UDC 517.925

ANALYTIC STRUCTURE

OF THE SOLUTION OF A DIFFERENTIAL EQUATION

WITH NONHOLOMORPHIC RIGHT-HAND SIDE IN THE CASE OF A CENTER

AND FOCUS

A. N. ERUGIN

§ 1. DEFINITIONS. NOTATION

We shall give the basic definitions and notation. By \(o(f(t))\) we shall denote infinitesimal quantities (i.q.) having a higher order of smallness than the order of smallness of the function \(f(t)\). Moreover, when speaking of the order of smallness of infinitesimal quantities, we shall have in mind the definitions given by us in [5].

We shall call a \(T\)-function

\[ T(t)=\sum_{n=1}^{\infty}(a_n\sin nt+b_n\cos nt), \]

where

\[ \sum_{n=1}^{\infty}(|a_n|+|b_n|)<+\infty, \]

and a \(K\)-function \(a_0+T(t)\), where \(a_0=\mathrm{const}\).

For the coefficients of the binomial series one can prove an estimate that we shall need for § 3.

Let \(\alpha_n=\mathrm{const}\), \(\alpha_n<\alpha_{n+1}\) \((n=1,2,\ldots)\); \(s\) an integer. Denote

\[ C_{n,s}=\frac{1}{s!}\prod_{l=0}^{s-1}(\alpha_n-l). \]

For \(C_{n,s}\) the estimate

\[ |C_{n,s}|\leq c\exp \alpha_n,\qquad c=\mathrm{const}, \tag{1.1} \]

is valid.

Further, integrating by parts, one can obtain formulas that will be needed for § 5. Let \(\vartheta_1=c(\vartheta)^{1/n}\); \(s,q,n,c=\mathrm{const}\); \(s\geq 0\), \(q,n,c>0\); \(s\) an integer. The following formulas are valid:

\[ \int \frac{\ln^s \vartheta}{\vartheta_1^q}\,d\vartheta = \frac{1}{\vartheta_1^{\,q-n}} \sum_{l=0}^{s} F_l^{s,q}\ln^{q-l}\vartheta+C, \]

\[ \left(\frac{\ln^s\vartheta}{\vartheta_1^q}\right)^{(l)} = \frac{1}{\vartheta_1^{\,q-ln}} \sum_{v=0}^{\min(l,s)} Q_{l,v}^{s,q}\ln^{\,l-v}\vartheta, \tag{1.2} \]

where

\[ F_l^{s,q}=-\frac{1}{(q-n)^{l+1}}\frac{n^{l+1}}{c^n}\frac{s!}{(s-l)!}, \]

\[ Q_{l,\nu}^{s,q}=-\frac{s!}{(s-\nu)!}\frac{Q_{l-\nu}^{q,c}}{n^{l-\nu}}c^{ln}, \]

where \(Q_{l-\nu}^{q,l}\) is the sum of all possible products of \(l-\nu\) elements from \(q+sn\) \((s=0,1,2,\ldots,l-1)\).

By definition, set

\[ Q_{0,0}^{s,q}=1;\qquad Q_{0,\nu}^{s,q}=0 \quad (1\le \nu \le s). \]

In what follows, when referring, for example, to Theorem 2 of paper [5] or to formula (4.5) of paper [3], we shall use, respectively, the notation: Theorem 2 [5], formula (4.5) [3].

§ 2. THE AIM OF THE INVESTIGATION. PRELIMINARY CONSIDERATIONS

The present paper is a generalization of the cited investigations of the analytic structure of the solution of the equation of integral curves of a system of type (2.1). The aim of the paper is to investigate the analytic structure of the solution of the equation of integral curves of the system

\[ x'=-y+\sum_{n=1}^{\infty}u_n^{(1)}(x,y)\equiv -y+X(x,y), \tag{2.1} \]

\[ y'=x+\sum_{n=1}^{\infty}u_n^{(2)}(x,y)\equiv x+Y(x,y), \]

where 1) the series \(X, Y\) converge absolutely and uniformly for \(x^2+y^2=r^2\le r_0^2=\mathrm{const}\); 2) \(u_n^{(l)}(r\cos\vartheta,r\sin\vartheta)=r^{\alpha_n^{(l)}}u_n^{(l)}(\cos\vartheta,\sin\vartheta)\); \(\alpha_n^{(l)}=\mathrm{const}\); \(1<\alpha_1^{(l)}<\alpha_2^{(l)}<\cdots<\alpha_n^{(l)}<\alpha_{n+1}^{(l)}\) \((n\ge1)\); \(u_n^{(l)}(\cos\vartheta,\sin\vartheta)\) are certain \(K\)-functions \((l=1,2)\).

