta_s)^3}<\rho^{3/4}. \]

Consequently, \(|\widetilde{\varphi}_{is}\Delta \widetilde H_{is}|\) is an infinitely small quantity as \(\rho\to0\),

Using (2.23), by means of analogous reasoning, for \(|u_{is}| \leqslant \rho_s,\ |v_{is}| \leqslant \rho_s\) we obtain the estimate

\[ \left|\frac{\partial^2 \widetilde H_{is}}{\partial z_{j_1s}\partial z_{j_2s}}\right| < \frac{\rho^{-7}(2m_s+1)^{4n+6}}{1-\Theta_s^4} \qquad (i,\ j_1,\ j_2=1,\ldots,n). \]

Hence, by Taylor’s formula,

\[ |\widetilde X_{is}| < 2n^2|\widetilde\varphi_{is}|^2 \left|\frac{\partial^2 \widetilde H_{is}}{\partial z_{j_1s}\partial z_{j_2s}}\right| \qquad (i,\ j_1,\ j_2=1,\ldots,n) \]

and \(|\widetilde X_{is}| \to 0\) as \(\rho \to 0\) for the same reasons as \(|\widetilde\varphi_{is}\Delta\widetilde H_{is}|\).

Further, for \(|u_{is}| \leqslant \rho_s,\ |v_{is}| \leqslant \rho_s\),

\[ \sum_{k=4m_s+1}^{\infty} \widetilde M_{is}^{k_i l_i} u_{js}^{k_j} v_{js}^{l_j} < \sum_{k=4m_s+1}^{\infty} \frac{k^{2n}\rho_s^k}{R_s^k} \qquad (i=1,\ldots,n) \]

and, by virtue of (2.31),

\[ \sum_{k=4m_s+1}^{\infty} \widetilde M_{is}^{k_i l_j} u_{js}^{k_j} v_{js}^{l_j} < \frac{(4m_s+1)^{\ln\rho}}{1-\Theta_s} \to 0 \quad \text{as } \rho \to 0. \]

In an analogous way we are convinced that the remaining functions entering the right-hand side of (2.10) are also infinitely small as \(\rho \to 0\). Choosing \(\rho\) so that \(|\Phi_{is+1}|<1\) for \(|u_{is+1}| \leqslant R_{s+1},\ |v_{is+1}| \leqslant R_{s+1}\), we obtain

\[ |M_{is+1}^{k_j,l_j}| < \frac{1}{R_{s+1}^k}. \]

This completes the proof of Lemma 2.1.

§ 3. Inequality (1.7) gives us a basis for induction with respect to \(s\). On the basis of Lemma 2.1 we conclude that the inductive assumptions I—III of § 2 hold for all natural \(s\). Let \(s\) tend to infinity. Then we obtain that there exists an analytic transformation \(T_\infty\)

\[ \begin{aligned} u_{i\infty} &= u_{i0}+\psi_{i\infty}(u_{j0},v_{j0}),\\ v_{i\infty} &= v_{i0}+\psi_{i\infty}(v_{j0},u_{j0}) \end{aligned} \qquad (i,\ j=1,\ldots,n), \]

defined on the set \(G_\infty=\bigcap_{k=1}^{\infty}G_k\), having an inverse on the torus \(\Gamma\), defined by the equations

\[ (\lambda_i-\mu_i)+H_{i\infty}(z_{j\infty})=0 \qquad (i,\ j=1,\ldots,n), \]

and bringing system (1.11) to the form

\[ \begin{aligned} \dot u_{i\infty} &= i\mu_i u_{i\infty},\\ \dot v_{i\infty} &= -i\mu_i v_{i\infty} \end{aligned} \qquad (i=1,\ldots,n). \tag{3.1} \]

At the same time \(G_\infty=T_\infty^{-1}\Gamma\). From (3.1) it follows that the torus \(\Gamma\) is invariant for system (3.1), and then the \(n\)-dimensional toroidal surface \(G_\infty\) is invariant for system (1.11).

Note that, since \(x_i=\overline{y}_i\) \((i=1,\ldots,n)\), we have \(u_{is}=v_{is}\) \((i=1,\ldots,n;\ s=0,1,\ldots,\infty)\).

