UDC 517.916 : 517.917

ANALYTIC THEORY AND PROBLEMS OF THE REAL THEORY OF DIFFERENTIAL EQUATIONS CONNECTED WITH THE FIRST METHOD AND WITH METHODS OF ANALYTIC THEORY

N. P. ERUGIN

A few words about the plan of the lectures1 that I am going to give. There will be two sections here:

I. Analytic theory of differential equations—its problems and certain methods of investigation in the most general sense.

II. Real theory—when we study solutions on the real axis \(t\). Methods for solving questions of stability theory, the problems at which we have arrived, possible modifications of known methods of the real theory, and a new approach to the solution of these problems on the basis of methodology developed in analytic theory. I shall speak about one method that arose in analytic theory, then migrated into the real theory, then returned in modified form to analytic theory, and is now again returning to the real theory.

I. ANALYTIC THEORY OF DIFFERENTIAL EQUATIONS

The task of any theory of differential equations is to classify the entire set of solutions according to certain properties of functions. Thus, different classifications are possible—they depend on the properties chosen. It is known what classification is carried out in the qualitative (real) theory and in stability theory. From the properties of the right-hand sides of differential equations we seek to determine what groups of solutions they have in the whole domain in which the right-hand sides are defined or in a part of it, for example in a neighborhood of some point. We also seek to determine to which class a given solution, subject to one or another condition, belongs. But different properties of functions may be taken as the basis of a classification. Here there also arises the question of the compatibility of certain properties of solutions. What properties of functions can be taken as the basis of classification in the analytic theory of differential equations? But first of all, what differential equations does the analytic theory of differential equations consider?

Equations of the form

\[ \frac{dy}{dz}=f(y,z),\quad y=(y_1,\ldots,y_n),\quad f=(f_1,\ldots,f_n), \]

where \(f(y,z)\) is analytic in some domain \(A(y,z)\). In a neighborhood of every point \((y_0,z_0)\) of this domain, \(f(y,z)\) is holomorphic, i.e., representable in the form of a series in positive powers of \((z-z_0)\), \((y_k-y_k^0)\) \((k=1,\ldots,n)\), \(y_0=(y_1^0,\ldots,y_n^0)\).

We do not consider those domains \(A(y,z)\) where \(f(y,z)\) will not be single-valued1.

Why do we consider single-valued \(f(y,z)\)? Because, analytically continuing the function \(y=y(z)\)—a solution—in this case we are dealing with one and the same equation

\[ \frac{dy}{dz}=f(y,z). \]

Thus, let \(f(y,z)\) be single-valued in the domain \(A(y,z)\). Take \((y_0,z_0)\in A(y,z)\). Then, by Cauchy’s theorem, we have a unique solution, holomorphic in a neighborhood of \(z_0\), possessing the property \(y\to y_0\) as \(z\to z_0\). Here \(z_0\) and \(y_0\) are complex or real. If \(y_0,z_0\) are real and \(f(y,z)\) is real for real \(y,z\), then \(y=y(z)\) will be real for real \(z\). Thus we have obtained a unique solution \(y=y(z)\) possessing the property \(y\to y_0\) as \(z\to z_0\). This solution is an analytic function in a neighborhood of the point \(z=z_0\), representable by a convergent Taylor series

\[ y=y_0+\sum_{k=1}^{\infty} a_k (z-z_0)^k. \]

One could, of course, have complicated the problem, and then the behavior of this function would in some sense have remained unclear even in a neighborhood of the point \(z=z_0\). A further study of its behavior in a neighborhood of the point \(z_0\) would have been required (but additional properties of \(f(y,z)\) would also have been required). We shall not do this; we shall assume that the behavior of \(y=y(z)\) is completely clear in a neighborhood of the point \(z_0\). But what will happen to \(y=y(z)\) under analytic continuation from a neighborhood of \(z=z_0\)? What shall we take as the basis for a classification of the whole set of analytic functions—solutions \(y=y(z)\)?

One may pose the following questions:

1. Are there, among the solutions of a given equation, multivalued solutions, or are they all single-valued?

2. If they are single-valued, are there \(y=y(z)\) having no singular points at all, if, for example, \(f(y,z)\) is an entire function without singular points in the finite part of the domain, for example, \(f(y,z)\) is a polynomial.

We note, however, that even in the presence of singular points of \(f(y,z)\), the solution \(y=y(z)\) may turn out to be an entire function.

1. Are there \(y=y(z)\) having only poles?

2. Are there among \(y=y(z)\) functions with essentially singular points?

3. If \(y=y(z)\) are multivalued, are there among them such that, as \(z\to z^*\), the function \(y(z)\) has no limit, where \(z^*\) is a singular point of the essentially singular type. If \(z^*\) is single-valued, then in this case it is simply essentially singular, according to the terminology of the theory of functions of a complex variable.

4. Are solutions possible whose singular points will not be isolated?

Let us note that even for single-valued \(f(y,z)\) the solution may turn out to be a multivalued function. For example, for the equations

\[ \frac{dy}{dz}=\frac{y}{2z} \quad \text{and} \quad \frac{dy}{dz}=\frac{1}{y} \]

the solution \(y=z^{1/2}\) is a multivalued (two-valued) function in a neighborhood of the point \(z=0\).

We have outlined a classification of the whole set of solutions, proceeding directly from the classification in the theory of functions of a complex variable. But this, of course, is not all. Another, finer classification is also possible. I shall speak about this later.

In general the question arises: if the solution \(y=y(z)\) has singular points, then what are they, and how can one construct this solution in a neighborhood of these singular points? This is the second problem, closely connected with the first. But the singular points of \(y=y(z)\) will be of different kinds—those which we see, as it were, immediately from the form of \(f(y,z)\), and those which, from the appearance of \(f(y,z)\), we cannot see directly. In what follows this last assertion will become clear.

§ 1. Fixed singular points

We shall consider one differential equation

\[ \frac{dy}{dz}=f(y,z) \tag{1.1} \]

with one unknown function \(y\).

We know that in a neighborhood of a point \(z_0\), \(y=y(z)\) is holomorphic if \(y \to y_0\) as \(z \to z_0\) and \(f(y,z)\) is holomorphic in a neighborhood of the point \((y_0,z_0)\). What point \(z_0\) can be singular for \(y=y(z)\)? For example, the following: if \(y \to y_0\) as \(z \to z_0\), but \(f(y,z)\) is not defined at the point \((y_0,z_0)\), or \(f(y,z)\) is not holomorphic at this point, or, perhaps, \(y=y(z)\) has no limit as \(z \to z_0\). Can such a solution exist, and if it can, how can it be constructed in a neighborhood of the point \(z_0\)?

Suppose, for example, that the equation (Briot and Bouquet) is given:

\[ \frac{dy}{dz} = \frac{by+az+\cdots}{z} = \frac{\varphi(y,z)}{z} = f(y,z), \tag{1.2} \]

where the constant \(b\ne 0\) and \(\varphi(y,z)\) is holomorphic in a neighborhood of the point \(y=z=0\). As we see, \(f(y,z)\) is not defined for \(y=0,\ z=0\). Here the conditions of Cauchy’s theorem are not satisfied in a neighborhood of the point \((0,0)\), but one may raise the question of the existence of solutions \(y=y(z)\to 0\) as \(z\to 0\). However, we immediately suspect the point \(z=0\) to be a singular point. Here the question is one of classifying solutions of this kind for the given equation. For example, for the equation

\[ \frac{dy}{dz}=n\frac{y}{z}, \qquad n \text{ an integer } >0, \]

we have the general solution \(y=cz^n\). As we see, all solutions have the indicated property and will be holomorphic in a neighborhood of the point \(z=0\). But this is not always so. Sometimes there are no such solutions; sometimes they exist, but not all of them: some may be holomorphic in a neighborhood of the point \(z=0\), while others will not be. If they are not holomorphic, then what can be said about them, and how can they be constructed in a neighborhood of the point \(z=0\)?

Poincaré was the first to investigate equation (1.2) in detail. He showed that if1 \(R(b)>0\), then solutions \(y=y(z)\to 0\) as \(z\to 0\) exist and are representable in powers of \(z\) and \(z^b\), or in powers of \(z\) and \(pz^b+kz^b\ln z\). The second case is possible only when \(b=n\) is an integer \(>0\).

Let us note that in the first case the function \(z^b\) is a solution of the shortened equation (1.2)

\[ \frac{dy}{dz}=b\,\frac{y}{z}. \]

We have the second case when equation (1.2) has the form

\[ \frac{dy}{dz}=\frac{by+az^b+\cdots}{z},\qquad b=n>0. \]

Here the function \(y=(a\ln z+b)z^b\) is also a solution of the shortened equation

\[ \frac{dy}{dz}=\frac{by+az^b}{z}. \]

More precisely, here we have the following result. In the first case, when \(R(b)>0\) and \(b\) is not an integer, there exists one holomorphic solution in a neighborhood of \(z=0\), \(y=y(z)\to 0\) as \(z\to 0\); the others will not be holomorphic, but are represented in the form of convergent series in positive powers of the quantities \(z\) and \(z^b\), \(y=y(z,z^b)\). In the second case, when \(b=n\) is a positive integer, either there is no such holomorphic solution, or there are infinitely many of them—all for sufficiently small initial values \(z_0\) and \(y_0\) in modulus. If there is no such holomorphic one-parameter family, then there is an infinite set of nonholomorphic solutions \(y=\Phi(z,(a\ln z+c)z^b)\to 0\) as \(z\to 0\), representable in the form of convergent series in positive powers of both arguments.

The Briot and Bouquet equation is a special case of the equation

\[ \frac{dy}{dz}=\frac{P(y,z)}{Q(y,z)}, \tag{1.3} \]

where \(P(y,z)\) and \(Q(y,z)\) are holomorphic in a neighborhood of the point \(y_0,z_0\), and \(P(y_0,z_0)=Q(y_0,z_0)=0\). We assume that \(P(y,z)\) and \(Q(y,z)\) have no common factor \(\varphi(y,z)\). Then the set of such points \((y_0,z_0)\) (in a neighborhood of which \(P(y,z)\) and \(Q(y,z)\) are holomorphic and vanish) is completely determined and consists of isolated points. The points \(z_0\) are called fixed, or immovable, singular points, although, as we have seen, they may turn out not to be singular even for solutions with the initial conditions \(y\to y_0\) as \(z\to z_0\). But here the problem is to study the set of solutions with such initial conditions and the character of the singularity for them at this point. Obviously, without diminishing the generality of the question, one may put \(y_0=z_0=0\). Under fairly general assumptions this equation is easily reduced to the Briot and Bouquet equation (1.2). In other cases it is reduced to another special case

\[ x^n\frac{dy}{dx}=a(x)+b(x)y+\sum_{i=2}^{\infty}\beta_i(x)y^i, \tag{1.4} \]

where \(n\) is an integer \(>1\); \(a(0)=0\), \(b(0)\ne 0\), and \(a(x)\), \(b(x)\), \(\beta_i(x)\) are holomorphic functions in a neighborhood of the point \(x=0\).

Without restricting generality, we shall assume1

\[ b(x)=\sum_{i=0}^{n-1} b_i x^i \qquad (b_i=\mathrm{const}). \tag{1.5} \]

Let \(a(x)=0\). Malmquist (1920) and Hukuhara (1949) showed that in this case, near the point \((0,0)\), in a certain sector about \(x=0\), the general solution of this equation can be represented in the form of the series

\[ y=\sum_{i=1}^{\infty}\varphi_i(x)t^i(x,c),\qquad \varphi_1(x)=1, \tag{1.6} \]

\[ t(x,c)=c\exp\left(\int_{\infty}^{x}\frac{b(x)-b_{n-1}x^{\,n-1}}{x^n}\,dx\right)\cdot x^{b_{n-1}}, \]

which converges absolutely and uniformly for sufficiently small \(|x|\) and \(|t(x,c)|\), where the functions \(\varphi_i(x)\) are asymptotically expandable in power series in \(x\) in the same sector. Thus, here the solution is represented in the form \(y=\Phi(x,t)\), where \(t\) is a solution of the reduced equation (1.4):

\[ \frac{dt}{dx}=\frac{b(x)\cdot t}{x^n}. \]

E. Trudo [51] indicated cases in which the solution of equation (1.4) will be holomorphic with respect to \(x\) and \(t(x,c)\). In the case \(a(x)\not\equiv0\), Malmquist and Hukuhara also obtained a convergent series of the form (1.6). In this case E. Trudo finds a formal solution of equation (1.4) in the form

\[ y=\sum_{i=0}^{\infty}\varphi_i(\tau)x^i, \tag{1.7} \]

\[ x^n\frac{d\tau}{dx}=(\delta_0+\delta_1x+\cdots+\delta_{n-1}x^{n-1})\tau +A_1x+\cdots+A_{n-1}x^{n-1}. \tag{1.8} \]

Here \(\delta_0,\ldots,\delta_{n-1}\) are polynomials in the coefficients of equation (1.4), \(A_1,\ldots,A_{n-1}\) are arbitrary constants, and \(\varphi_i(\tau)\) are holomorphic functions in a neighborhood of \(\tau=0\). A solution of equation (1.4) tending2 to zero as \(x\to0\) in a certain sector can, in that sector, be represented in the form

\[ y=\sum_{i=0}^{s}\varphi_i(\tau(x))x^i+O(x^{s+1}), \tag{1.9} \]

where \(s\) is any positive integer, and \(\tau(x)\) is the unique solution of equation (1.8), for arbitrarily fixed \(A_1,\ldots,A_{n-1}\), tending to zero as \(x\to0\) in the same sector. If in (1.9) \(\tau(x)\) is the general solution of equation (1.8), then in the form (1.9) one can represent in a ce-

in which sectors the general solution of equation (1.4) is represented. The series (1.7) can converge only for one system of values \(A_1,\ldots,A_{n-1}\), but it is unknown whether it will always be found.

In a neighborhood of the point \(x=0\) different sectors arise, in which the solutions of equation (1.4) are representable in different ways and behave differently; therefore there appears the problem of analytic continuation of the solution from one sector into another, which is what characterizes the multivaluedness of solutions in a neighborhood of the point \(x=0\).

As we see, the representation of solutions through solutions of auxiliary truncated equations appeared in Poincaré’s work in the analytic theory of differential equations. This method of representing solutions was then greatly developed in works on the real theory in Lyapunov’s theory of stability \(^{1)}\). But both these famous authors thus obtained solutions in the form of convergent series \(^{2)}\). Later, as we shall see, this method of representing solutions through solutions of auxiliary equations began to be used again in the analytic theory by Malmquist, Trjitzinsky, and Hukuhara.