It is not hard to show that in the present case the qualitative picture is one of three types: center, focus, or center-focus. Lyapunov in [1] studies system (2.1) in the case of analytic \(X,Y\) and shows that the qualitative picture is either of center type or of focus type. Further, in [1] the analytic structure of the solution of system (2.1) in the case of a center is studied. The analytic structure of the solution of the equation of integral curves of this system in the case of a focus is studied in papers [3, 4]; a special case of system (2.1) is studied in [2]. The present paper is a generalization of investigations [1—4], concerning system (2.1) with holomorphic right-hand sides, to the case of nonholomorphic right-hand sides of the system; moreover, we note that here a detailed exposition [6] is given.

The right-hand sides of system (2.1) begin with specific terms of first order. However, we note that in paper [7] it is shown that if system (2.1) begins with terms of order \(r^\alpha\), \(\alpha=\mathrm{const}\), then there are two cases: in case I the equation of integral curves is obtained in a form similar to the case now being studied, while in case II the equation of integral curves has specific peculiarities; therefore the analytic structure of the solution in this case differs essentially from the analytic structure of the solution in the case we are now studying.

Passing to polar coordinates, we obtain the equation of the integral curves of system (2.1) in the form

\[ \frac{dr}{d\vartheta}=\sum_{n=1}^{\infty} Q_n(\vartheta) r^{\bar{\alpha}_n}=Q(r,\vartheta), \tag{2.2} \]

where \(\bar{\alpha}_n=\mathrm{const}\), \(1<\bar{\alpha}_1<\bar{\alpha}_2<\cdots<\bar{\alpha}_n<\bar{\alpha}_{n+1}\) \((n\geq 1)\); the \(Q_n\) are certain \(K\)-functions; the series \(Q(r,\vartheta)\) converges absolutely and uniformly for \(r\leq r_0\).

We shall seek the solution of equation (2.2) in the form

\[ r=C\left(1+\sum_{n=1}^{\infty} u_n(\vartheta) C^{\gamma_n}\right), \tag{2.3} \]

where \(\gamma_n=\mathrm{const}\); \(1<\gamma_1<\gamma_2<\cdots<\gamma_n<\gamma_{n+1}\) \((n\geq 1)\); the \(u_n\) are certain \(2\pi\)-periodic functions; \(C\) is an arbitrary constant. Substituting (2.3) into (2.2), we shall have

\[ r'=C\sum_{n=1}^{\infty} u'_n(\vartheta) C^{\gamma_n} =\sum_{n=1}^{\infty} Q_n C^{\bar{\alpha}_n} \left(1+\sum_{n=1}^{\infty} u_n C^{\gamma_n}\right)^{\bar{\alpha}_n} = \]

\[ =\sum_{n=1}^{\infty} Q_n C^{\bar{\alpha}_n} \left(1+\sum_{s=1}^{\infty} U_{n,s} C^{\beta_s}\right) =\sum_{n=1}^{\infty} U_n C^{\delta_n}, \tag{2.4} \]

where \(U_{n,s}\) are polynomials in \(u_l\) \((l\leq s)\); the \(\beta_s\) are linear combinations of \(\gamma_n\) with integer nonnegative coefficients, with \(l\leq s\); \(U_n\) are polynomials in \(Q_n\), \(U_{u,v}\), with \(u,v\leq n\); the \(\delta_n\) are linear combinations of \(\alpha_u,\beta_v\) with integer nonnegative coefficients, with \(u\leq n,\ v\leq n\). It is not difficult to see that, in order to determine the \(u_n\), we obtain an infinite system of differential equations, and, in order to determine the \(\gamma_n\), an infinite system of algebraic equations, from which \(u_n,\gamma_n\) are determined successively:

\[ u'_n=U_n,\qquad 1+\gamma_n=\delta_n. \tag{2.5} \]

Moreover, it is most convenient to obtain \(u_n\) from system (2.5) with the initial data \(u_n(\vartheta_0)=0\), since then we shall have \(r(\vartheta_0)=C\). In this case there are two possibilities:

1) the functions \(u_n\) are obtained as \(2\pi\)-periodic functions;

2) the functions \(u_n\), beginning with the function \(u_m\), are obtained as nonperiodic, with \(u_m\) being obtained in the form \(u_m=c\vartheta+V(\vartheta)\), \(c=\mathrm{const}\ne0\), while \(V\) is a certain \(2\pi\)-periodic function.