Denote by \(\Lambda^{(0)}\) the set of points \((\mu_1,\ldots,\mu_n)\) whose coordinates satisfy the equations

\[ (\lambda_i-\mu_i)+H_{i0}(z_{j0})=0 \qquad (i,j=1,\ldots,n) \]

for \(|z_{j0}|<\frac14\rho^2\). Put

\[ \Lambda^{(s)}=\Lambda_{\rho^3,\,3(2m_{s-1}+1)}^{(s-1)} \qquad (s=1,2,\ldots), \]

where \(\Lambda_{\varepsilon,N}\) denotes the set of points \(\Lambda\) for which

\[ |k_1\mu_1+\cdots+k_n\mu_n|> \frac{\varepsilon}{(|k_1|+\cdots+|k_n|)^{n+1}} \]

for all integers \(k_i\) with \(|k_1|+\cdots+|k_n|<N\).

Consideration of the \(s\)-th step in the applied method of successive approximations shows that the set of admissible \(\mu_1,\ldots,\mu_n\) for which the system of the \(s\)-th approximation is defined is \(\Lambda^{(s)}\). Consider the sets

\[ \Omega^{(s)}=\Lambda^{(s-1)}\setminus\Lambda^{(s)} \qquad (s=1,2,\ldots). \]

Using the arithmetical lemma ([3], p. 167), it is not difficult to show that

\[ \operatorname{mes}\Omega^{(s)}<L\rho\sigma_s\operatorname{mes}\Lambda^{(0)} \qquad (s=1,2,\ldots), \]

where

\[ L>0,\qquad \sigma_1=\sum_{m=1}^{3(2m_0+1)}\frac1{m^2},\qquad \sigma_s=\sum_{3(2m_{s-2}+1)}^{3(2m_{s-1}+1)}\frac1{m^2} \qquad (s=2,3,\ldots). \]

Consequently,

\[ \operatorname{mes}\Lambda^{(s)}>(1-L\rho\tau_s)\operatorname{mes}\Lambda^{(0)} \qquad (s=1,2,\ldots), \tag{3.2} \]

where

\[ \tau_s=\sum_{m=1}^{3(2m_{s-1}+1)}\frac1{m^2}. \]

Let \(Z^{(s)}\) \((s=0,1,\ldots,\infty)\) be the set of points of the space \(z_{is}\) that is the image of the set \(\Lambda^{(s)}\) under the transformation defined by the equations

\[ (\lambda_i-\mu_i)+H_{is}(z_{js})=0 \qquad (i,j=1,\ldots,n). \]

It follows from (1.8) that

\[ \operatorname{mes}Z^{(s)}=(J^{-1}+\delta_s(\rho))\operatorname{mes}\Lambda^{(s)},\qquad \lim_{\rho\to0}\delta_s(\rho)=0 \qquad (s=0,1,\ldots,\infty). \]

Hence

\[ \operatorname{mes}Z^{(0)}<(J^{-1}+|\delta_0(\rho)|)\operatorname{mes}\Lambda^{(0)}, \tag{3.3} \]

\[ \operatorname{mes}Z^{(\infty)}>(J^{-1}-|\delta_\infty(\rho)|)\operatorname{mes}\Lambda^{(\infty)}. \tag{3.4} \]

Letting \(s\) tend to infinity in (3.2), we obtain

\[ \operatorname{mes}\Lambda^{(\infty)}>(1-D\rho)\operatorname{mes}\Lambda^{(0)} \tag{3.5} \]

for some positive \(D\).

From (3.3), (3.4), and (3.5) we have

\[ \frac{\operatorname{mes} Z^{(\infty)}}{\operatorname{mes} Z^{(0)}}>(1-\delta(\rho)),\qquad \lim_{\rho\to 0}\delta(\rho)=0, \]

and then

\[ \lim_{\rho\to 0}\frac{\operatorname{mes} Z^{(\infty)}}{\operatorname{mes} Z^{(0)}}=1. \]

It follows from the last relation that, as \(\rho\to 0\), the measure of the set of points of the \(\rho\)-neighborhood of the origin of coordinates of system (1.11) lying on invariant toroidal surfaces tends to the measure of this neighborhood.

Finally, we note that system (1) is transformed into the system (1.11) corresponding to \(s=0\) by means of the transformations (1.1). As a result we obtain the theorem.

Theorem. Let \(V_\varepsilon\) be an \(\varepsilon\)-neighborhood of the origin of coordinates for system (1). Then, for any \(\chi>0\), one can indicate an \(\varepsilon_0>0\) such that, for all \(0<\varepsilon<\varepsilon_0\), the measure of the subset \(v_\varepsilon\) free of points belonging to \(n\)-dimensional invariant toroidal surfaces satisfies the inequality

\[ \frac{\operatorname{mes} v_\varepsilon}{\operatorname{mes} V_\varepsilon}<\chi. \]

Remark. We assumed the incommensurability of \(\lambda_1,\ldots,\lambda_n\). However, for the validity of the theorem it suffices to restrict oneself to the assumption
\(k_1\lambda_1+\cdots+k_n\lambda_n\ne 0\) when
\(|k_1|+\cdots+|k_n|\le 2m_0\).