We shall show below that recently, beginning with Hukuhara’s work, this method of constructing solutions has again found application in the real theory, in the theory of stability. It is true that now asymptotic expansions of solutions appear in this way. But in one way or another this method makes it possible to find, in a certain sense, the coordinate system in which we can better see the behavior of the solutions of the differential equations under consideration. Sometimes these auxiliary functions \(\tau(x)\) are generated by our desire formally to satisfy the differential equations under consideration. And it is not clear to us whether we have chosen \(\tau(x)\) well in doing this, especially in those cases when we obtain an asymptotic expansion of solutions and cannot sufficiently well (and sometimes cannot at all) estimate the approximate value of the solution or see a qualitative picture of the behavior of the solutions. Perhaps the coordinate system \(x\) and \(\tau(x)\) that we often obtain is far from the best for considering the solution. It is known how, beginning with Poincaré, asymptotic expansions of solutions of linear equations were obtained in a neighborhood of an irregular singular point of the independent variable \(t=\infty\). These expansions were obtained in the form of formal solutions in the form of power series. Here it was not possible to obtain a good estimate of the approximate solutions on the infinite interval \(t\geq t_0\), if a segment of the series was taken as the approximate solution [55]. But later, on the basis of Lyapunov’s method, it proved possible to construct a method of successive approximations that made it possible to obtain the solution in the form of a series uniformly convergent on the interval \(t\geq t_0\). And here exact estimates of the modulus of the difference between the solution and the first \(n\) terms of the series on the interval \(t\geq t_0\) are easily obtained. Here, apparently, the best coordinate system for representing the solutions has been found. Sometimes this coordinate system is also generated in the theory of nonlinear equations by some method of successive approximations. For example, this is what we have in the works of A. Erugin in the study of systems of the form

\[ \frac{dx}{dt}=y+\varphi(x,y),\qquad \frac{dy}{dt}=\psi(x,y), \tag{1.10} \]

\[ \frac{dx}{dt}=x-\varphi(x,y),\qquad \frac{dy}{dt}=x+y+\psi(x,y), \tag{1.11} \]

\(^{1)}\) See, for example, Lyapunov’s work [53].

\(^{2)}\) I shall dwell below on those cases when Lyapunov obtained a formal solution in the form of a series, about the convergence of which he could say nothing.

\[ \varphi(x,y)=\sum_{n=2}^{\infty}\sum_{m=0}^{n} b_{n-m,m}x^{\,n-m}y^m, \]

\[ \psi(x,y)=\sum_{n=2}^{\infty}\sum_{m=0}^{n} c_{n-m,m}x^{\,n-m}y^m \]

or

\[ \frac{dx}{dt}=-y+\sum_{n=1}^{\infty}\varphi_n^{(1)}(x,y)=X(x,y), \]

\[ \frac{dy}{dt}=x+\sum_{n=1}^{\infty}\varphi_n^{(2)}(x,y)=Y(x,y), \]

where: 1) the series \(X\) and \(Y\) converge absolutely and uniformly for \(x^2+y^2<r_0^2\);

2) \(\varphi_n^{(l)}(r\cos\vartheta,r\sin\vartheta)=r^{\alpha_n^{(l)}}\varphi_n^{(l)}(\cos\vartheta,\sin\vartheta);\ \alpha_n^{(l)}=\mathrm{const};\ 1<\alpha_1^{(l)}<\alpha_2^{(l)}<\cdots\).

Here in the first and second cases one obtains\(^1\) asymptotic expansions that are entirely different and clear in their analytic structure.

In some cases the method of successive approximations, arising in functional analysis, can generate a convenient form for representing solutions. It is no accident that I have dwelt on these representations of solutions in the analytic theory of differential equations. This was not an end in itself. I shall return to this later and show what significance this has in the real theory of differential equations, in questions of qualitative theory and stability theory, in the study of the behavior of solutions in a neighborhood of the point \(t=\infty\). I shall show this by the example of the works of E. Grudo, who applies this method in the real theory, in the theory of integral and integro-differential equations, and also in the theory of equations with a delayed argument. I shall not touch upon the analytic theory of linear systems, which, as is known, has been developed to a high degree thanks to the method of Lappo-Danilevsky. Nor shall I touch upon the connection of this theory with the real theory of linear systems. This is a separate large topic. I shall consider only the analytic theory of nonlinear systems.

§ 2. Moving singular points

For the equation \(y'=y^2\), the solution

\[ y=\frac{y_0}{y_0x_0-xy_0+1} \]

with initial conditions \(y=y_0\) for \(x=x_0\) has a pole at the point

\[ x=\frac{x_0y_0+1}{y_0}. \]

The position of this singular point (and its character) is not visible from the differential equation itself. Such singular points are called moving singular points. True, we could at once choose initial values \(y\to\infty\) as \(x\to x_0\) (arbitrary), and then, if such a solution

\(^1\) A. Erugin, IFZh, vol. II, No. 2, 1959; Dokl. AN BSSR, vol. V, No. 5, 1961; Ukr. Mat. Zh., No. 2, 1961. The starting point in the development of this method is the method constructed in the analytic theory.

exists, then \(x_0\) will be a singular point. In other words, the division of singular points into movable and fixed ones is of a conditional character. The number of movable singular points may be finite or infinite, and their nature may be very complicated. All this is also the subject of study of the analytic theory of differential equations. It is easy to see the great importance of the theory of movable singular points. If, moreover, one is speaking about the analytic theory of differential equations, then among the movable singular points one should distinguish the following points:

1. Single-valued, when, upon going around this point \(z_0\), the solution \(y(z)\) does not change its value.

2. Multivalued.

3. Those for which, as \(z \to z_0\), the solution \(y(z)\) (vector or scalar) has a limit, finite or infinite.

4. Those for which, as \(z \to z_0\), the solution \(y=y(z)\) has no limit. In this case we shall call the singular point \(z_0\) a point of indeterminacy or of essentially singular type.

5. Isolated singular points and nonisolated ones.

It may, of course, happen that the point \(z_0\) will belong simultaneously to classes 1 and 4. Then it will be an essentially singular point. Or it belongs simultaneously to classes 1 and 3, with \(y(z)\to\infty\) as \(z\to z_0\). Then it will be a pole.

It may happen that the point \(z_0\) is singular in the sense that, as \(z\to z_0\), we have \(y\to y_0\) (possibly \(y_0=\infty\)), but after a change of variables the new initial values are such that, in a neighborhood of these new initial values, the right-hand sides of the equations are holomorphic and application of Cauchy’s theorem is possible. Then the nature of the singular point \(z_0\) can easily be clarified, if the change of variables was more or less simple. Let us give examples.

Consider the Riccati equation

\[ y' = a(x)y^2 + b(x)y + c(x), \tag{2.1} \]

where \(a(x)\), \(b(x)\), and \(c(x)\) are entire functions or simply polynomials. Let \(x_0\) be a singular point such that \(y\to\infty\) as \(x\to x_0\). To study the singular point \(x=x_0\), introduce a new unknown function \(y=\frac{1}{u}\), \(u\to0\) as \(x\to x_0\).

For \(u\) we obtain the equation

\[ u' = -a(x)-b(x)u-c(x)u^2. \tag{2.2} \]

The required solution, according to Cauchy’s theorem, has the form

\[ u=-a(x_0)(x-x_0)+\sum_{k=2}^{\infty}\alpha_k(x-x_0)^k,\qquad \alpha_k\text{ constants}. \]

Hence we see that, for the solutions \(y=y(x)\), the point \(x_0\) will be a pole.

Let us now consider the special case of the Briot and Bouquet equation:

\[ xy' = y + \sum_{k+l\ge 2}^{\infty} a_{kl}x^k y^l,\qquad a_{kl}\text{ constants}. \]

Introduce a new unknown \(y=ux\). For \(u\) we obtain the equation

\[ \frac{du}{dx}=\sum_{k+l\ge 2} a_{kl}x^{k+l-2}u^l. \]

Let this series converge in the domain

\[ |x| \leqslant r,\qquad |u| \leqslant r. \tag{r} \]

By Cauchy’s theorem we have a solution \(u(x)\to c\) as \(x\to 0\), holomorphic in a neighborhood of the point \(x=0\), where \(c\) is an arbitrary constant from the domain of convergence \((r)\),

\[ u=c+\sum_{k=1}^{\infty} a_k(c)x^k . \]

Hence we obtain

\[ y=cx+\sum_{k=1}^{\infty} a_k(c)x^{k+1}. \]

This solution will be general\(^1\) in the domain

\[ |x|\leqslant R,\qquad |y|\leqslant R \tag{R} \]

for sufficiently small \(R\). It follows that all solutions beginning in the domain \((R)\) will have the property \(y\to 0\) as \(x\to 0\) and will be holomorphic in a neighborhood of the point \(x=0\). It also follows that there is no such \(u(x)\) for which \(y=u(x)\cdot x\to 0\), while \(u(x)\) either tends to \(\infty\) or has no limit as \(x\to 0\).

Thus, a simple change of variables has made it possible here, on the basis of Cauchy’s theorem, to obtain all solutions possessing the property \(y\to 0\) as \(x\to 0\), and to clarify the analytic character of the point \(x=0\).

Consider one equation with one unknown:

\[ \frac{dy}{dx}=\frac{P(x,y)}{Q(x,y)}=f(x,y), \tag{2.3} \]

where \(P(x,y)\) and \(Q(x,y)\) are polynomials in \(y\) with analytic coefficients.

We note that among fixed singular points there may be points of the essentially singular type, i.e., such that \(y\) has no limit as \(x\to x_0\). For example, for the equation

\[ \frac{dy}{dx}=-\frac{y}{x^2} \]

the point \(x=0\) will be a fixed singular point (which we find, as indicated above, from the equations \(P(x,y)=0\), \(Q(x,y)=0\), i.e. here \(x=0\), \(y=0\)). The general solution of the equation under consideration will be

\[ y=ce^{1/x}, \]

whence we see that \(x=0\) will be a fixed essentially singular point.

Movable singular points \(x_0\) of equation (2.3) are obtained by assuming that there is a solution \(y=y(x)\) such that, as \(x\to x_0\), either \(y\to\infty\), or \(y\to y_0\) and \(f(x,y)\to\infty\), or \(y\) has no limit. We are interested in the last possibility. In this connection there is a remarkable theorem of Painlevé, which states: solutions of equation (2.3) do not have movable singular points of the essentially singular type.

This theorem considerably simplifies the investigation of the question of the character of movable singular points of equation (2.3). It has been shown that movable

\(^1\) That is, for any sufficiently small \(y^*\), \(x^*\), one can find \(c^*\) such that there will be a solution \(y=u(c^*,y^*,x^*,x)\to y^*\) as \(x\to x^*\). This is easy to prove.

the singular points \(x_0\) of equation (2.3) can only be algebraic or algebraic polar ones, i.e., such that in a neighborhood of them the solution is representable as a series in powers of \((x-x_0)^{1/n}\), \(n>1\) a positive integer, and the number of negative powers of this quantity can be only finite. True, the question of the number of movable singular points and of their location remains difficult here as well. But already for a system of two equations or for one equation of second order there is no theorem analogous to Painlevé’s theorem; therefore the question of movable singular points for such equations at once becomes much more complicated. Moreover, nonisolated movable singular points also appear here, and consequently the question of movable singular points becomes altogether entangled.

Let us note that if real solutions of the system

\[ \frac{dy}{dx}=f(x,y), \tag{2.4} \]

are considered, where \(f(x,y)\) is a continuous vector-function in a domain \(D(x,y)\), then one can speak only of the nearest singular point \(x\) to the right and to the left of the point \(x_0\), in whose neighborhood the solution \(y=y(x)\) has been constructed. Here \(x\) is singular only in the sense that beyond \(x\) the solution \(y=y(x)\) cannot be continued. This singular point \(x^*\) is always such that as \(x\to x^*\) the solution \(y(x)\) tends to the boundary of the domain \(D(x,y)\) of continuity of the function \(f(x,y)\). This is only a certain analogue of Painlevé’s theorem on the absence of movable singular points of indefiniteness. But here the singular point \(x^*\) may also be such that, as \(x\to x^*\), \(y(x)\) has no limit, but only tends to the boundary of the domain of continuity of \(f(x,y)\); for example, it tends to the sphere \(y_1^2+\cdots+y_n^2=b^2\), if this sphere is the boundary of the domain of continuity of \(f(x,y)\). But if \(y\) is not a vector but a scalar, then as \(x\to x^*\) (a singular point) one always has \(y\to \bar y\) (finite or not). Here this acquires the character of a qualitative geometric theory of the location of integral curves as a whole.

Suppose, for example, that in equation (2.4) \(y\) is a vector of second order, \(y=(y,z)\), and \(f(x,y,z)\) is a continuous vector-function in the whole infinite space. Assume now that \(x=x^*\) is a singular point of the solution \(y=y(x),\ z=z(x)\). Then as \(x\to x^*\) the point \((y,z)\) (the projection of the integral curve onto the plane \(x=x^*\)) recedes to infinity. We distinguish here the cases when 1) the point \((y,z)\) recedes to infinity along a straight line (asymptotically); 2) the point \((y,z)\) asymptotically approaches some curve \(\Phi(y,z)\) going off to infinity; 3) the point \((y,z)\) recedes to infinity along a spiral.

Sometimes it is precisely the methods of the analytic theory that make it possible to elucidate this qualitative real picture as a whole. I shall not dwell on this in the general case, although one could speak about it as well [37]. But I shall describe the qualitative real picture for some classical Painlevé equations, which has been discovered in the last 5–6 years. It should also be noted that, remaining within the position of only the real theory, we lose, for example, such results as Malmquist’s theorem:

Given equation (2.3), where \(P\) and \(Q\) are polynomials in \(y\) with analytic coefficients in \(x\).

If equation (2.3) is not a Riccati equation, then every one-valued integral of it is a rational function.

If we remain in the position of the real theory, then the solution

\[ y=\frac{x+2}{(x^2-1)(x+3)}, \]

constructed in a neighborhood of the point \(x=0\), is regarded as nonexistent for \(x>1\), since it cannot be continued beyond the point \(x=1\), for \(y\to\infty\) as \(x\to 1\). But if we wish to consider this solution also outside the interval \((-1,+1)\), then we must splice it together with the solution (the same function) defined outside the interval \((-1,+1)\). This is somewhat artificial.

In general, the question of classifying the entire set of solutions of a given system, or of a single equation, loses its completeness if the right-hand sides of the given system are analytic while we remain in the position of the real theory. But the question of classifying the entire set of solutions even for the simplest equations is far from complete and is very important for the theory of differential equations, for example for the classical Painlevé equations. For some of these equations, for example for the second, a complete solution has been obtained to the question of rational solutions\({}^{1)}\), which has already altered the established view that all solutions of Painlevé equations are new transcendental functions. In general, rational solutions, for example of the form

\[ y=\frac{x-1}{x^2+1}\quad \text{(imaginary poles),} \]

and such problems as the domain of convergence of series representing the desired solution escape us, since we do not see the nearest singular points if we remain at the point of view of the real theory.

§ 3. Painlevé Equations

Painlevé and Gambier carried out a profound investigation of movable singular points, considering equations of the form

\[ \frac{d^2w}{dz^2}=R(z,w,w'), \tag{3.1} \]

where \(R\) is a rational function of \(w\) and \(w'\) and analytic in \(z\). They found the necessary conditions that \(R\) must satisfy in order that there be no singular movable multivalued points. As Painlevé showed, for example, such equations are

\[ w''=6w^2+z, \qquad \mathrm{I} \tag{3.2} \]

\[ w''=2w^3+zw+a,\qquad a\text{ is a parameter}, \qquad \mathrm{II} \tag{3.3} \]

\[ w''=\frac{1}{w}\left(\frac{dw}{dz}\right)^2-\frac{1}{z}\frac{dw}{dz} +\frac{1}{z}(\alpha w^2+\beta)+\gamma w^3+\frac{\delta}{w}, \qquad \mathrm{III} \tag{3.4} \]

\[ w''=\frac{1}{2w}\left(\frac{dw}{dz}\right)^2+\frac{3}{2}w^3+4zw^2+2(z^2-\alpha)w+\frac{\beta}{w}, \qquad \mathrm{IV} \tag{3.5} \]

\[ w''=\left(\frac{1}{2w}+\frac{1}{w-1}\right)\left(\frac{dw}{dz}\right)^2-\frac{1}{z}\frac{dw}{dz}+ \]

\({}^{1)}\) It is hardly possible to obtain this result more easily, apart from serious results obtained in analytic theory by Wittich, Yablonskii, and Vorob'ev.

\[ +\frac{(w-1)^2}{z^2}\left(\alpha w+\frac{\beta}{w}\right)+\frac{\gamma w}{z}+\delta\frac{w(w+1)}{w-1}, \qquad \mathrm{V} \tag{3.6} \]

\[ w''=\frac{1}{2}\left(\frac{1}{w}+\frac{1}{w-1}+\frac{1}{w-z}\right)\left(\frac{dw}{dz}\right)^2 -\left(\frac{1}{z}+\frac{1}{z-1}+\frac{1}{w-z}\right)\frac{dw}{dz}+ \]

\[ +\frac{w(w-1)(w-z)}{z^2(z-1)^2} \left[ \alpha+\frac{\beta z}{w^2} +\frac{\gamma(z-1)}{(w-1)^2} +\frac{\delta z(z-1)}{(w-z)^2} \right] \qquad \mathrm{VI} \tag{3.7} \]

These are the so-called Painlevé equations I–VI.