§ 3. THE CASE OF A CENTER

We proceed to study the case when the functions \(u_n\) are obtained as periodic. We shall show that in this case the series (2.3) converges for all \(\vartheta\) and for sufficiently small \(C\) uniformly with respect to \(\vartheta\), and that this series gives the analytic structure of the solution of the equation of the integral curves of system (2.1). Since the functions \(u_n\) are periodic, it is clear that the solutions of this equation are periodic, and we have a center.

Applying the theorem from § 12 of [1], we obtain that if the series \(Q(r,\vartheta)\) converges absolutely and uniformly for \(r\leq r_0\) and for all \(\vartheta\), then there are constants \(\overline Q_n\), \(|Q_n(\vartheta)|\leq \overline Q_n\) \((n=1,2,\ldots)\), such that the series

\[ \sum_{n=1}^{\infty}\overline Q_n(re)^{\overline a_n}\equiv \overline Q(r) \]

converges absolutely and uniformly for \(r\leq r'_0\leq r_0\). Suppose that there exists a series convergent for \(C\leq C_0\),

\[ \overline r = C\left(1+\sum_{n=1}^{\infty}\overline u_n C^{\nu_n}\right) \equiv C(1+\overline S(C)), \tag{3.1} \]

where \(\overline u_n=\mathrm{const}\), \(|u_n(\vartheta)|\leq \overline u_n\) \((n=1,2,\ldots;\ \vartheta\geq \vartheta'_0)\).

Substituting \(\overline r\) into (2.2), applying estimate (1.1), and taking into account the remark made at the beginning of this paragraph, we shall have, for \(\vartheta\geq \vartheta'_0\) and \(C\leq C_0\leq r'_0\leq r_0\),

\[ U(C,\vartheta_0) = \sum_{n=1}^{\infty}U_n C^{\delta_n} \leq C\sum_{n=1}^{\infty}\overline Q_n(Ce)^{\overline a_n} \left(1+\sum_{n=1}^{\infty}\overline S^{s}\right) = \]

\[ = \sum_{n=1}^{\infty}\overline U_n C^{\hat n} \equiv \overline U(C), \tag{3.2} \]

where \(\overline U_n=\mathrm{const}\), \(|U_n(\vartheta)|\leq \overline U_n\) \((n=1,2,\ldots)\).

Let, by definition, \(\overline u_n=2\pi \overline U_n\). It is not difficult to see that, since \(u_n=\int_{\vartheta_0}^{\vartheta} U_n\), \(|U_n|\leq \overline U_n\) \((n=1,2,\ldots)\), we shall have \(|u_n(\vartheta)|\leq \overline u_n\) \((n=1,2,\ldots)\).

Indeed, \(u_1=\int_{\vartheta_0}^{\vartheta} Q_1 e^{\overline a_1}\,d\vartheta\), and \(Q_1\) is a \(T\)-function, because under the condition of this case \(u_n\) \((n>1)\) are periodic functions; therefore the integrands are \(T\)-functions, i.e.

\[ |u_1| \leq e^{\overline a_1} \left|\int_{\vartheta_0}^{\vartheta_0+2\pi} Q_1\,d\vartheta\right| \leq e^{\overline a_1}2\pi \overline Q_1 \leq 2\pi \overline U_1, \tag{3.3} \]

since one may assume that \(c>1\), and \(\overline U_1=c\overline Q_1 e^{\overline a_1}\) (\(c\) from (1.1)). Similarly we obtain that also \(|u_n(\vartheta)|\leq \overline u_n\) \((n>1)\).