Appendix. Below we consider a sequence of positive integers \(\{m_k\}\) \((k=0,1,\ldots)\), \(m_{k+1}=2m_k\), and positive constants \(\alpha\) and \(\rho<1\).

Lemma 1. There exists \(M(\alpha)\) such that, for all \(m_0>M(\alpha)\),

\[ \prod_{k=0}^{\infty}(2m_k+1)^{\frac{\alpha\ln\rho}{2m_k+1}}>\rho^{1/2}. \]

Lemma 2. There exist \(M(\alpha)\) and \(\rho_0(\alpha)\) such that, for \(m_0>M(\alpha)\) and for all \(\rho<\rho_0(\alpha)\),

\[ \frac{(2m_k+1)^{\alpha\ln\rho}} {1-(2m_k+1)^{\frac{\ln\rho}{2m_k+1}}} >\rho^{1/4}. \]

Proof of Lemma 1. The series

\[ \sum_{k=0}^{\infty}\frac{\ln(2m_k+1)}{2m_k+1} \]

converges, and its sum is less than

\[ \frac{1}{2\alpha}, \]

if \(m_0>M(\alpha)\). Then

\[ \prod_{k=0}^{\infty}(2m_k+1)^{\frac{\alpha\ln\rho}{2m_k+1}}>\rho^{1/2}. \]

Proof of Lemma 2. Since

\[ \frac{(2m_0+1)^{\alpha\ln\rho}} {1-(2m_0+1)^{\frac{\ln\rho}{2m_0+1}}} < \frac{(2m_0+1)^{\alpha\ln\rho}} {1-\exp\left\{\frac{\ln\rho}{2m_0+1}\right\}} = \frac{(2m_0+1)^{\alpha\ln\rho}} {1-\rho^{\frac{1}{2m_0+1}}}, \]

it follows that

\[ \frac{(2m_0+1)^{\alpha\ln\rho}} {1-(2m_0+1)^{\frac{\ln\rho}{2m_0+1}}} <\rho^{1/4}, \]

if

\[ \alpha\ln\rho \ln(2m_0+1) < \frac14\ln\rho+\ln\left(1-\rho^{\frac{1}{2m_0+1}}\right). \]

This inequality is valid if \((2m_0+1)>e^{1/\alpha}\), \(\rho<\rho_1\), where \(\rho_1\) is the smallest root of the equation

\[ \ln\left(1-\rho^{\frac{1}{2m_0+1}}\right)=\frac34\ln\rho. \]

Next,

\[ \operatorname{sign}\frac{d}{dx}\left(\frac{x^{\alpha\ln\rho}}{1-\rho^{1/x}}\right) = -\operatorname{sign}\left[ \alpha-\frac{\rho^{1/x}}{x(1-\rho^{1/x})} \right]. \]

The equation

\[ \alpha=\frac{\rho^{1/x}}{x(1-\rho^{1/x})} \]

defines a function \(\rho(1/x)\to e^{-1/\alpha}\) as \(x\to\infty\). Consequently, there exists \(\rho_2\) such that, for all \(\rho<\rho_2\), \(x\ge 2m_0+1\), the function \(x^{\alpha\ln\rho}(1-\rho^{1/x})^{-1}\) decreases monotonically with \(x\). Then, for all \(k\ge 1\),

\[ \frac{(2m_k+1)^{\alpha\ln\rho}} {1-(2m_k+1)^{\frac{\ln\rho}{2m_k+1}}} < \frac{(2m_k+1)^{\alpha\ln\rho}} {1-\rho^{\frac{1}{2m_k+1}}} < \frac{(2m_0+1)^{\alpha\ln\rho}} {1-\rho^{\frac{1}{2m_0+1}}} <\rho^{1/4}, \]

if \(\rho<\rho_0=\min\{\rho_1,\rho_2\}\).

References

1. Kolmogorov A. N. Dokl. Akad. Nauk SSSR, 98, No. 4, 1954.
2. Kolmogorov A. N. International Mathematical Congress in Amsterdam. Moscow, Fizmatgiz, 1961.
3. Arnol’d V. I. Uspekhi Mat. Nauk, 18, issue 6, 1963.
4. Malkin I. G. Theory of Stability of Motion. Moscow–Leningrad, GITTL, 1952.

Received by the editors
May 18, 1967

Leningrad State University
named after A. A. Zhdanov