After eliminating equations of the form (3.1) that obviously have movable multivalued singular points, 50 classes were found. But, as Painlevé and Gambier showed, among them only the Painlevé equations I, ..., IV are fundamental. The remaining 44 equations either, by simple transformations of the unknown functions and the independent variable, are reduced to the fundamental ones or to linear equations, or their solutions are expressed in terms of known functions, or in terms of solutions of a first-order equation \(w'=f(z,w)\). Further, Painlevé showed that the solutions of all six fundamental equations have only movable poles of the first or second order. Thus, for example, the first Painlevé equation has only movable poles with principal part \((z-z_0)^{-2}\), where \(z_0\) is the movable singular point determined by the initial values. We may choose \(z_0\) arbitrarily; then one more arbitrary parameter remains, \(a\)—the coefficient of \((z-z_0)^4\). Equation II has only movable poles of the first order with principal part \(\varepsilon (z-z_0)^{-1}\), \(\varepsilon^2=1\), and, for fixed \(z_0\), the coefficient of \((z-z_0)^3\) will be arbitrary. If, however, the solution is defined by the initial values \(w(z_1)=w_1,\ w'(z_1)=w'_1\), then \(z_0\) and the second parameter are determined. In a neighborhood of the point \(z_0\) the solution is easily constructed in the form of a series. We have analogous results for the remaining four Painlevé equations. These results were obtained at the beginning of the twentieth century, and the investigation of the solutions of the Painlevé equations was considered complete at that point. The opinion took shape that the solutions of the Painlevé equations are new transcendental functions (Painlevé transcendents), in the sense that they cannot be expressed in terms of solutions of first-order differential equations of the Fuchs class (equations of the form

\[ A_0(w,z)w'^n+A_1(w,z)w'^{\,n-1}+\ldots+A_n(w,z)=0, \tag{3.8} \]

having no movable multivalued singular points (Golubev, p. 78, 201)), linear equations of any order, and solutions of one another. But in 1952 [4, 37] the following questions were posed:

1. What can be said about those equations of the form (3.1) that are eliminated by Painlevé’s method as equations obviously having multivalued movable singular points? What is the nature of these singular points, and how can one construct a solution in a neighborhood of them?

2. The solutions of the Painlevé equations I, II, IV are meromorphic functions, i.e.

\[ w(z)=\frac{P(z)}{Q(z)}, \tag{3.9} \]

where \(P(z)\) and \(Q(z)\) are entire functions. But how are \(P(z)\) and \(Q(z)\) to be found? In other words, this concerns the representation of solutions \(w\) in the whole domain of existence.

1. If we have a movable pole at the point \(z_0\), will this solution have other poles as well, how many, and what is their arrangement in the plane of

$z$. What is the role of the second arbitrary parameter (the coefficient of $(z-z_0)^4$ or $(z-z_0)^3$)?

1. The transcendence of the solutions of Painlevé equations may also be called into question.

Painlevé’s method of isolating classes of equations without movable critical singular points was also applied to equations

\[ w^{(n)}=R\left(w^{(n-1)},\ldots,w',w,z\right),\quad n\geqslant 3, \]

where $R$ is a rational function of $w^{(n-1)},\ldots,w',w$, analytic in $z$. But here, even for $n=3$, only the first results have been obtained (Chazy, Garnier, Bureau). Let us also note that for $n=3$ there may be movable singular lines.

§ 4. Systems of equations¹

Since 1952, systems of the form

\[ \frac{dx}{dz}=P(x,y,z),\quad \frac{dy}{dz}=Q(x,y,z), \tag{4.1} \]

began to be considered by us, where $P$ and $Q$ are rational functions of $x$ and $y$ with integral² coefficients with respect to $z$. Such systems, even in the simplest cases, are much more general than equations (3.1), since equations (3.1) can be written in the form (4.1), whereas equations (4.1) generally cannot be reduced to a single equation of the form (3.1). For example, the system

\[ \frac{dx}{dz}=y^m,\quad \frac{dy}{dz}=P(x,z),\quad m>2, \tag{4.2} \]

where $P(x,z)$ is a polynomial in $x$ of degree $n>2$, cannot be written in the form (3.1) with $R$ rational in $w$ and $w'$.

Let, in the system (4.1), $P$ and $Q$ be polynomials in $x$ and $y$, whose coefficients are integral functions of $z$. Subject these polynomials to the conditions:

\[ P_1(x,\xi,z)=\xi^{m_1}P(x,\xi^{-1},z), \]

\[ P_1(x,0,z)=\mathrm{const}\ne 0, \]

\[ Q_1(x,\xi,z)=\xi^{n_1}Q(x,\xi^{-1},z), \]

\[ Q_2(\eta,y,z)=\eta^{n_2}Q(\eta^{-1},y,z), \tag{4.3} \]

\[ Q_2(0,y,z)=\mathrm{const}\ne 0, \]

\[ P_2(\eta,y,z)=\eta^{m_2}P(\eta^{-1},y,z). \]

Here $m$ and $n$ are integers. On the basis of conditions (4.3), we distinguish the cases:

I. $m_1-n_1+2=0,\ m_1\geqslant 0$ or $m_1=0,\ 2-n_1\geqslant 0$,

\[ P_1(x,0,z)=\mathrm{const}\ne 0; \tag{4.4} \]

II. $n_2>0,\ n_2-m_2+2=0$ or $n_2=0,\ 2-m_2\geqslant 0$,

\[ Q_2(0,y,z)=\mathrm{const}\ne 0; \tag{4.5} \]

III. $m_1>0,\quad m_1-n_1+2>0,$

\[ Q_1(x,0,z)\ \text{is bounded in every finite domain } A(x,z); \tag{4.6} \]

¹ Section 4 was prepared by A. Yablonskii.
² Or analytic.

IV. \(n_2>0,\quad n_2-m_2+2>0,\)
\(P_2(0,y,z)\) is bounded in every finite domain \(B(y,z)\).
\[ \tag{4.7} \]

For system (4.1), since 1952 the following problems have been considered [4, 37]:

1. To find conditions under which there are no solutions with a movable singular point \(z_0\) of essentially singular type, i.e., such that as \(z\to z_0\) either \(y(z)\) or \(x(z)\) has no limit (finite or infinite).

2. To give a method for constructing solutions in a neighborhood of these singular points \(z_0\). It turned out that the method for solving these problems must be different in cases I, II, III, IV.

It has been proved that when conditions I and II are fulfilled, system (4.1) has no movable singular points of essentially singular type. This system has such movable singular points \(z_0\) that

\[ x(z)\to x_0\ \text{(finite)},\quad y(z)\to\infty \quad \text{as } z\to z_0^{1)}, \tag{4.8} \]

and also such that

\[ x(z)\to\infty,\quad y(z)\to y_0\ \text{(finite)} \quad \text{as } z\to z_0^{2)}. \tag{4.9} \]

This system may also have still other singular points such that

\[ x(z)\to\infty,\quad y(z)\to\infty \quad \text{as } z\to z_0. \tag{4.10} \]

Solutions (4.8) and (4.9) are easily constructed. A method has also been created for constructing solutions (4.10).

The existence of solutions (4.10) must be established by means of an additional investigation. Let us now consider the special case of system (4.1)

\[ \frac{dx}{dz}=P(y,z),\qquad \frac{dy}{dz}=Q(x,z), \tag{4.11} \]

where \(P(y,z)\) and \(Q(x,z)\) are polynomials, respectively of degrees \(n\) and \(m\), in \(y\) and \(x\), with constant coefficients at the highest powers. The remaining coefficients are entire functions of \(z\). For this system conditions III and IV are fulfilled. It has been proved that this system has only movable singular points \(z_0\) of the type

\[ x(z)\to\infty,\quad y(z)\to\infty \quad \text{as } z\to z_0. \tag{4.12} \]

For these solutions a special construction method\(^{3)}\) has been created, making it possible to obtain all such solutions without additional assumptions concerning their properties (single-valuedness or multivaluedness of one type or another).

This method, in particular, makes it possible to obtain for the Painlevé I and II equations the same result as that obtained by Painlevé, i.e., it makes it possible to prove that the Painlevé equations have only movable singular points that are poles of the first and second orders.

Let us also consider the system

\[ \frac{dx}{dz}=P(x,y),\qquad \frac{dy}{dz}=Q(x,y), \tag{4.13} \]

where \(P\) and \(Q\) are polynomials with real constant coefficients, satisfying conditions III and IV.

\(^{1)}\) These solutions are generated only by conditions I.
\(^{2)}\) These solutions are generated only by conditions II.
\(^{3)}\) But up to now there has not existed a method for constructing a solution in a neighborhood of the singular point \(z=z_0\), if it is of essentially singular type.

The theorem has been proved. For any real singular point \(z_0\) of a real solution of equations (4.13) we have

\[ x(z)\to \infty,\qquad y(z)\to \infty \quad \text{as } z\to z_0, \tag{4.14} \]

i.e. \(z_0\) cannot be a singular point of the essentially singular type.

In other words, the projection of this integral curve onto the \(x\)- and \(y\)-plane cannot be a spiral. These solutions can be constructed by the method indicated. If the polynomials \(P(x,y)\) and \(Q(x,y)\) satisfy conditions (4.4) and (4.5), then we have the previous result, i.e. we have the solutions (4.8) and (4.9) and, perhaps, (4.12), which is established by an additional investigation. But there can be no movable singular point of the essentially singular type.

One may ask the question: can solutions of the system (4.1) fail to have movable singular points of the essentially singular type when conditions I, II, III, IV are violated? It turns out that they can. For example, this is the case for the system\(^1\)

\[ \frac{dx}{dz}=x^2+1,\qquad \frac{dy}{dz}=y^3,\qquad y=(z-c_1)^{-\frac12}(-2)^{-\frac12};\qquad x=\tg(z-c_2). \]

Here we have

\[ n_2=0,\qquad n_2-m_2+2=0, \]

\[ n_1=3,\qquad m_1=0,\qquad m_1-n_1+2=-1<0, \]

i.e. the condition \(m_1-n_1+2>0\) is violated. But there also exist systems (4.1), where \(P\) and \(Q\) are polynomials, for which conditions I, II, III, IV are violated and whose solutions have movable singular points of the essentially singular type. It has been established that this can also occur in those cases where \(P\) and \(Q\) do not contain \(z\).

We have just considered the system (4.11) from the point of view of the presence or absence of movable singular points of the essentially singular type.

Wittich [31] found equations of the form

\[ w'=\sum_{k=0}^{n} a_k(z)w^{\,n-k}, \]

where the \(a_k(z)\) are polynomials having at least one single-valued nonrational solution and, for every finite \(z=z_0\), having solutions with a pole at this point. It has been established that equations satisfying these properties are Painlevé equations (the first and second) or equations whose solutions are expressible in terms of elliptic and elementary functions.

Goursat considered the problem of isolating classes of systems of equations without movable multivalued singularities (in both unknowns). He indicated one such system

\[ \frac{dy}{dz}+ay+bx+c-y(a_2y+b_2x+c_2)=0, \]

\[ \frac{dx}{dz}+a_1y+b_1x+c_1-x(a_2y+b_2x+c_2)=0. \]

\[ \text{} \]

\(^1\) True, this system has separated variables: \(\dot{x}=P(x)\), \(\dot{y}=Q(y)\), and such equations in general, according to Painlevé’s theorem, do not have movable singular points of the essentially singular type.

R. Garnier [34] considered from this point of view the system1

\[ \frac{dx}{dz}=\frac{P_M(x,y)}{T_N(x,y)}=P(x,y),\qquad \frac{dy}{dz}=\frac{Q_M(x,y)}{T_N(x,y)}=Q(x,y), \]

where \(P\) and \(Q\) are homogeneous rational functions of degree \(m=M-N\). Such systems are obtained from the more general ones

\[ \frac{dx}{dz}=R_1(x,y,z),\qquad \frac{dy}{dz}=R_2(x,y,z), \]

where \(R_1, R_2\) are rational functions of \(x,y\) with identical denominators and with coefficients analytic in \(z\), as simplified by Painlevé’s method.

A. Yablonskii [35], for the system

\[ \frac{dx}{dz}=a_1(z)y^2+b_1(z)y+c_1(z);\qquad \frac{dy}{dz}=a_2(z)x^2+b_2(z)x+c_2(z), \]

where \(a(z)\ne0\), \(b(z)\) and \(c(z)\) are analytic functions, found necessary and sufficient conditions under which the solutions of this system have only movable simple poles, i.e., in a neighborhood of a singular point \(z_0\) we have

\[ x=\frac{a_{-1}}{z-z_0}+\sum_{k=0}^{\infty} a_k (z-z_0)^k,\qquad y=\frac{b_{-1}}{z-z_0}+\sum_{k=0}^{\infty} b_k (z-z_0)^k . \]

To obtain these conditions, A. Yablonskii introduces new variables \(x_1, y_1\) and \(z_1\)

\[ x=x_1+\frac{b_2}{2a_2},\qquad y=y_1+\frac{b_1}{2a_1},\qquad z=\int_{z_0}^{z_1} a_1(t)\,dt, \]

after which he obtains the equations

\[ \frac{dx}{dz}=y^2+\varphi(z),\qquad \frac{dy}{dz}=f(z)x^2+\psi(z),\qquad f(z)\ne0. \]

Then the indicated necessary and sufficient conditions are obtained in the form

\[ 2f'''(z)f^2(z)-10f''(z)f'(z)f(z)+9f'^3(z)=0, \]

\[ \varphi(z)=b\,\sqrt[3]{f(z)},\qquad \psi(z)=a\,\sqrt[3]{f^2(z)} \tag{4.15} \]

with arbitrary constants \(a\) and \(b\).

Let us note that A. Yablonskii’s system differs from the equations considered by Painlevé and Gambier [2, 3], as well as by Garnier [34] (1960) and Bureau [47] (1963). A. Yablonskii also considered [36] a system of the form

\[ \frac{dx}{dz}=\frac{P_1(y,z)}{Q_1(y,z)},\qquad \frac{dy}{dz}=\frac{P_2(x,z)}{Q_2(x,z)}, \tag{4.16} \]

where \(P(u,z)\) and \(Q(u,z)\) are polynomials in \(u\) with coefficients analytic in \(z\). He found necessary and sufficient conditions under which this system has no movable multivalued singular points. And it is interesting that, in general, when these conditions are fulfilled, the solutions turn out—

are expressed either in terms of elementary, elliptic functions, or in terms of Painlevé transcendents—solutions of the first and second equations.