Next, substituting in (3.2) \(\overline S(C)=(C^{-1}\overline r-1)\) and taking (2.5) into account, from (3.2) we shall have

\[ 2\pi \overline U(C) = C\overline S(C) = \overline r-C = \frac{2\pi cC}{2C-\overline r} \sum_{n=1}^{\infty}\overline Q_n(Ce)^{\overline a_n} \equiv \]

\[ \equiv \frac{2\pi cC}{2C-\overline r}\,\overline Q(C) = \frac{2\pi cC^2}{2C-\overline r}\,\overline Q(C). \tag{3.4} \]

Hence we obtain an equation for \(\overline r\):

\[ \overline r^{\,2}-3C\overline r+2C\bigl(1+\pi c\overline Q(C)\bigr)=0. \tag{3.5} \]

It is not hard to see that of the two solutions of equation (3.5) the one suitable for us is

\[ \bar r=\frac{C}{2}\left(3-\left(1-8\pi c\bar Q(C)\right)^{\frac12}\right) =\frac{C}{2}\left(2+\sum_{n=1}^{\infty}s_n\bar Q^n\right), \tag{3.6} \]

where

\[ s_n=4^n(\pi c)^n(n!)^{-1}\prod_{s=1}^{n-1}(2s-1). \]

Therefore from equation (3.5) we can obtain \(\bar r\) in the form of a series in powers of \(C\), convergent for sufficiently small \(C\), with constant positive coefficients. On the other hand, from the method of obtaining equation (3.5) it is clear that the resulting series is a majorant for (2.3) for \(C\leq C_0\leq z_0'\leq z_0\) and for \(\vartheta\geq\vartheta_0\). Hence we obtain that the series (2.3), for sufficiently small \(C\) and for \(\vartheta\geq\vartheta_0'\), converges absolutely and uniformly with respect to \(\vartheta\), and this series gives the analytic structure of the solution in the case of a center.

§ 4. THE EQUATION IN THE CASE OF A FOCUS

Let now \(u_m=c\vartheta+V(\vartheta)\), \(c=\mathrm{const}\ne0\). Make the change of variables by the formula

\[ r=z\left(1+\sum_{n=1}^{m-1}u_n(\vartheta)z^{\gamma_n} +V(\vartheta)z^{\gamma_m}\right)\equiv zS(z,\vartheta). \tag{4.1} \]

Substituting (4.1) into (2.2), taking into account that \(u_n=u_n(u_1,\ldots,u_{n-1})\), we shall have

\[ r'=z'S(z,\vartheta)+zS'_{\vartheta}(z,\vartheta) =z'\left(S+z\frac{\partial S}{\partial z}\right) +z\frac{\partial S}{\partial\vartheta} \]

\[ =\sum_{n=1}^{\infty}Q_n(zS)^{\bar\alpha_n} =\sum_{n=1}^{m}U_nz^{\delta_n}+o(z^{\delta_m}), \tag{4.2} \]

\[ z'\left(S(z,\vartheta)+\frac{\partial S}{\partial z}\right) +z\frac{\partial S}{\partial\vartheta} =cz^{\delta_m}+\sum_{n=m+1}^{\infty}\tilde U_nz^{\tilde\delta_n}, \tag{4.3} \]

\[ z'=cz^{\delta_m}+o(z^{\delta_m}). \tag{4.4} \]

It is easy to see that we have a focus. It is necessary to note that, studying the cases indicated in § 2, we established that in case 1) we have a center, and in case 2) a focus, i.e., we have either a center or a focus, as in the holomorphic case.

It is not hard to see that, without loss of generality, one may assume that \(c<0\), since for \(\vartheta\,\mathrm{sign}\,c\to-\infty\), \(z\to0\). Taking into account \(\delta_m>1\) and the structure of the coefficients \(Q_n,\ u_n,\ U_n,\ \tilde U_n\), the equation of the integral curves of system (2.1) in the case of a focus is obtained in the form

\[ z'=-cz^{1+a_0}+\sum_{n=1}^{\infty}Z_n(\vartheta)z^{1+a_0+a_n} \equiv -cz^{1+a_0}+Z(z,\vartheta), \tag{4.5} \]

where \(z,\vartheta\) are generalized polar coordinates, \(Z_n(\vartheta)=c_n+\overline Z_n,\ c_n=\mathrm{const}\), \(\overline Z_n\) are certain \(T\)-functions, and the series \(Z(z,\vartheta)\) converges absolutely and uniformly for \(z\le z_0=\mathrm{const}\) and for all \(\vartheta\).

Denote
\[ l_s=\sum_{n=s}^{\infty} Z_n z^{\alpha_n},\qquad I=\int_{\vartheta_0}^{\vartheta} l_1\,d\vartheta; \]
\[ s_0=-(c\vartheta_0+z_0^{-\alpha_0}a_0^{-1});\qquad \vartheta_1^{\alpha_0}=-ca_0\vartheta=c^{\alpha_0}\vartheta,\qquad \vartheta_{1,0}^{\alpha_0}=-ca_0\vartheta_0; \]
\[ \gamma_1=(s_0+I)\vartheta_1^{-\alpha_0};\qquad \overline\gamma_1=-\alpha_0\gamma_1; \]
\[ a_q=\overline a_q(-1)^q a_0^q,\qquad \overline a_q=(q!\,\alpha_0^q)^{-1}(-1)^q\prod_{l=0}^{q-1}(1-l\alpha_0). \]

Applying these notations in integrating equation (4.5), we obtain
\[ z^{-\alpha_0}=ca_0(\vartheta-\vartheta_0)+z_0^{-\alpha_0}-a_0 I, \]
\[ \left. \begin{aligned} z&=\vartheta_1^{-1}(1+\overline\gamma_1)^{-\alpha_0^{-1}} =\vartheta_1^{-1}\left(1+\sum_{q=1}^{\infty}\overline a_q\,\overline\gamma_1^{\,q}\right)\\ &=\vartheta_1^{-1}\left(1+\sum_{q=1}^{\infty}a_q\gamma_1^{q}\right) \equiv \vartheta_1^{-1}(1+\gamma_2). \end{aligned} \right\} \tag{4.6} \]

It is not difficult to show that, similarly to the holomorphic case [3], as \(\vartheta\to\infty\) one has \(\gamma_2\to0\). Taking this and formulas (4.6), (1.2) into account, one can obtain estimates, valid for sufficiently large \(\vartheta\),
\[ |I|\le \mathrm{const}\,\vartheta_1^{\alpha_0-\alpha_1},\qquad |\gamma_1|\le \mathrm{const}\,\vartheta_1^{-\alpha_1},\qquad |\gamma_2|\le \mathrm{const}\,\gamma_1^{\alpha_1}\quad (\alpha_1<\alpha_0); \]
\[ |I|\le \mathrm{const},\qquad |\gamma_1|\le \mathrm{const}\,\vartheta_1^{-\alpha_0},\qquad |\gamma_2|\le \mathrm{const}\,\vartheta_1^{-\alpha_0}\quad (\alpha_1>\alpha_0); \]
\[ |I|\le \mathrm{const}\ln\vartheta,\qquad |\gamma_1|\le \mathrm{const}\,\vartheta^{-1}\ln\vartheta,\qquad |\gamma_2|\le \mathrm{const}\,\vartheta^{-1}\ln\vartheta\quad (\alpha_1=\alpha_0); \]
\[ \left|\int_{\infty}^{\vartheta}\frac{Z_s(\vartheta)}{\vartheta_1^n}\gamma_2^l\,d\vartheta\right| \le \frac{\mathrm{const}}{\vartheta_1^{\,n+l\alpha_1-\alpha_0}} \quad (n+l\alpha_1\ne\alpha_0,\ \alpha_1<\alpha_0); \]
\[ \left|\int_{\infty}^{\vartheta}\frac{Z_s(\vartheta)}{\vartheta_1^n}\gamma_2^l\,d\vartheta\right| \le \frac{\mathrm{const}}{\vartheta_1^{\,n+(l-1)\alpha_0}} \quad (n+l\alpha_0\ne\alpha_0,\ \alpha_1>\alpha_0); \]
\[ \left|\int_{\infty}^{\vartheta}\frac{Z_s(\vartheta)}{\vartheta_1^n}\gamma_2^l\,d\vartheta\right| \le \frac{\mathrm{const}\,\ln^l\vartheta}{\vartheta_1^{\,n+(l-1)\alpha_0}} \quad (n+l\alpha_0\ne\alpha_0,\ \alpha_1=\alpha_0), \]
where \(n,l\) are nonnegative integers, and \(Z_s\) are \(K\)-functions. If a function
\[ \gamma=\mathrm{const}\,\vartheta_1^{-\alpha}+O(\vartheta_1^{-\alpha}),\qquad \alpha=\mathrm{const}>0, \]
and a \(T\)-function \(\overline Z(\vartheta)\) are given, then, applying Theorems 1 and 2 [5] and formulas (1.2), one can obtain the more precise estimate
\[ \left|\int_{\infty}^{\vartheta}\frac{\overline Z}{\vartheta_1^n}\gamma^l\,d\vartheta\right| \le \frac{\mathrm{const}}{\vartheta_1^{\,n+l}}. \]