Let us dwell on this in more detail. A. Yablonskii showed that if system (4.16) has no movable many-valued singular points, then by a linear transformation of the unknown functions and a change of the independent variable it is reduced to one of the following systems:

\[ \text{1)}\qquad \frac{dx}{dz}=\frac{1}{y}+\frac{1}{y-\alpha},\qquad \frac{dy}{dz}=\frac{1}{x}+\frac{1}{x-\beta}, \tag{4.17} \]

where \(\alpha\) and \(\beta\) are constants;

\[ \text{2)}\qquad \frac{dx}{dz}=p_2(z)y^2+p_1(z)y,\qquad \frac{dy}{dz}=x, \tag{4.18} \]

where either \(p_1, p_2\) are constants, or \(p_1(z), p_2(z)\) are related by the relation

\[ -6v(z)p_1(z)+cv''(z)-6p_2(z)v^2(z)+ \]

\[ +\left(\frac{a}{\sqrt{b}}\int_{z_0}^{z} p_2^{2/5}(z)\,dz+b\right)p_2^{4/5}(z)=0, \]

\[ v(z)=\frac{1}{50p_2^3(z)} \left[6p_2'^2(z)-5p_2''(z)p_2(z)-25p_1(z)p_2^2(z)\right] \]

with constants \(a\) and \(b\);

\[ \text{3)}\qquad \frac{dx}{dz}=p_3(z)y^3+p_2(z)y^2+p_1(z)y,\qquad \frac{dy}{dz}=x, \tag{4.19} \]

where either \(p_1, p_2, p_3\) are constants, or they satisfy two relations

\[ 3p_1(z)p_3^2(z)-12p_2^2(z)p_3(z)-7p_3'^2(z)+6p_3''(z)p_3(z)- \]

\[ -3bp_3^{8/3}(z)\left[ac\int_{z_0}^{z}p_3^{1/3}(z)\,dz+bc^2\right]=0, \]

\[ 9p_1(z)p_2(z)p_3^2(z)+2p_2^3(z)p_3(z)-9p_2''(z)p_3^2(z)+18p_2'(z)p_3'(z)p_3(z)- \]

\[ -9p_2(z)p_3''(z)p_3^2(z)+p_2(z)p_3'^2(z)-27ac^3p_3^{13/6}(z)=0. \]

with constants \(a, b, c\), and \(\alpha\);

\[ \text{4)}\qquad \frac{dx}{dz}=0,\qquad \frac{dy}{dz}=\frac{P(x,z)}{Q(x,z)}, \tag{4.20} \]

where \(P\) and \(Q\) are polynomials, and \(Q(x,z)\) either does not depend on \(x\), or the simple roots of \(Q(x,z)=0\) with respect to \(x\) are constant;

\[ \text{5)}\qquad \frac{dx}{dz}=\frac{1}{y-\alpha},\qquad \frac{dy}{dz}=\sum_{k=1}^{m}\frac{1-n_k}{n_k}\,\frac{1}{y-\beta_k}, \tag{4.21} \]

where \(m<4\), \(\alpha\) and \(\beta_k\) are constants, and (4.21) by means of the first integral

\[ y-\alpha=c_1\prod_{k=1}^{m}(x-\beta_k)^{-\left(1-\frac{1}{n_k}\right)} \]

reduces to the Briot and Bouquet equation

\[ \left(\frac{dx}{dz}\right)^N=p(x); \]

b)

\[ \frac{dx}{dz}=y^2+\varphi(z),\qquad \frac{dy}{dz}=f(z)x^2+\psi(z), \tag{4.22} \]

where \(f(z)\not\equiv 0\), \(\varphi(z)\), \(\psi(z)\) are a solution of the system

\[ 2f'''f^2-10f''f'f+9f'^2f=0, \tag{4.22_1} \]

\[ 3\varphi'f-f'\varphi=0,\qquad 3\psi'f-2f'\psi=0. \tag{4.22_2} \]

Equation \((4.22_1)\) has the first integral

\[ 2f''f^{-2}-3f'^2f^{-3}=K. \tag{4.22_3} \]

This makes it possible to obtain some particular solutions of equations \((4.22_1)\), \((4.22_2)\).

In 1953 R. A. Smith [39] considered the Liénard equation

\[ \frac{d^2w}{dz^2}+f(w)\frac{dw}{dz}+q(w)=p(z), \tag{4.23} \]

where \(f\) and \(q\) are polynomials of degrees \(n\) and \(m\), respectively. He showed that, for \(n>m\), for every point \(z=z_0\) distinct from a singularity of \(p(z)\), one can specify a one-parameter family of solutions representable in the form

\[ w(z)=\sum_{k=-1}^{\infty} a_k(z-z_0)^{k/n},\qquad a_{-1}\ne 0. \tag{4.24} \]

If a movable singular point of a solution of equation \((4.23)\) is attainable along a path of finite length, then it is of type \((4.24)\). It is proved that there can also be movable singularities which are limiting for points of type \((4.24)\). The existence of non-isolated singularities is proved by an example. Singularities of type \((4.24)\) and their points of accumulation exhaust the movable singularities of equation \((4.23)\) for \(n>m\). Equation \((4.23)\) is considered also by Sugiyama Shohei [40], without assuming \(n>m\). Here, for the case \(n>m-1\) (i.e. \(m>n\)), the existence of a singular point \(z=z_0\) of the form \((4.24)\) is proved under the assumption that, as \(z\to z_0\) along a certain path, \(w\to\infty\). For \(n<m-1\), and under the assumption \(w\to\infty\) as \(z\to z_0\), the solution obtained is

\[ w=\sum_{k=-1}^{\infty} a_k(z-z_0)^{\frac{k}{m-n-1}}. \tag{4.25} \]

It is shown that if one of the following conditions is satisfied: 1) \(n\geq 2,\ m-n-1=0\); 2) \(n=1,\ m=0,1,2,3,4\); 3) \(n=0,\ m=2,3\), then \(z_0\) will be a pole. Kimura Toshihusa [41] considered the equation

\[ \frac{d^2y}{dz^2}=\frac{P(z,y,y')}{Q(z,y,y')}, \tag{4.26} \]

where \(P\) and \(Q\) are polynomials in \(y,y'\) with coefficients analytic in \(z\):

\[ P=\Pi_p(z,y)y'^p+\cdots+\Pi_0(z,y),\qquad Q=K_q(z,y)y'^q+\cdots+K_0(z,y). \tag{4.27} \]

Putting in (4.26) \(y=\dfrac{1}{u}\), we obtain the equation

\[ \frac{d^{2}u}{dz^{2}}=\frac{M(z,u,u')}{N(z,u,u')}. \tag{4.28} \]

Introduce the function \(y=G(z)\), which is a root of \(Q_1(z,y)\), if
\(Q(z,y,y')=Q_1(z,y)Q_2(z,y,y')\), where \(Q_1\) and \(Q_2\) are polynomials in \(y\) and \(y'\), and the roots of \(Q_2\) with respect to \(y\) depend on \(y'\). Let also \(y=g(z)\), \(y'=h(z)\), where \(P(z,g(z),h(z))=0\), \(Q(z,g(z),h(z))=0\).

Suppose that \(z=z_0\) is not a fixed singularity. Then, if \(z=z_0\) is a singularity of the solution \(y=y(z)\), \(y(z_0)\ne G(z_0)\), \((y(z_0), y'(z_0))\ne (g(z_0),h(z_0))\), and for \(p>q+2\) \(y_0\) is finite, while for \(p\le q+2\) \(y_0\) and \(y'_0\) are finite, then the singular point \(z_0\) is algebraic. Concerning non-algebraic singular points it is established: when \(p>q+2\), \(y(z)\) cannot have a movable singularity of essential type, but \(y'(z)\) can have one, and as \(z\to z_0\), \(y(z)\to b\), where \(b=G(z_0)\), while \(y'(z)\) is not defined. When \(p\le q+2\), singularities of essential type may occur both for \(y(z)\) and for \(y'(z)\), and for a sequence \(\{z_n\}\to z_0\), if \(\{y(z_n)\}\to b\), where \(b\ne G(z_0)\), then \(\{y'(z_n)\}\to\infty\); if, however, \(\{y'(z_n)\}\to c\ne\infty\), then \(\{y(z_n)\}\to b=G(z_0)\). In a neighborhood of movable essentially singular points \(z=z_0\), \(y=y(z)\) assumes all values with the exception of a finite number. The same is true for the case when \(z=z_0\) is an essentially singular point for \(y'(z)\). These results partly overlap with the results of Painlevé [42].

One says that the analytic function \(y=y(z)\) possesses Iversen’s property in a domain \(D\) if it has the following property.

Let \(y=y(z)\) be holomorphic at the point \(z_1\in D\). Choose an arbitrary path \(L\) from the point \(z_1\) to \(z_0\), lying in \(D\). Then \(y=y(z)\) can be analytically continued along some path \(L_1\), connecting \(z_1\) and \(z_0\), lying in \(D\) and arbitrarily close to \(L\).

Kimura Toshihusa [43] showed that the solutions of equation (4.26) possess Iversen’s property if \(p>q+2\); moreover, in this case the function inverse to the solution also possesses this property. If the solutions possess Iversen’s property in a neighborhood of \(z=z_0\), then no movable singular line (consisting of movable singular points) passes through the point \(z=z_0\).

Kimura Toshihusa [44] also considered the equation

\[ P_m(z,y,y')y''^{\,m}+P_{m-1}(z,y,y')y''^{\,m-1}+\cdots+P_0(z,y,y')=0, \]

where \(P_k(z,y,y')\) \((k=1,2,\ldots,m)\) are polynomials in \(y\) and \(y'\), of degree \(p_k\) with respect to \(y'\). The solutions of this equation possess Iversen’s property if

\[ p_j>p_m-2(m-j). \]

B. P. Bogoslovskii in 1966 [38] considered the equation

\[ \frac{d^2x}{dz^2}=a_0x^k+f(x,z), \]

where \(a_0\) is a constant; \(k\) is a positive integer, and \(f(x,z)\) is a polynomial in \(x,z\) of degree \(k_1<k\) with respect to \(x\). According to the properties of system (4.11) (conditions (4.6), (4.7)), here there exists a path \(z\to z_0\) along which \(|x|\to\infty\), \(|x'|\to\infty\), if \(z_0\) is a movable singular point of the solution.

B. P. Bogoslovskii constructs asymptotic expansions of solutions in a neighborhood of singular points \(z_0\), finds conditions under which they will also be convergent, and distinguishes classes of these equations whose movable singular points co-

for which there will be only algebraic ones, i.e., in a neighborhood of these points \(z_0\) the solutions expand in series in powers of \((z-z_0)^{1/n}\).

A. Yablonskii [54] in 1967 considered the system

\[ \frac{dx}{dz}=\sum_{k=q}^{\tilde q} a_k(z)x^k y^{n-k},\qquad \frac{dy}{dz}=\sum_{k=t}^{\tilde t} b_k(z)x^k y^{m-k}, \]

for which the conditions of the works [4, 37] are not fulfilled. He found conditions which \(n, m, q, \tilde q, t, \tilde t\) must satisfy in order that there exist a family of solutions with an algebraic movable singularity \(z=z_0\), independently of the values of the functions \(a_k(z)\) and \(b_k(z)\), and also under certain relations among them; moreover, the presence of such singularities does not exclude other movable singularities. He indicated conditions under which there is a movable singular point of logarithmic type, and gave examples of movable singular points of the essential type (isolated and non-isolated). However, the theory of movable singular points of this system is far from complete.

I have already pointed out that, for constructing\(^1\) solutions of system (4.1) possessing property (4.12), a special method has been created. This method makes it possible to determine the nature of the movable singular points \(z_0\). With the help of this method it was possible to give a way of constructing integral curves for \(t>0\) in a neighborhood of an equilibrium point of focus type for the system [52]

\[ \dot{x}=P(x,y),\qquad \dot{y}=Q(x,y), \]

and sometimes also to reveal the qualitative picture of a focus type in various cases of complicated specification of the functions \(P(x,y)\) and \(Q(x,y)\). We shall not dwell on other applications of the methods of the analytic theory to problems of the qualitative theory of differential equations as a whole or locally, when we see no other ways of solving these problems. However, as I have already noted above, we shall dwell on those problems, still very difficult today, of qualitative theory and stability theory which, as has turned out recently, can be solved with the aid of certain methods developed in the analytic theory of differential equations. But this is another program, which will open after a survey of the so-called first method in qualitative theory and stability theory.

§ 5. Development of the Theory of Painlevé Equations

We have just considered the question of extending the class of equations that was studied by Painlevé, Gambier, and Garnier, and of new methods for studying these differential equations, allowing one to obtain all the old results as well as many new ones. And now I shall pass to the theory of the principal Painlevé equations, which has been developed over the last 10 years.

First let us consider the Painlevé I and II equations, i.e., equations (3.2) and (3.3). As I have already said, Painlevé proved that equation (3.2) has only movable poles \(z_0\) with principal part \((z-z_0)^{-2}\), and consequently the general solution of equation (3.2) is a meromorphic function

\[ w=\frac{P(z)}{Q(z)}. \tag{5.1} \]

\(^1\) This phrase and what follows are N. Erugin’s text.

Equation (3.3) also has only movable poles of the first order, with principal part

\[ \varepsilon (z-z_0)^{-1},\qquad \varepsilon^2=1. \tag{5.2} \]

It was also believed that all solutions of these equations are certain new transcendental functions (Painlevé transcendents), in the sense that they are not rational functions, solutions of linear equations with analytic coefficients, or equations of the first order

\[ P(w',w,z)=0, \tag{5.3} \]

where \(P\) is a polynomial1 in \(w'\), \(w\). True, there were no rigorous arguments here; these were hypotheses. The theory of Painlevé equations was considered complete. But, as I have already noted above, in 1952 a number of new questions were posed concerning the Painlevé equations. These questions set the task of constructing a classification of the entire set of solutions of each of the Painlevé equations. What, then, was done in this direction after 1952?

In 1956 H. Davis [12] considered the question of the pole nearest to the point \(z_0\) (not a singular point) by means of re-expansions of the solution into a Taylor series and with the use of computing machines. He also considered the question of the asymptotics of the distribution of poles. On the first of these questions he did not obtain clear conclusions; some of the assertions on the second question turned out to be true only under additional assumptions (which are realizable), as was pointed out by A. Yablonskii [8, 13, 14]. It was H. Davis who stated the assertion that the real poles of solutions of the second Painlevé equation come closer together inversely proportionally to the square root of their distance from the origin, independently of the value of the parameter \(\alpha\). However, this assertion does not take into account the fact, still unknown at that time, that there are possible solutions having no poles on the real half-axis, or on the negative half-axis, or no real poles at all.

The complete solution of the problem of the location of the poles \(z_k\) in the plane \(z\) consists in finding \(z_k\) as functions of the initial values

\[ w(z_0)=w_0 \quad \text{and} \quad w'(z_0)=w'_0 \tag{5.4} \]

or as functions of one pole \(z_0\) and the value of the second free parameter—the coefficient of \((z-z_0)^4\) or \((z-z_0)^3\). A solution of such a problem was first given for the Riccati equation [48]

\[ y'=a(z)y^2+b(z)y+c(z), \tag{5.5} \]

where \(a(z)\), \(b(z)\), and \(c(z)\) are entire functions. It is precisely here that the function

\[ z_0=\varphi(y_0), \tag{5.6} \]

was found and studied, where \(z_0\) is the movable pole of the solution of equation (5.5) with initial value \(y(0)=y_0\). In [49] it is proved that an implicit function \(\tau=\tau(y)\), defined by the equality \(F(\tau,y)=0\), where \(F(\tau,y)\) is an entire function, can have only algebraic singular points \(\overset{*}{y}\). This makes it possible to prove that the function (5.6) also can have only algebraic singular points \(\overset{*}{y}_0\), and sometimes the function (5.6) has no singular points at a finite distance at all. For example, this will be the case when \(a(z)\ne 0\). This makes it possible to draw certain conclusions concerning the solutions of the second equation

Painlevé, i.e., with respect to the solution of equation (3.3). As was indicated above, the solutions of this equation are meromorphic functions having only simple poles with residue \(\pm 1\).