We shall henceforth call these estimates the estimates of § 4.

Applying the method constructed by us in studying system (2.1) with holomorphic right-hand sides [3], one can find terms of arbitrarily high order of smallness in the asymptotic expansion of \(\gamma_2\), which, as in the holomorphic case, we obtain through the quantities
\(h_s, \vartheta_1^{-l}\ln^s\vartheta_1,\ H_{s,l}\vartheta_1^{-l}\ln^s\vartheta_1\)
(\(s\) integer, \(s \geqslant 0,\ l \geqslant a_0s\)), where \(h_{s,l}=\mathrm{const}\), and \(H_{s,l}\) are certain \(K\)-functions. Taking into account that the method for obtaining the asymptotic expansion of \(\gamma_2\) is described in [3], our principal aim below is to study the specific features that distinguish the case of nonholomorphic right-hand sides of system (2.1) from the case of holomorphic right-hand sides of system (2.1). We note here that, in studying the analytic structure of the solution of the equation of integral curves of system (2.1) with nonholomorphic right-hand sides, one must distinguish several cases in which the analytic structure of the solution, i.e., the structure of \(\gamma_2\), assumes its own specific form; moreover, one of these cases is essentially similar to the case of holomorphic right-hand sides of system (2.1). Our principal aim below is to study the structure of \(\gamma_2\) in these cases.

The approximate expression for \(\gamma_l\) (\(l=1,2\)), obtained with accuracy up to infinitesimal quantities of order \(\vartheta_1^{-s}\), will be denoted by \(\gamma_{l,s}\). Linear combinations of the numbers \(a_0, a_1, a_2,\ldots\) with integer nonnegative coefficients will be denoted below by \(S\), and linear combinations of the numbers \(a_1,a_2,\ldots,a_n\) with integer nonnegative coefficients will be denoted below by \(S_n\).

§ 5. ANALYTIC STRUCTURE OF THE SOLUTION OF THE EQUATION OF INTEGRAL CURVES OF SYSTEM (2.1) IN THE CASE OF A FOCUS

In investigating the analytic structure of the solution of equation (4.5), one must distinguish two principal cases:

\[ \begin{aligned} \mathrm{I.}\quad & a_0<a_1;\\ \mathrm{II.}\quad & a_m \leqslant a_0<a_{m+1}\quad (m=1,2,\ldots)\quad (m\geqslant 1,\ \text{integer}),\\ & 1)\quad a_m=a_0\quad (m=1,2,\ldots)\quad (m\geqslant 1,\ \text{integer}),\\ & 2)\quad a_m<a_0\quad (m=1,2,\ldots)\quad (m\geqslant 1,\ \text{integer}),\\ & \qquad \text{a) } S_{n_0}=a_0\quad (n_0\geqslant 1,\ \text{integer}),\\ & \qquad \text{b) } S_n\ne a_0\quad (n=1,2,\ldots)\quad (n\geqslant 1,\ \text{integer}). \end{aligned} \]

Case I. Applying the estimates of § 4, it is not difficult to obtain that the integral \(I(\vartheta)=\)

\[ =\int_{\infty}^{\vartheta} l_1\,d\vartheta \]

converges for \(\vartheta\geqslant \vartheta_0\), i.e., for \(\vartheta\geqslant \vartheta_0\) we shall have

\[ I=\int_{\infty}^{\vartheta} l_1(\vartheta)\,d\vartheta +\int_{\vartheta_0}^{\infty} l_1(\vartheta)\,d\vartheta =\bar I(\vartheta)+C',\qquad C'=\mathrm{const}, \]

\[ \gamma_{1,a_0}=(s_0+C')\vartheta_1^{-a_0} +\vartheta_1^{-a_0}\bar I(\vartheta) =(s_0+C')\vartheta_1^{-a_0}+o(\vartheta_1^{-a_0}). \]