In [15] the second equation is studied for \(a=-\dfrac{1}{2}\)

\[ \frac{d^2 w}{dz^2}=2w^3+zw-\frac{1}{2} \tag{5.7} \]

and for \(a=\dfrac{1}{2}\)

\[ \frac{d^2 w}{dz^2}=2w^3+zw+\frac{1}{2}. \tag{5.8} \]

It is precisely there that it is shown that all solutions of the equation

\[ \frac{dw}{dz}=-\frac{z}{2}-w^2 \tag{5.9} \]

are solutions of equation (5.7), and all solutions of the equation

\[ \frac{dw}{dz}=\frac{z}{2}+w^2 \tag{5.10} \]

are solutions of equation (5.8). Thus, for \(a=\dfrac{1}{2}\) and \(a=-\dfrac{1}{2}\) we obtain a class of solutions of equation (3.3) in the form (5.1), where the integral functions \(P(z)\) and \(Q(z)\) are easily found, since (5.7) and (5.8) are particular cases of equation (5.5). This class of solutions is determined by initial values \(z_0, w_0\) and \(w_0'\), satisfying respectively equations (5.9) and (5.10). And, in addition, from this we can obtain the following conclusions:

1. There exist solutions of equation (3.3) (at least for \(a=-\dfrac{1}{2}\) and \(a=\dfrac{1}{2}\)) having an infinite number of poles on the half-axis \(z>0\), all of whose residues are \(+1\) (in the case \(a=-\dfrac{1}{2}\)) or \(-1\) (in the case \(a=\dfrac{1}{2}\)).

2. The asymptotic distance between the poles \(z_n\) in these classes of solutions has the form\(^1\)

\[ \delta_n= \frac{\pi\sqrt{2}\left[1+nO\left(\frac{1}{n}\right)\right]}{\sqrt{z_n}}, \qquad \lim_{n\to\infty}\left|\,nO\left(\frac{1}{n}\right)\right|=A>0. \]

1. Not all solutions of equation (3.3) are new transcendental functions in the sense indicated above, since for \(a=-\dfrac{1}{2}\) and \(a=\dfrac{1}{2}\) there are solutions satisfying equations (5.9) and (5.10).

\(^1\) Thus H. Davis’s theorem ceases to be conditional.

Interesting investigations in this direction were carried out by A. I. Yablonskii and A. P. Vorob'ev. We shall now discuss them.

In 1950 N. Wittich [5] showed that equations

\[ F\left(w^{(n)},\, w^{(n-1)},\, \ldots,\, w',\, w,\, z\right)=0, \]

polynomial with respect to all their arguments and with one term of highest degree with respect to \(w, w', \ldots, w^{(n)}\), cannot have entire transcendental solutions. It is obvious that all the Painlevé equations, except the third for \(\gamma=\alpha=0\), the fifth for \(a=0\), and the sixth for \(a=0\), have this property.

It follows from this theorem that any solution of the first, second, and fourth Painlevé equations with a finite number of poles is a rational function (H. Wittich [6]). Thus, the question of the number of poles of the solutions of the indicated Painlevé equations is resolved by investigating the existence of rational solutions. The question of rational solutions was first considered by H. Schubart in 1956. He proved that the first equation has no rational solutions\(^1\). Concerning the second equation it is shown here that, apart from \(w=0\) for \(\alpha=0\) and \(w=\pm \dfrac{1}{z}\) for \(\alpha=\mp 1\), respectively, there are no other rational solutions for any \(\alpha\).

A. Yablonskii in 1959 showed that this assertion is false. Namely, he found rational solutions for \(\alpha=\pm2, \pm3, \pm4, \pm5\), and proved that, in general, rational solutions are possible only for integral values of \(\alpha\) and only one for each \(\alpha\). A method for constructing possible rational solutions was also indicated here. But there was no proof of the existence of rational solutions for all integral \(\alpha\). Let us formulate these assertions more precisely.

Theorem of A. Yablonskii. In order that equation (3.3) have a rational solution, it is necessary and sufficient that the system of differential equations

\[ P''Q-2P'Q'+PQ''=0, \]

\[ P'''Q-3P''Q'+3P'Q''-PQ'''-z(P'Q-PQ')-\alpha PQ=0 \tag{5.11} \]

have a polynomial solution \(P_\alpha, Q_\alpha\) \((P_\alpha Q_\alpha \ne 0)\), and if such a solution exists, then the rational solution of equation (3.3) has the form

\[ w_\alpha=\frac{P'_\alpha}{P_\alpha}-\frac{Q'_\alpha}{Q_\alpha}. \tag{5.12} \]

A. Vorob'ev in 1965 continued this investigation and obtained the following theorem:

If \(P_m(z)\) and \(Q_m(z)\) are a polynomial solution, for \(\alpha=m\), of the system (5.11), then \(P_{m+1}(z), Q_{m+1}(z)\), determined by the formulas

\[ P_{m+1}(z)=Q'_m, \qquad P_m Q_{m+1}=zQ_m^2+4Q_m'^2-4Q_m Q''_m, \]

are a polynomial solution of the system (5.11) for \(\alpha=m+1\).

This proved the existence of rational solutions of equation (3.3) for all integral \(\alpha\) and gave a method\(^2\) for constructing them. Thus

\(^1\) Thereby he proved that all solutions of the first Painlevé equation have an infinite number of movable poles.

\(^2\) If \(w(z)\) is a solution of equation (3.3), then \(-w(z)\) is a solution of the equation \(w''=2w^3+zw-\alpha\), and therefore we have solutions for all integral \(\alpha\).

It was shown that the second Painlevé equation has nontranscendental solutions which are rational functions or solutions of first-order equations (5.7), (5.8). Thereby it was also shown that some solutions of this equation have a finite number of poles, while others have an infinite number of poles and with only one residue, \(+1\) or \(-1\). A. Yablonskii [8], for each integer \(\alpha\), found the number of positive and negative residues. Namely, he showed that for every \(\alpha\) any solution distinct from a rational one has an infinite number of poles with residue both \(+1\) and \(-1\). The number of positive residues of the poles of rational solutions is determined by the formula \(l=\alpha(\alpha-1)/2\), and of negative ones by \(r=\alpha(\alpha+1)/2\).

N. Lukashevich showed that every solution of the fourth Painlevé equation [17] has an infinite number of poles with residue \(+1\) and \(-1\), if it is not a solution of some Riccati equation or rational. This exactly coincides with property II of the Painlevé equation. The first Painlevé equation, however, indeed has only transcendental solutions in the sense that they are not solutions of linear equations (of any order with integral coefficients), and \(w\) is not a solution of an equation \(w'=\psi(w,z)\), where \(\psi(w,z)\) is a rational function of \(w\) with integral coefficients in \(z\). Let us prove this last assertion. Write the first Painlevé equation as \(w''=6w^2+z\). We know that \(w\) has the form

\[ w=\frac{1}{(z-z_0)^2}+\varphi(z), \]

where \(\varphi(z)\) is regular in a neighborhood of the point \(z=z_0\). Substitute this \(w\) into the equation

\[ w'=\psi(w,z)= \frac{\displaystyle\sum_{k=0}^{n} a_k(z)w^k} {\displaystyle\sum_{k=0}^{m} b_k(z)w^k}, \tag{5.13} \]

where \(a_k(z)\) and \(b_k(z)\) are integral functions of \(z\). Here on the left \(z=z_0\) is a pole of third order, and on the right of order \(2n-2m\); but for integer \(n\) and \(m\) it is impossible that \(2n-2m=3\), which proves the assertion.\(^1\) Thus, the investigations noted here destroyed the opinion that all solutions of the Painlevé equations are new transcendental functions. But the classification of solutions of the Painlevé equations has, of course, only begun.

§ 6. Representation of solutions of Painlevé equations in the whole domain of existence, and entire solutions of nonlinear differential equations

It is known that solutions of linear equations are representable in the whole domain of existence. For nonlinear equations we do not have this. But, as we see, the general form of the solutions of the first and second Painlevé equations is quite simple—they are meromorphic functions of the form (5.1). It is necessary, therefore, only to be able to construct the entire functions \(P(z)\) and \(Q(z)\). In 1958, in [28], a solution of the first Painlevé equation (3.2) was constructed

\(^1\) We shall also prove that the second Painlevé equation only for \(\alpha=\pm \dfrac{1}{2}\) has a solution coinciding with a solution of an equation of the form (5.13), which can only be (5.9) and (5.10).

throughout the whole domain of existence, i.e. entire functions \(P(z)\) and \(Q(z)\) were found in terms of the initial conditions

\[ w(z_0)=w_0,\qquad w'(z_0)=w'_0 \tag{6.1} \]

or in terms of a root of the equation

\[ Q(z_0)=0, \tag{6.1_1} \]

where

\[ w(z)=\frac{P(z)}{Q(z)}. \tag{6.2} \]

From this one can also obtain the pole \(z_0\) as a function of the initial values \(w_0\) and \(w'_0\). It is not difficult to indicate also the nearest pole \(z_0\) to an initial nonsingular point \(z_1\), if the solution is taken under the conditions

\[ w(z_1)=w_1,\qquad w'(z_1)=w'_1 . \tag{6.3} \]

All solutions of the second Painlevé equation (3.3) in the form (6.2) were constructed by A. Yablonskii in 1958 [29].

Let us show in more detail how the solution \(w\) in the form (6.2) was obtained for the first Painlevé equation. First it was shown that \(w\) can be obtained in the form

\[ w=\frac{s'^2-ss''}{s^2}, \]

where \(s\) is a solution of the equation

\[ ss^{\mathrm{IV}}-4s'''s' + 3s''^2 + zs^2=0. \tag{6.4} \]

Obviously, the solution \(s\) contains four arbitrary constants. In order to satisfy the conditions (6.1), one must take only certain solutions of equation (6.4)—not the general solution.

Proceeding from the general properties of solutions of the first Painlevé equation, the suspicion arose that the solutions \(s\) of equation (6.4) satisfying condition (6.1) would be entire. This was indeed proved.

A. Yablonskii solved in the same way the problem of representing the solution \(w\) of the second Painlevé equation in the form (6.2), but in his case, instead of equation (6.4), there appeared the equation

\[ 2(u'^2-uu'')\,u u^{\mathrm{IV}} + u^2u'''^2 + 2uu'u''u''' - \]

\[ {}-4u'^2u''' - 2uu'''^3 + 3u'^2u''^2 + 4z(u'^2-uu'')^2 + \tag{6.5} \]

\[ {}+4\alpha u(u'^2-uu'')\sqrt{u'^2-uu''}=0, \]

where

\[ w=\frac{\sqrt{u'^2-uu''}}{u}=\frac{P(z)}{Q(z)}. \]

But in 1964 A. Yablonskii [30] showed that all solutions of equations (6.4) and (6.5), as well as of system (5.11), are entire. In this work A. Yablonskii also indicated nonlinear differential equations of the second order all of whose solutions are entire functions, i.e. have no singular points at finite distance.

In 1965 [45] A. Yablonskii found necessary and sufficient conditions under which all solutions of the equation

\[ w''^{\,2}+B(z)w''w'+c(z)w''w+a(z)w'^{\,2}+ \tag{6.6} \]

\[ +b(z)w'w+M(z)w^2=0 \]

with meromorphic coefficients will be entire functions.

Let us also note that in 1957 A. A. Zinger and Yu. V. Linnik [46] proved that a positive definite solution of a certain nonlinear differential equation arising in mathematical statistics is entire. It is interesting that the proof of this assertion is based on probability theory. It would be of interest to construct a proof of this assertion by methods of the theory of differential equations. In doing so, other results will probably also be obtained along the way.

§ 7. Qualitative picture of the arrangement of integral curves of the first and second Painlevé equations in the large

An interesting classification of real solutions of the first and second Painlevé equations was carried out by A. Yablonskii. In [13, 14] A. Yablonskii considers the question of regular solutions of the equations

\[ y''=f(x,y). \]

A solution \(y=y(x)\) is called regular in the domain \(x_0\leq x<\infty\) if it exists in this domain and is twice continuously differentiable. From the general theorems obtained here by A. Yablonskii with respect to

\[ y''=6y^2-x, \tag{7.1} \]

the following follows.

Let \(D\) be the domain \(x>0,\ y>0\). Through every point \(M(x_0,y_0)\) of the domain \(D\) there passes one1 regular solution \(L\), asymptotically approaching the curve \(y=\sqrt{x/6}\) from below. All the other solutions passing through the point \(M\) with a smaller slope than \(L\) intersect the axis \(y=0\), while those with a larger one have a vertical asymptote.

This shows that through every point \(M\) of the domain \(D\) there passes one solution \(L\) having no movable poles in the domain \(x\geq x_0\). Solutions passing through the point with a slope greater than that of \(L\) apparently have a movable pole in the domain \(x\geq x_0\). With respect to the second Painlevé equation

\[ y''=2y^3+xy+a \tag{7.2} \]

A. Yablonskii obtained the following.

Through every point

\[ M(x_0,y_0)\in D \qquad (0<x<\infty,\quad -\infty<y<\infty) \]

there passes a unique regular solution \(y=y_M(x)\). For \(a>0\) this solution asymptotically approaches from below the branch \(y=y(x)\), lying in \(D\), of the odd function

\[ 2y^3+xy+a=0 \quad\text{and}\quad \lim_{x\to\infty} y_M'(x)=0. \]

For \(\alpha=0\), the regular solution lying in the upper half-plane approaches \(y=0\) from above, and the one lying in the lower half-plane—from below.

A. Yablonskii studied real solutions of the second Painlevé equation also in the domain \(x<0\). He constructed asymptotic expansions of regular solutions of the first and second Painlevé equations and investigated the qualitative picture of the behavior of real solutions of these Painlevé equations as a whole.

§ 8. The III, IV, V, VI Painlevé equations

N. A. Lukashevich studied the III, IV, V, and VI Painlevé equations. He showed that there exists a number of relations among the parameters entering into these equations for which the general solution, or some class of solutions, are elementary functions, or are expressed in the form of quadratures of elementary functions, or are simply expressed through solutions of the Riccati or Bessel equations, or through Weber–Hermite functions and Hermite polynomials. He also showed that the III equation may have many-valued fixed singular points, which is not consistent with the opinion expressed earlier in [50] that all solutions of this equation are single-valued functions. N. A. Lukashevich studied the III and IV Painlevé equations in greater detail.

Let us consider the III Painlevé equation, or, in our notation, equation (3.4). Here \(z=0\) will, generally speaking, be a fixed singular point. N. A. Lukashevich found necessary and sufficient conditions under which there exist solutions of equation III that are holomorphic at the point \(z=0\) or have a pole at this point. Such solutions may be unique, or may constitute a one-parameter family. He then also found necessary and sufficient conditions for the presence at the point \(z=0\) of a critical pole (i.e., the point \(z=0\) will be multivalued for the solution, and \(w\to\infty\) as \(z\to0\)). The solutions of equation III for which the point \(z=0\) is a pole or a holomorphic point are meromorphic functions, i.e.

\[ w=\frac{v(z)}{u(z)}, \]

where \(u(z)\) and \(v(z)\) are entire functions. N. A. Lukashevich showed that all single-valued solutions of equation III, with the exception of the elementary ones indicated earlier, will have an infinite number of simple poles, generally speaking with residues equal both to \(1/\sqrt{\gamma}\) and to \(-1/\sqrt{\gamma}\). But when certain other relations among the parameters are satisfied, the poles will have residues only \(1/\sqrt{\gamma}\), or only \(-1/\sqrt{\gamma}\), or, finally, under certain relations among the parameters, these solutions will have movable poles of second order with residues equal to zero. He also indicated conditions under which equation III may have rational solutions. But the question of the actual existence and the set of rational solutions for equation III remains open. The question of a general representation of the solutions and of a complete classification of the solutions of equation III also remains open. All solutions of the IV Painlevé equation are meromorphic functions having only simple poles with residue \(\pm1\). N. A. Lukashevich showed that every solution of equation IV has movable singular points—poles. If this solution is not rational and is different from the one-parameter family of solutions of a certain Riccati equation, then it has an infinite number of residues equal both to \(+1\) and to \(-1\). The solutions belonging to the indicated family of solutions of the Riccati equation have residues either only equal to \(+1\) or

—1. Despite the sharp difference between equation IV and equation II, we see that this latter, rather complicated phenomenon for them, coincides. This is curious.

II. ON SOME QUESTIONS OF THE FIRST METHOD

§ 1. Lyapunov’s first method and works adjoining it

Consider a system of \(n\) differential equations

\[ \dot{x}=xP(t)+f(t,x),\qquad x=(x_1,\ldots,x_n),\qquad f=(f_1,\ldots,f_n), \tag{1.1} \]

where \(P(t)\) is a matrix of order \(n\), and

\[ f_k=\sum P^{(k)}_{m_1\ldots m_n}(t)x_1^{m_1}\ldots x_n^{m_n},\qquad m_1+\cdots+m_n>1 \tag{1.2} \]

are convergent series in a neighborhood of \(x=0\). I shall not specify the other, known assumptions. Here \(x=0\) is a solution.

If \(t\) does not enter the right-hand sides of equations (1.1), i.e. \(P^{(k)}_{m_1\ldots m_n}\) are constants, then we write this in the form

\[ \dot{x}=xP+f(x),\qquad P\text{ is a real constant matrix.} \tag{1.3} \]

Let us recall how the question of stability of the zero solution \(x=0\) of system (1.3) is solved according to Lyapunov. If the real parts \(a_k\) of all characteristic numbers (c.n.) \(\lambda_k=a_k+b_k i\) of the matrix \(P\) are negative, then the zero solution \(x=0\) of equations (1.3) is asymptotically stable, and the general solution in a neighborhood of \(x=0\) and for \(t\ge t_0\) can be obtained in the form

\[ x_s= \sum_{m_1+\cdots+m_n\ge 1} L_s^{(m_1\ldots m_n)}(t)\alpha_1^{m_1}\ldots \alpha_n^{m_n} e^{\sum_{i=1}^{n}m_i a_i t}, \qquad (s=1,\ldots,n), \tag{1.4} \]

where \(L_s(t)\) are polynomials in \(t,\sin b_k t,\cos b_k t\) \((k=1,\ldots,n)\), and \(\alpha_1,\ldots,\alpha_n\) are arbitrary constants small in modulus. These series converge for \(t\ge t_0\). And instead of (1.4) one could write also as follows:

\[ x_s= \sum_{m_1+\cdots+m_n\ge 1} L_s^{(m_1\ldots m_n)}(t)\alpha_1^{m_1}\ldots \alpha_n^{m_n} e^{\sum_{k=1}^{n}\lambda_k m_k t}. \tag{1.5} \]

Here \(L_s^{(m_1\ldots m_n)}(t)\) are now either polynomials in \(t\) or constants. If all elementary divisors of the matrix \(P\) are simple and there are no relations

\[ m_1\lambda_1+\cdots+m_n\lambda_n=\lambda_j\qquad (j=1,\ldots,n), \tag{1.6} \]

where \(m_1,\ldots,m_n\ge 0\) are integers and \(m_1+\cdots+m_n\ge 2\), then \(L_s^{(m_1\ldots m_n)}(t)\) are constants.

Poincaré (1879) was the first to obtain such a representation of the general solution, and then Lyapunov (1888) did so in the case when all \(\lambda_1,\ldots,\lambda_n\) are distinct and their real parts are of one sign. But Lyapunov (1892) obtained the general solution in the form (1.4) for \(t\ge t_0\) also for system (1.1) in the case when the corresponding system of the first approximation

\[ \dot{x}=xP(t) \tag{1.7} \]

regular and when all the characteristic numbers \(a_k\) \((k=1,\ldots,n)\) of system (1.7) are negative in the sense of Lyapunov\(^1\). Thus we have this for system (1.3), when the real parts of all the characteristic numbers of the matrix \(P\) are negative.

Here Lyapunov no longer requires the assumption that all
\(a_k=R(\lambda_k)\) are distinct.

Let us note that from this the following assertion follows under the assumptions made concerning \(a_1,\ldots,a_n\).

For system (1.1) there exists a transformation

\[ x_k=\sum_{m_1+\cdots+m_n\geq 1} L_s^{(m_1\ldots m_n)}(t)y_1^{m_1}\cdots y_n^{m_n}, \tag{1.8} \]

where the series on the right converge for \(t\geq t_0\) and for \(|y_k|<a\) \((k=1,\ldots,n)\), which transforms system (1.1) into the system

\[ \dot y_k=a_k y_k \quad (k=1,\ldots,n). \tag{1.9} \]

The general solution of system (1.1) is also represented in the form of series in the general solutions of system (1.9). We noted above that for system (1.3), under the condition that all elementary divisors of the matrix \(P\) are simple and there are no relations (1.6), the coefficients of the transformation (1.8), \(L_s(t)\), will be constant.

Zigel considered the question of a transformation of this kind with constant \(L_s(t)\) for system (1.3) under the condition of simple elementary divisors of the matrix \(P\) and also in the case when the real parts \(a_k=R(\lambda_k)\) are not necessarily of one sign. He proved the existence of an analytic transformation (1.5) with constant \(L_s(t)\), transforming system (1.3) into system (1.9), only under the condition that there are no relations (1.6) and

\[ |m_1\lambda_1+\cdots+m_n\lambda_n-\lambda_j|\geq \frac{2n}{m^\nu}, \quad m=\lambda_1+\cdots+\lambda_n, \tag{1.10} \]

where \(\nu>0\) is some constant. Zigel showed that, except for a set of \(\lambda_1,\ldots,\lambda_n\) of measure zero, these conditions are always fulfilled. V. Pliss [4] showed that if, for all \(m_1,\ldots,m_n\) for which

\[ m_1\lambda_1+\cdots+m_n\lambda_n=\lambda_j, \tag{1.11} \]

some constants are equal to zero, while for the remaining ones we have

\[ \left|\sum_{i=1}^{n} m_i\lambda_i-\lambda_j\right|>\frac{1}{m^\nu}, \quad \nu\geq 2, \tag{1.12} \]

then there exists a convergent transformation

\[ x_i=y_i+\sum_{k=2}^{\infty}\psi_k^{(i)}(y_1,\ldots,y_n), \tag{1.13} \]

where \(\psi_k(y_1,\ldots,y_n)\) are homogeneous polynomials, transforming the system

\[ \dot x_k=\lambda_k x_k+f_k(x_1,\ldots,x_n), \tag{1.14} \]

\[ f_k=\sum_{k_1+\cdots+k_n\geq 2} A_{k_1\ldots k_n}x_1^{k_1}\cdots x_n^{k_n} \tag{1.15} \]

\(^1\) Here \((-a_k)\) are Lyapunov characteristic numbers, and \(L_s^{(m_1\ldots m_n)}(t)\) are continuous functions whose characteristic numbers are not greater than zero.

into the system

\[ \dot y_k=\lambda_k y_k\quad (k=1,\ldots,n). \]

Here, of course, there arises a problem of distinction similar to the problem of distinguishing a center from a focus: to find cases when the Pliss constants indicated here are equal to zero, or to find criteria for the case when the right-hand sides of equations (1.3) are polynomials and when the vanishing of a finite number of the Pliss constants indicated here entails the vanishing of all these quantities. But if the real parts \(\lambda_k\) are not all of one sign, then we do not have a general solution \(^{1)}\) of the original system in the form (1.5) for \(t\geq t_0\) or for \(t\leq t_0\). We shall obtain such a representation of the solution only on some interval \(|t-t_0|\leq T\), the larger the smaller the moduli \(|a_k|\). Hence it follows that, in this case, the form of the solution (1.5) with constant \(L_s(t)\) for system (1.3) is not convenient for investigating the behavior of solutions as \(t\to\infty\). We are not even certain of its existence in the case when the Siegel conditions (1.10) are not fulfilled. And such a representation certainly ceases to exist if the relations (1.6) hold and the Pliss constants are not equal to zero. These difficulties of estimating the convergence of the series (1.5) and of their existence disappear if we take the general case of real parts \(\lambda_k\) of one sign (and, possibly, nonsimple elementary divisors) and allow \(L_s(t)\) to be nonconstant. In this case we shall again have the transformed system (1.12).

Thus, the general solution of system (1.3) will be obtained in the form of series in the general solutions of system (1.12), but the coefficients of these series will be functions of \(t\).

We shall return later to those cases when the relations (1.6) hold, and shall indicate another approach to the construction of solutions in this case.

§ 2. Doubtful cases

Those cases when all the characteristic numbers of system (1.7) are negative and zero \(^{2)}\) were called doubtful by Lyapunov. In this case the character of the behavior of the general solution, or of the whole set of integral curves near the origin of coordinates, is determined by the coefficients of the nonlinear terms in the right-hand sides of equations (1.1). Lyapunov considered exhaustively that doubtful case of system (1.3) in which one characteristic number of the constant matrix \(P\) is zero (and all the others have negative real part). Such a system can be written (after a certain transformation which does not change the asymptotic character \(^{3)}\) of the behavior of the integral curves in a neighborhood of the origin) in the form

\[ \frac{dx}{dt}=X,\qquad \frac{dx_s}{dt}=p_{s1}x_1+\cdots+p_{sn}x_n+p_sx+X_s, \tag{2.1} \]

where \(X,X_1,\ldots,X_n\) are holomorphic functions of the variables \(x,x_1,\ldots,x_n\) in a neighborhood of the point \(x=x_1=\cdots=x_n=0\), whose expansions begin with terms of not lower than second order. Here the matrix \(P=\|p_{kl}\|\) has characteristic numbers only with negative real parts. Lyapunov resolves the question of the stability of the zero solution in this case as follows.

From the equations

\[ p_{s1}x_1+\cdots+p_{sn}x_n+p_sx+X_s=0\quad (s=1,\ldots,n) \tag{2.2} \]

\(^{1)}\) But this Siegel transformation will be significant in the investigation of the question of the topological equivalence of the qualitative picture of two different systems of equations.

\(^{2)}\) And there is at least one zero one.

\(^{3)}\) As \(t\to\infty\).

we find \(x_k=x_k(x)\) \((k=1,\ldots,n)\) as holomorphic functions, and in a unique way. Substituting these values \(x_k\) into \(X\), we obtain

\[ X(x_1,\ldots,x_n,x)=gx^m+g_1x^{m+1}+\cdots,\qquad g\ne0. \tag{2.3} \]

In this case Lyapunov constructs a function \(V(x,x_1,\ldots,x_n)\) that solves the stability problem by the second method, and arrives at the conclusion: if \(m\) is an even number, then the zero solution of system (2.1) will be unstable; if \(m\) is odd, then for \(g>0\) the zero solution is unstable, while for \(g<0\) it is asymptotically stable. But it may happen that \(X(x_1(x),\ldots,x_n(x),x)\equiv0\). Then Lyapunov finds an integral of system (2.1)

\[ x=c+f(x_1,\ldots,x_n,c), \tag{2.4} \]

where \(f\) is a holomorphic function of \(x_1,\ldots,x_n,c\) in a neighborhood of the point \(x_1=\cdots=x_n=c=0\), vanishing at this point, and \(c\) is an arbitrary constant (small). In this case, as Lyapunov shows, there exists a curve \(x_k=x_k(c)\), \(x=x(c)\), passing through the origin and consisting of rest points of system (2.1). On each of the surfaces (2.4) there is one of these rest points. Each solution of system (2.1) that begins near the origin remains on one of the integral surfaces (2.4) and asymptotically approaches the rest point lying on this surface¹). Thus, here Lyapunov did not find the general solution in a neighborhood of the origin or of the rest points of which the above-mentioned curve consists. But this can be done, i.e., on the basis of Lyapunov’s first method one can construct the general solution in a neighborhood of the indicated rest points [3], which will also give us the general solution in a neighborhood of the origin of equations (2.1). Here we have non-asymptotic stability of the origin. But up to now the general solution in a neighborhood of the origin has not been constructed in the case when we have (2.3). A method for constructing the general solution in a neighborhood of the origin in this case as well has now been outlined. A paper by E. Grudo will be published on this subject. Here the general solution is constructed by a method developed in the analytic theory; namely, the general solution is obtained through the solution of the auxiliary equation

\[ \frac{dy}{dt}=gy^m+hy^{2m-1},\qquad g,\ h\ \text{constants}, \]

which in form coincides with those we indicated earlier when mentioning the works of Malmquist, Trjitzinsky, and Hukuhara.

Hukuhara was the first to use such a construction of the general solution in the real domain. We shall speak about this in more detail later. Let us note the following here as well. In the case under consideration, in a neighborhood of the origin for \(t>t_0\), we could construct an \(n\)-parameter family of solutions according to Lyapunov in the form

\[ x_k= \sum_{m_1+\cdots+m_n=1}^{\infty} L_s^{(m_1\ldots m_n)}(t)a_1^{m_1}\cdots a_n^{m_n} e^{\sum_{i=1}^{n}m_i\lambda_i t} \qquad (k=1,\ldots,n), \]

\[ \tag{2.5} x= \sum_{m_1+\cdots+m_n=1}^{\infty} L^{(m_1\ldots m_n)}(t)a_1^{m_1}\cdots a_n^{m_n} e^{\sum_{i=1}^{n}m_i\lambda_i t}, \]

¹) In the case of three equations (2.1), B. N. Skachkov showed what qualitative picture of the arrangement of integral curves on this surface (2.4) is possible.

where \(L_s^{(m_1\ldots m_k)}(t)\) and \(L^{(m_1\ldots m_n)}(t)\) have the same properties as in formulas (1.5). But this is not the general solution. In the cited work of É. Groud, a solution with \(n+1\) arbitrary constants was obtained, but this is only an asymptotic expansion. However, such a construction of solutions makes it possible to find the whole family of solutions of the system (2.1): \(x=x(t)\to 0\), \(x_s=x_s(t)\to 0\) as \(t\to\infty\). This family may turn out to be \(n+1\)-parametric also in the case of instability of the zero solution. Let us also note that this is the case when the relations\(^1\) (1.6) can hold. The Siegel and Pliss transformation does not exist here. Indeed, if it did exist, then we would not obtain \(x\to0\), \(x_k\to0\) as \(t\to\infty\), since in the Pliss series (1.13) one of the \(y_k\), namely that which corresponds to \(\lambda=0\), reduces to the constant \(y=\alpha e^{0t}=\alpha\). In any case this will be so if in the Pliss series there is at least one term containing only \(y\), i.e. if they do not vanish when \(y_1=\cdots=y_n=0\) (these variables \(y_k=\alpha_k e^{\lambda_k t}\to0\) as \(t\to\infty\), since \(R(\lambda_k)<0\)). But if we consider the case when we have the integral (2.4), then the Pliss series (1.13) probably exist, since in this case we shall have

\[ x\to x(\alpha)=\alpha+\sum_{k=2}^{\infty}\beta_k\alpha^k,\qquad x_\nu\to x_\nu(\alpha)=\sum_{k=2}^{\infty}\beta_\nu^{(k)}\alpha^k \]

as \(t\to\infty\), in view of the fact that \(y_k=\alpha_k e^{\lambda_k t}\to0\) as \(t\to\infty\). Will they always exist in this case? Recall that in Pliss’s theorem the matrix \(P=\{0,\lambda_1,\ldots,\lambda_n\}\), i.e. it is purely diagonal. As for Siegel’s condition (1.12), it will be fulfilled here, since \(R(\lambda_k)<0\) \((k=1,\ldots,n)\); therefore, for a large value of \(m_1+\cdots+m_n+m\) we shall have

\[ |m_1\lambda_1+\cdots+m_n\lambda_n-\lambda_j| = |m_1\lambda_1+\cdots+(m_j-1)\lambda_j+\cdots+m_n\lambda_n| \gg \]

\[ \gg |m_1+\cdots+m_j-1+\cdots+m_n|\tau \gg \frac{1}{|m_1+\cdots+m_n+m|}, \]

where \(\tau=\min |R(\lambda_k)|\) \((k=1,\ldots,n)\).