Substituting \(\gamma_{1,a_0}\) into (4.6), we obtain

\[ \gamma_{2,a_0}=a_1(s_0+C')\vartheta_1^{-a_0}+o(\vartheta_1^{-a_0}). \]

Substituting \(\gamma_{2,\alpha_0}\) in \(\tilde I\), taking into account formulas (4.6), (1.2), and applying Theorem 1 [5], we obtain

\[ \tilde I=c_1 F_0^{0,\alpha_1}\vartheta_1^{\alpha_0-\alpha_1} +o\left(\vartheta_1^{\alpha_0-\alpha_1}\right); \]

\[ \gamma_{1,\alpha_1} =(s_0+C')\vartheta_1^{-\alpha_0} +c_1 F_0^{0,\alpha_1}\vartheta_1^{-\alpha_1} +o\left(\vartheta_1^{-\alpha_1}\right), \]

\[ \gamma_{2,\alpha_1} =a_1(s_0+C')\vartheta_1^{-\alpha_0} +a_1c_1F_0^{0,\alpha_1}\vartheta_1^{-\alpha_1} + \]

\[ +a_2(s_0+C')^2\vartheta_1^{-2\alpha_0} +o\left(\vartheta_1^{-\alpha_1}\right) +o\left(\vartheta_1^{-2\alpha_0}\right). \]

It is not difficult to see that, continuing in this way and applying formulas (1.2) and Theorems 1 and 2 [5], we obtain \(\gamma_2\) in the form

\[ \gamma_2= \sum_{n=1}^{s} h_n\vartheta_1^{-\gamma_n} + \sum_{n=1}^{s}\overline H_n\vartheta_1^{-\delta_n} +o\left(\vartheta_1^{-\gamma_s}\right) \equiv \]

\[ \equiv \sum_{n=1}^{s} H_n\vartheta_1^{-\gamma_n} +o\left(\vartheta_1^{-\gamma_s}\right), \tag{5.1} \]

where \(\gamma_n,\delta_n=\mathrm{const}\), \(\gamma_n<\gamma_{n+1}\), \(\delta_n<\delta_{n+1}\) \((n=1,2,\ldots)\), with \(\gamma_n,\delta_n\) certain combinations of \(S\); \(h_n\) are certain constants; \(\overline H_n\) are certain \(T\)-functions. Further, \(s_0+C'\) is a certain constant, taking its own value on each integral curve, and therefore it may be taken as an arbitrary constant \(C\). It is not difficult to see that the analytic structure of \(\gamma_1\) is similar to the analytic structure of \(\gamma_2\).

Indeed, continuing the indicated process, we shall multiply, add, and integrate terms of the form \(Z(\vartheta)\vartheta^{-s}\), where \(Z\) is a certain \(K\)-function, \(s=\mathrm{const}\); by virtue of Theorems 1 and 2 [5] and formulas (1.2), we shall obtain terms of the same type, i.e. \(\gamma_1,\gamma_2\) are obtained in the form (5.1), and from the method of obtaining \(\gamma_1,\gamma_2\) it is clear that these series are asymptotic.

Continuing the indicated process sufficiently far, we shall obtain \(\gamma_1,\gamma_2\) with accuracy up to terms of arbitrarily high order; however, recurrence formulas for determining \(H_n\), analogous to the recurrence formulas for \(h_{s,l},H_{s,l}\) in the holomorphic case, are not obtained in this case, because there is no algorithm for obtaining \(\gamma_n\), since there is no algorithm for obtaining \(\alpha_n\). In order to obtain these formulas, one must study special cases, and it is necessary to have the function \(\alpha_n=\alpha(n)\). This remark is also valid for case II.

Case II. 1) is similar to the case of holomorphic right-hand sides of system (2.1), which is studied in papers [3—5]; therefore we shall not study it in detail, but shall only examine the distinctive features of this case in comparison with case I and with the case of holomorphic right-hand sides of system (2.1).