But what shall we have in the case when the matrix \(P\) is not diagonal, i.e. has non-simple elementary divisors? It is necessary somehow to modify the construction of the series so that they exist and converge, as we obtain this, for example, according to Lyapunov, if the elementary divisors of the matrix \(P\) are non-simple, all \(R(\lambda_k)<0\), and we construct series (1.5) or (1.8), in which the \(L_s(t)\) are not constants. But using Lyapunov’s apparatus directly, in the case of the presence of the integral (2.4) we construct this solution, as shown in [3], under all circumstances.

Lyapunov also studied such a system (1.3), in which the coefficient matrix \(P=\|p_{kl}\|\) has two purely imaginary characteristic roots, and the real parts of the remaining characteristic roots are negative. Such a system can be written in the form

\[ \frac{dx}{dt}=-\lambda y+X,\qquad \frac{dy}{dt}=\lambda x+Y, \]

\[ \frac{dx_s}{dt}=p_{s1}x_1+\cdots+p_{sn}x_n+\alpha_s x+\beta_s y+X_s, \tag{2.6} \]

where \(X\), \(Y\), \(X_s\) are holomorphic functions in a neighborhood of the point \(x=y=x_1=\cdots=x_n=0\), containing no constant or linear terms; \(p_{kl}\), \(\alpha_s\), \(\beta_s\)—

\(^1\) Since one \(\lambda=0\), \(\lambda_k+m\lambda=\lambda_k\), \(m\ge2\).

real numbers and the real parts of all characteristic roots of the matrix \(P=\|p_{kl}\|\) are negative. Lyapunov reduces the study of this system to one in which \(X\) and \(Y\) vanish for \(x=y=0\). Further, this system (2.6) is investigated in the following way. First one poses the question of the existence of a family of solutions of the form

\[ x=x^{(1)}c+x^{(2)}c^2+\cdots,\qquad y=y^{(1)}c+y^{(2)}c^2+\cdots, \]

\[ x_s=x_s^{(1)}c+x_s^{(2)}c^2+\cdots, \tag{2.7} \]

where \(c\) is an arbitrary constant and the coefficients \(x^{(s)},\ y^{(s)},\ x_k^{(s)}\) are periodic functions of

\[ \tau=\frac{(t-t_0)\lambda}{1+h_2c^2+\cdots} \]

with period \(2\pi\), while the constants \(h_2,\ h_3,\ldots\) are to be determined. If such formal series are found, then, as Lyapunov showed, they will also be convergent for sufficiently small \(|c|\). In this way a family of periodic solutions of (2.6) is found. In this case, using Lyapunov’s first method, one can also construct the general solution of system (2.6) in a neighborhood of the origin of coordinates [3].

It will follow from this that the zero solution of system (2.6) is non-asymptotically stable, which was proved by Lyapunov with the aid of the second method. If, however, it turns out that there is no formal solution of the form (2.7), then, as Lyapunov showed, the question of the stability of the zero solution is decided on the basis of the second method. But in this case the general solution in a neighborhood of the origin has not yet been constructed in the form of convergent series. The construction of the general solution in this case can be carried out on the basis of methods of the analytic theory (and as Fuchs constructed the general solution in analogous cases). Namely, here the general solution is obtained through the solutions of a certain auxiliary equation in the form of asymptotic expansions.

What can be said about the existence of Poincaré series? Here the equalities (1.11) will be satisfied. Indeed, since we have \(i\lambda,\ -i\lambda\), then \(\lambda_k+mi\lambda-mi\lambda=\lambda_k\). Here we shall also have condition (1.12), since

\[ |m_1\lambda_1+\cdots+m_n\lambda_n+Mi\lambda-Ni\lambda-\lambda_j|= \]

\[ =|m_1\lambda_1+\cdots+(m_j-1)\lambda_j+\cdots+m_n\lambda_n+(M-N)i\lambda|\geq \]

\[ \geq |m_1+\cdots+(m_j-1)+\cdots+m_n|\tau \geq \frac{1}{m_1+\cdots+m_n+M+N} \]

for large values of \(m_1+\cdots+m_n+M+N\). Thus, if the formal Poincaré series exist, then they also converge.

We shall also suppose that the matrix \(P=\|p_{kl}\|\) in equations (2.6) is diagonal. Denote the new variables in Poincaré’s formulas (1.13) by \(\xi,\ \eta,\ y_1,\ldots,y_n\), where \(\xi,\ \eta\) correspond to the characteristic roots \(i\lambda,\ -i\lambda\), i.e. \(\xi=\alpha e^{i\lambda t}\), \(\eta=\beta e^{-i\lambda t}\), and \(\alpha\) and \(\beta\) are small constants.

Suppose that we have the case where there is no formal solution (2.7), i.e. we have either asymptotic stability or instability of the zero solution. Then, if in the Poincaré series there is at least one term containing \(\xi\) or \(\eta\), then they cannot exist, since the terms containing \(y_k=a_k e^{\lambda_k t}\) \((k=1,\ldots,n)\) vanish as \(t\to\infty\).

Let us now have the case where there exists a family of periodic solutions (2.7). If the formal Poincaré series now exist, then as \(t\to\infty\) the solutions asymptotically approach a periodic solution with period\(^{1)}\) \(2\pi/\lambda\). From this we see that, for the existence of series

\(^{1)}\) Since as \(t\to\infty\) in the Poincaré series only terms with variables \(\xi\) and \(\eta\), having period \(2\pi/\lambda\), remain.

For Pliss it is necessary that the period \(\tau\) of the periodic solutions (2.7) be equal to \(2\pi/\lambda\), and this is possible only in the case when the Lyapunov constants \(h_2, h_3, \ldots\), determining the period \(\tau\), are equal to zero: \(h_2=h_3=\ldots=0\). One may think that these equalities and the equalities determining the existence of periodic solutions (2.7) are equivalent to requiring that the Pliss constants appearing in the construction of his series be equal to zero, which is a necessary condition (and in this case also a sufficient one) for the existence of the Pliss series. If the elementary divisors of the matrix \(P\) are not simple, then one can make the same remarks as in the preceding case.

For systems of the form

\[ \frac{dx}{dt}=-\lambda y+X(x,y),\qquad \frac{dy}{dt}=\lambda x+Y(x,y) \tag{2.8} \]

or

\[ \frac{dx}{dt}=y+X(x,y),\qquad \frac{dy}{dt}=Y(x,y), \tag{2.9} \]

where \(X(x,y), Y(x,y)\) are series without constant and linear terms, in the case of a focus A. Erugin developed a special method [5] for the asymptotic construction of solutions in the form \(r=r(\vartheta)\). This method makes it possible to obtain solutions and to clarify their properties in a neighborhood of the origin also in the case of nonanalytic right-hand sides of the differential equations. The point of departure in this method is again a method that appeared in the analytic theory, in the theory of movable singular points. True, here it was necessary to construct an additional complicated apparatus for the investigation. One could continue the consideration of such systems from the real theory of differential equations, where the methods of investigation developed in the analytic theory are now applied. We shall not do this.

Can one construct Pliss series for equations (2.8)? Obviously, if here we have the case of a focus, then the Pliss series will not exist. But if we have the case of a center and the period is equal to \(2\pi/\lambda\) independently of the choice of the integral curve, then the Pliss series are possible. Thus, in addition to the vanishing of the constants determining the existence of a center, the constants \(h_2, h_3, \ldots\), which generate the dependence of the period on the arbitrary constant \(c\), must also be equal to zero.

§ 3. Systems of the form \(\dot{x}=f(x,t)=f(x,t+\omega)\)

We shall now touch upon one Lyapunov method for the investigation of such systems whose right-hand sides are periodic functions of \(t\). Namely, consider the system\(^1\)

\[ \frac{dx}{dt}=X,\qquad \frac{dx_s}{dt}=p_{s1}x_1+\ldots+p_{sn}x_n+p_sx+X_s \tag{3.1} \]

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

where \(X\) and \(X_s\) are series in \(x,x_1,\ldots,x_n\) without constant and linear terms, whose coefficients are periodic functions of \(t\) with period \(\omega\). The real parts of the characteristic roots of a constant real matrix

\(^1\) Such a system is obtained from a system of \(n+1\) equations whose right-hand sides are periodic functions of \(t\) with period \(\omega\), and whose first approximation has one zero characteristic root, while the remaining characteristic roots are negative.

\(P=\|p_{kl}\|\) are negative. \(p_s\) are real constants. Lyapunov showed that two cases may occur here:

1. The system has a formal solution of the form

\[ \begin{aligned} x&=c+u^{(2)}c^2+u^{(3)}c^3+\cdots,\\ x_s&=u_s^{(1)}c+u_s^{(2)}c^2+\cdots, \end{aligned} \tag{3.2} \]

where \(u^{(l)}, u_s^{(l)}\) are functions periodic with period \(\omega\), and \(c\) is an arbitrary constant. Lyapunov shows that the series (3.2) will also be convergent for small \(|c|\).

1. There is no such family of periodic solutions.

The second case is detected by a finite number of operations. If it is detected, then, applying the second method, Lyapunov shows that here there will be either asymptotic stability of the zero solution, or instability. But until now we have not had a general solution in a neighborhood of the zero solution here. Quite recently a method has been outlined for constructing the general solution in this case on the basis of methods of the analytic theory (É. I. Grudo), through solutions of an auxiliary equation in asymptotic form. And the fact that we obtain solutions only in asymptotic form may indicate that we construct the auxiliary differential equation too crudely and uniformly, through whose solutions we obtain the general solution in the form of a formal series.

In A. Erugin’s method the asymptotics is obtained not in the form of formal series, but by a certain method of successive approximations, which seems quite natural; however, an asymptotic expansion is also obtained,\(^1\) though of an entirely different structure. Here all analytic elementary functions of increasing orders of smallness have been found, of which the asymptotic series consists. Let us recall that Lyapunov always obtained convergent series, although, to be sure, not in the case of a focus or a node.

What do we have in the case when there is a periodic solution (3.2)? In this case Lyapunov proves the asymptotic stability of the solutions (3.2) and thereby obtains non-asymptotic stability of the zero solution. But in this case one can also obtain a general solution in a neighborhood of the origin of coordinates, from which the same conclusion will follow [3].

We now pass to the most difficult case, when the system after transformation has the form\(^2\)

\[ \frac{dx}{dt}=-\lambda y+X(x;\ y,\ x_1,\ \ldots,\ x_n,\ t), \]

\[ \frac{dy}{dt}=\lambda x+Y(x,\ y,\ x_1,\ \ldots,\ x_n,\ t) \]

\[ \frac{dx_s}{dt}=p_{s1}x_1+\cdots+p_{sn}x_n+p_sx+ \tag{3.3} \]

\[ +q_s y+X_s(x,\ y,\ x_1,\ \ldots,\ x_n,\ t), \]

where \(X, Y\) and \(X_s\) are series in \(x, y, x_1,\ldots,x_n\) without constant and linear terms, with coefficients periodic with period \(\omega\) in \(t\).

\(^1\) Incidentally, it may be that they are also convergent, since we know of no examples in which they diverge.

\(^2\) This is the case in which the right-hand sides of the system are periodic with period \(\omega\) in \(t\), and the first approximation has two zero characteristic exponents, while the remaining ones are negative in Lyapunov’s sense.

The quantities \(p_{kl}, p_s\) and \(q_s\) are real constants, and the real parts of all the characteristic roots of the matrix \(\lVert p_{kl}\rVert\) have negative values.

How does Lyapunov study this system? Let us dwell on this in more detail, since these systems have been little studied. We shall consider the difficulties that arise here and the possibilities for overcoming them that can be seen here.

Lyapunov, putting

\[ x=r\cos\vartheta,\qquad y=r\sin\vartheta, \tag{3.4} \]

brings system (3.3) to the form

\[ \frac{dr}{dt}=rR,\qquad \frac{d\vartheta}{dt}=\lambda+\Theta, \]

\[ \frac{dx_s}{dt}=p_{s1}x_1+\cdots+p_{sn}x_n+(p_s\cos\vartheta+q_s\sin\vartheta)r+X_s, \tag{3.5} \]

where in \(X_s\) the variables \(x\) and \(y\) have been replaced according to (3.4), and \(R\) and \(\Theta\) are holomorphic functions of the quantities \(r\) and \(x_s\), vanishing for \(r=x_1=\cdots=x_n=0\), in which the coefficients are finite sums of sines and cosines of integral multiples of \(\vartheta\), with coefficients \(\omega\)-periodic with respect to \(t\). The \(X_s\) will be of the same kind. Considering \(r\) and \(x_s\) as functions of the independent variables \(\vartheta\) and \(t\), we obtain the system

\[ \frac{\partial r}{\partial t}+(\lambda+\Theta)\frac{\partial r}{\partial\vartheta}=rR, \]

\[ \frac{\partial x_s}{\partial t}+(\lambda+\Theta)\frac{\partial x_s}{\partial\vartheta} =p_{s1}x_1+\cdots+p_{sn}x_n+(p_s\cos\vartheta+q_s\sin\vartheta)r+X_s \tag{3.6} \]

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

For this system Lyapunov raises the question of the existence of a formal solution of the form

\[ r=c+u^{(2)}c^2+u^{(3)}c^3+\cdots, \]

\[ x_s=u_s^{(1)}c+u_s^{(2)}c^2+\cdots\qquad (s=1,\ldots,n), \tag{3.7} \]

where \(u^{(l)}\) and \(u_s^{(l)}\) are finite sums of sines and cosines of \(\vartheta\) with \(\omega\)-periodic coefficients as functions of \(t\). If after a finite number of steps it turns out that there is no such solution and the first nonperiodic function \(u^{(l)}\) has the form

\[ u^{(l)}=g(t)+v(t,\vartheta), \tag{3.8} \]

where \(g\) is a nonzero constant, and \(v\) is a finite series of sines and cosines of integral multiples of \(\vartheta\) with coefficients periodic with respect to \(t\), then Lyapunov decides the question of stability of the zero solution of system (3.3) by the second method. But the question becomes difficult if series of the form (3.7) can formally satisfy system (3.6) in such a way that all \(u^{(l)}(t,\vartheta)\), \(u_s^{(l)}(t,\vartheta)\) turn out to be periodic as functions of \(t\) (or if the first nonperiodic function does not have the indicated form, which is possible only when the number \(\lambda\omega/\pi\) is rational).