Applying the indicated method, we obtain \(\gamma_2\) in the form

\[ \gamma_2= \sum_{q=0}^{s}\sum_{l=l_q}^{l_{q,s}} h_{q,l}\frac{\ln^q\vartheta}{\vartheta_1^{\gamma_{q,l}}} + \sum_{q=0}^{s-1}\sum_{l=l_q}^{l_{q,s}} \overline H_{q,l}(\vartheta) \frac{\ln^q\vartheta}{\vartheta_1^{\delta_{q,l}}} + \]

\[ +o\left(\frac{1}{\vartheta_1^{\gamma_{s,l_{s,s}}}}\right), \tag{5.2} \]

where \(h_{q,l}\) are certain constants; \(\overline H_{q,l}\) are certain \(T\)-functions; \(\gamma_{q,l},\delta_{q,l}\) are certain combinations of \(S\), with
\[ \alpha_0 q\leqslant \gamma_{q,l_q},\qquad \gamma_{q,l_{q,s}}<s\alpha_0,\qquad (q+1)\alpha_0\leqslant \delta_{q,l_q}, \]

\(\alpha_0 s > \delta_{q,l}/q,s\). Terms with logarithms appear when integrating terms of the form \(\operatorname{const}\,\vartheta^{-1}\). It is not hard to see that in case I, in the expansion into a series of the integrand \(l_1(\vartheta)\), no terms of this kind occur, whereas here they do occur if \(H_{1,1}\ne 0\).

The present case is more complicated than case I, since the integral \(\int_{\infty}^{\vartheta} l_1 d\vartheta\) does not converge. Therefore one must first integrate successively \(I(\vartheta)\), obtain increasingly accurate approximations for \(\gamma_1,\gamma_2\), and then write \(I(\vartheta)\) in a form similar to (4.5) [3]:

\[ I(\vartheta)=\bar I(\vartheta)+I_2(\vartheta)+I_1(\vartheta)+\sum_{s=1}^{3} C_s, \]

where

\[ \bar I=c_1 F_0^{0,\alpha_1}\vartheta_1^{\alpha-\alpha_1}+\ldots+\operatorname{const}\ln\vartheta, \]

\[ I_2=\int_{\infty}^{\vartheta} L(\vartheta)\,d\vartheta,\quad L=l_1-l_{m+1}-\bar I',\quad I_1=\int_{\infty}^{\vartheta} l_{m+1}\,d\vartheta, \]

\[ C_1=\int_{\vartheta_0}^{\infty} l_{m+1}\,d\vartheta,\quad C_2=\int_{\vartheta_0}^{\infty} L\,d\vartheta,\quad C_3=-\bar I(\vartheta_0), \]

and the order of smallness of the function \(L\) is equal to the order of smallness of the function \(\vartheta^{-(1+s)}\) \((s=\operatorname{const}>0)\). The term \(\gamma_1\), whose order of smallness is equal to the order of smallness of the function \(\vartheta^{-1}\), is obtained, respectively, in the form \(C\vartheta_1^{-\alpha_0}\), \(C=s_0+\sum_{s=1}^{3} C_s\).

Further investigation is essentially similar to case I, apart from the fact that a greater complication arises because of the presence of terms with logarithms.

Case II. 2a) is similar to case II. 1), and case II. 2b) is similar to case I.

The basic properties of the asymptotic expansion of the solution of equation (4.5) are similar to the corresponding properties of the asymptotic expansion of the solution of equation (2.1) [3], but it is more expedient to study these properties in greater detail in a separate paper.

References

1. Lyapunov A. M. The General Problem of the Stability of Motion. GITTL, 1950.
2. Erugin A. N. Izvestiya AN BSSR, series of physical-technical sciences, No. 4, 23—32, 1958.
3. Erugin A. N. Izvestiya AN BSSR, series of physical-technical sciences, No. 1, 27—42, 1960.
4. Erugin A. N. Izvestiya AN BSSR, series of physical-technical sciences, No. 2, 16—26, 1960.
5. Erugin A. N. Izvestiya AN BSSR, series of physical-technical sciences, No. 1, 37—41, 1961.
6. Erugin A. N. DAN BSSR, 5, No. 5, 191—193, 1961.
7. Erugin A. N. DAN BSSR, 5, No. 7, 284—285, 1961.

Received by the editors
August 5, 1965

Leningrad Branch
of the V. A. Steklov
Mathematical Institute