In all previous cases we regarded \(u^{(l)}\) and \(u_s^{(l)}\) only as functions of \(t\). And if, under the formal satisfaction of the given system, they turned out to be periodic, then the series (3.7) also converged for small \(|c|\). Now they may turn out to be periodic functions of \(t\) (and with respect to \(\vartheta\) they are periodic by themselves), but we cannot specify such a \(c_0>0\) that, for

for \(|c|<c_0\) they will converge. The question of the convergence of the series (3.7), if they are obtained as periodic1 functions of \(t\), remains difficult to this day.2 Lyapunov usually gives examples for all points of his theory. Here he did not indicate an example of such a system (3.3) for which the series (3.7) would be obtained as periodic functions of \(t\) and would converge3 uniformly with respect to \(\vartheta\) and \(t\) for \(|c|<c_0\). But, following Poincaré, Lyapunov gives an example in which the formal series (3.7) are obtained as periodic functions of \(t\). It is precisely such a case that we encounter when a canonical system is considered.

Let, for example,

\[ \frac{dx}{dt}=-\lambda y-\frac{\partial F}{\partial y},\qquad \frac{dy}{dt}=\lambda x+\frac{\partial F}{\partial x}, \tag{3.9} \]

where \(F(x,y,t+\omega)=F(x,y,t)\) is a function holomorphic with respect to \(x\) and \(y\) in a neighborhood of the origin and has no terms below the third dimension, i.e.

\[ F(x,y,t)=\sum_{m_1+m_2\geq 3} F^{(m_1m_2)}(t)x^{m_1}y^{m_2}, \tag{3.10} \]

\[ F^{(m_1m_2)}(t+\omega)=F^{(m_1m_2)}(t) \]

or

\[ F(x,y,t)=\sum_{m=3}^{\infty}P_m(x,y,t),\qquad P_m=\sum_{k+l=m}x^k y^l\varphi_{kl}(t). \tag{3.11} \]

Replacing \(x\) and \(y\) according to formulas (3.4), we obtain for \(r\), as a function of \(\vartheta\) and \(t\), the equation

\[ \frac{\partial r}{\partial t} +\left(\lambda+\frac{1}{r}\frac{\partial F}{\partial r}\right) \frac{\partial r}{\partial\vartheta} =-\frac{1}{r}\frac{\partial F}{\partial\vartheta}, \tag{3.12} \]

where

\[ F=F(r\cos\vartheta,\ r\sin\vartheta,\ t) \tag{3.13} \]

and therefore

\[ \frac{\partial F}{\partial r} =\frac{\partial F}{\partial x}\cos\vartheta +\frac{\partial F}{\partial y}\sin\vartheta, \]

\[ \frac{\partial F}{\partial\vartheta} =-\frac{\partial F}{\partial x}r\sin\vartheta +\frac{\partial F}{\partial y}r\cos\vartheta. \]

Satisfying this equation by a series of the form

\[ r=c+u_2c^2+u_3c^3+\cdots,\qquad u_l=u_l(\sin\vartheta,\cos\vartheta,t), \tag{3.14} \]

we find the \(u_k\) in the form of polynomials in \(\sin\vartheta,\cos\vartheta\), whose coefficients will be \(\omega\)-periodic functions of \(t\). Thus, we always find a formal solution of system (3.9) in the form (3.14), where \(\vartheta\) is a solution of the equation

\[ \dot{\vartheta}=\lambda+\sum_{m=3}^{\infty}mr^{m-2}P_m(\cos\vartheta,\sin\vartheta,t). \tag{3.15} \]

But we do not know how to prove the convergence of the series (3.14) for small \(|c|\). We shall return again to the convergence of the series (3.7).

And now let us suppose that we have succeeded in proving the convergence of the series1 (3.7). Let us see what conclusions can be drawn from this. In the previous cases, when \(u^{(l)}\) and \(u_s^{(l)}\) were determined only as functions of \(t\) and turned out to be periodic, the series (3.7) furnished us with a one-parameter family of periodic solutions. Now we cannot assert this, since \(u^{(l)}\) and \(u_s^{(l)}\) are functions of two arguments \(\vartheta\) and \(t\), \(2\pi\)-periodic in the first argument and \(\omega\)-periodic in the second argument. Nevertheless, it will be a bounded family of solutions, and we have, uniformly with respect to \(t\),

\[ r^2+x_1^2+\cdots+x_n^2 \to 0 \quad \text{as } c\to 0. \tag{3.16} \]

True, one must be certain that \(\vartheta=\vartheta(t)\) is defined for all finite \(t\). But we have this, since, substituting (3.7) into (3.5), we obtain

\[ \frac{d\vartheta}{dt}=\lambda+\Theta(\vartheta,t), \tag{3.17} \]

where \(\Theta(\vartheta+2\pi,t)=\Theta(\vartheta,t)\), \(\Theta(\vartheta,t+\omega)=\Theta(\vartheta,t)\), and

\[ \left|\frac{\partial \Theta(\vartheta,t)}{\partial \vartheta}\right| \le M \]

is a constant.

Thus, let the series (3.7) converge. Then the function \(\vartheta=\vartheta(t)\) is also defined for all finite \(t\). Under this assumption Lyapunov introduces new variables

\[ r=z+u^{(2)}z^2+u^{(3)}z^3+\cdots, \]

\[ x_s=z_s+u_s^{(2)}z^2+u_s^{(3)}z^3+\cdots . \tag{3.18} \]

In the new variables the system of equations has the solution \(z=c\) (\(c\) arbitrary constant), \(z_1=\cdots=z_n=0\).

Lyapunov proves by the second method the asymptotic stability of this solution, whence follows the nonasymptotic stability of the zero solution of the original system. But in this case one can also obtain the general solution in a neighborhood of the origin of coordinates of the original system [3], whence the nonasymptotic stability of the zero solution of the original system will also follow.

Thus, with respect to system (3.3) we have the following. We seek a formal solution (3.7). If at the \(m\)-th step we discover that there is no such solution and the first nonperiodic function with respect to \(t\) has the form (3.8), then we have either asymptotic stability of the zero solution or instability. This is obtained by the second method, but the general solution in a neighborhood of the origin of coordinates has not yet been constructed in this case. If the formal solution (3.7) exists and it has been possible to prove the convergence of these series, then one can prove, using the second method, that the zero solution of the original system is nonasymptotically stable. But in this case one can also construct the general solution in a neighborhood of the origin of coordinates [3].

What questions remain open here?

1. What may happen in the case when, in finding the formal solution (3.7), the first nonperiodic function has a form different from

(3.8) (which is possible, as Lyapunov showed, only in the case when the number \(\lambda\omega/\pi\) is rational)?

1. Is it possible to indicate a system for which we have a formal solution (3.7) and the series converges for \(|c|<c_0\)? (Lyapunov did not indicate such a system.)

2. How is the problem to be solved in the case when the formal series (3.7) exists but diverges for all \(c\ne 0\)?

3. Is it possible in all these cases to avoid the first method when solving questions of stability of the zero solution? This seems to us unlikely.

4. What is the role of the irrationality of the number \(\lambda\omega/\pi\) in solving the question of stability of the zero solution? In other words, does the irrationality of this number affect only the apparatus of investigation (the choice of the form of the convergent series), or the fact of stability?

5. Is a formal series (3.7) possible that is periodic with respect to \(t\), but not with period \(\omega\), or nonperiodic (quasiperiodic), and converges for \(|c|<c_0\)?

The following answers can be given to these questions:

1. Cases are possible when the first nonperiodic function has a form different from (3.8), and in this case the zero solution may turn out to be nonasymptotically stable (which is impossible, as we have seen, in the case (3.8)).

2. One can indicate such systems for which the formal series (3.8) converge for \(|c|<c_0\).

3. There exist such systems for which the formal series (3.8) exist but do not converge. In this case one can construct series of the form (3.8), where the coefficients will not be periodic with respect to \(t\) (they will be quasiperiodic), or will be periodic, but not with period \(\omega\), and will converge for \(|c|<c_0\).

4. One can indicate systems for which the fact of irrationality of the number \(\lambda\omega/\pi\) has no significance in solving the question of stability of the zero solution, but there also exist such systems in which, for rational \(\lambda\omega/\pi\), the zero solution ceases to be stable. All this is shown in [3]. We shall only give some further examples.

Consider the system

\[ \frac{dx}{dt} = \bigl(-\lambda y + A(x,y)\bigr) \left[ 1+\sum_{k=2}^{\infty} \left( A_k(x,y)\sin\frac{kt}{\omega} + B_k(x,y)\cos\frac{kt}{\omega} \right) \right], \tag{3.19} \]

\[ \frac{dy}{dt} = \bigl(\lambda x + B(x,y)\bigr) \left[ 1+\sum_{k=2}^{\infty} \left( A_k(x,y)\sin\frac{kt}{\omega} + B_k(x,y)\cos\frac{kt}{\omega} \right) \right], \]

where \(A_k(x,y)\) and \(B_k(x,y)\) may be, for example, polynomials, and \(A(x,y)\) and \(B(x,y)\) are series having no constant or linear terms. Let

\[ \frac{\partial \omega(x,y)}{\partial x}\bigl(-\lambda y + A(x,y)\bigr) + \frac{\partial \omega}{\partial y}\bigl(\lambda x + B(x,y)\bigr) \equiv 0. \tag{3.20} \]

Suppose that here the function \(\omega(x,y)\) is holomorphic in a neighborhood of the origin. Then [1]

\[ \omega(x,y)=c, \tag{3.21} \]

integral of system (3.19), and the integral curves of system (3.19) in a neighborhood of the origin of coordinates will be closed, contracting to the origin of coordinates as \(|c| \to 0\). The zero solution of system (3.19) will be nonasymptotically stable independently of the relation between \(\lambda\) and \(\omega\).

Let us consider also the system

\[ \begin{aligned} \frac{dx_k}{dt}={}&X_k(x_1,\ldots,x_n)+\sum_{\nu=1}^{\infty}\bigl(X_k^{(\nu)}(x_1,\ldots,x_n)\sin \nu t \\ &+Y_k^{(\nu)}(x_1,\ldots,x_n)\cos \nu t\bigr), \end{aligned} \tag{3.22} \]

\[ X_k(0,\ldots,0)=X_k^{(\nu)}(0,\ldots,0)=Y_k^{(\nu)}(0,\ldots,0)=0 \]

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

Let \(\omega(x_1,\ldots,x_n)\) be a solution of the equations

\[ \sum_{k=1}^{n}\frac{\partial \omega}{\partial x_k}\,X_k(x_1,\ldots,x_n)=0, \tag{3.23} \]

\[ \sum_{k=1}^{n}\frac{\partial \omega}{\partial x_k}\,X_k^{(\nu)}(x_1,\ldots,x_n)=0, \tag{3.24} \]

\[ \sum_{k=1}^{n}\frac{\partial \omega}{\partial x_k}\,Y_k^{(\nu)}(x_1,\ldots,x_n)=0 \tag{3.25} \]

\[ (\nu=1,2,\ldots). \]

If among all these equations there are found \(n\) independent ones, i.e. with a determinant formed from the coefficients of \(\partial\omega/\partial x_1,\ldots,\partial\omega/\partial x_n\) different from zero, then \(\omega\equiv c\) (a constant), and, consequently, there is no nontrivial common solution \(\omega\) of these equations.

Suppose now that equation (3.23) and \(m-1\) equations \((m<n)\) among the others are independent, while all the remaining ones are consequences\(^1\) of these, i.e. are linear sums of these basic \(m\) equations. Then we have two cases:

1. In the known way, we form from these equations a closed system consisting of \(n\) equations. In this case we again have only \(\omega\equiv c\).

2. The closed system will consist of \(N<n\) equations. Then we have \(n-N\) independent common solutions \(\omega_1(x_1,\ldots,x_n),\ldots,\omega_{n-N}(x_1,\ldots,x_n)\), which will be integrals of system (3.22)

\[ \omega_1(x_1,\ldots,x_n)=c_1,\ldots,\omega_{n-N}(x_1,\ldots,x_n)=c_{n-N}. \tag{3.26} \]

Corollary. If among the integrals (3.26) there is one\(^2\) \(\omega(x_1,\ldots,x_n)\to 0\) as \(x_1^2+\cdots+x_n^2\to 0\), then for all small \(|c|\) the surface

\[ \omega(x_1,\ldots,x_n)=c \tag{3.27} \]

\(^1\) Or, perhaps, there will simply be only \(m\) equations (3.23), (3.24), (3.25) in all, and all the remaining \(X_k^{(\nu)}\equiv 0\), i.e. we have a finite sum in equation (3.22) with the number of terms \(m-1\).

\(^2\) We assume that the integral \(\omega(x_1,\ldots,x_n)\) contains all independent variables: \(x_1,\ldots,x_n\). If among (3.26) there is no such integral, then perhaps one can compose from them an integral possessing this property: \(\omega=\Phi(\omega_1,\ldots,\omega_{n-N})\).

will be bounded, then the zero solution of system (3.22) will be stable. The integral \(\omega(x_1,\ldots,x_n)\) has this property if \(\omega(x_1,\ldots,x_n)\) is a positive definite function in the sense of Lyapunov.

Remark 1. If on the right in equations (3.22) \(\nu\) runs through a finite number of values \(N\), and \(2N+1<n\), then one can indicate a condition which \(X_k^{(\nu)}\) and \(Y_k^{(\nu)}\) must satisfy in order that equations (3.23), (3.24), (3.25) have nontrivial common solutions \(\omega(x_1,\ldots,x_n)\). This condition consists in requiring that the closed system of equations corresponding to equations (3.23), (3.24), (3.25) consist of \(M<n\) equations.

Remark 2. In particular, \(X_k(x_1,\ldots,x_n)\) may be linear homogeneous functions with constant coefficients. We shall always arrive at such a case when the right-hand sides of the given system are functions periodic with respect to \(t\)¹ with period \(2\pi\); for example, under the corresponding assumption, we arrive at equations (3.3). In this case all \(n-1\) independent integrals of equation (3.23) are easily found, as are those relations between \(X_k^{(\nu)}\) and \(Y_k^{(\nu)}\) for which equations (3.23), (3.24), and (3.25) have common solutions \(\omega(x_1,\ldots,x_n)\).

One may indicate particular cases of system (3.22) for which the series (3.7) will converge.

Conclusion. Each time we have system (1.1) and, using some Lyapunov method, have a solution of the question of stability of the zero solution, we also obtain the general solution in a neighborhood of the origin of coordinates. Moreover, if the stability is asymptotic and is determined by the first approximation, or if we have nonasymptotic stability, then the general solution is obtained by means of Lyapunov’s first method. If, however, we have a critical case and asymptotic stability or instability, then the general solution is obtained in the form of asymptotic series through solutions of auxiliary differential equations, in general constructed in the analytic theory of differential equations (Fuchs, Grudo) or developed (A. Erugin) on the basis of a certain method of successive approximations [8].

Let us consider systems whose right-hand sides are functions periodic in \(t\), and we have critical cases:

1. One c. e. is equal to zero, the remaining ones are negative.
2. Two c. e. are equal to zero, the remaining ones are negative.

The first case differs in no way from the general one characterized by us. The second case occupies a special place. If we do not have a formal solution (3.7), then we obtain the same as in the general case, i.e., we obtain the solution through the solution of an auxiliary equation asymptotically.

If we have a formal solution (3.7) and it is proved that these series converge, then we have the same as in the general case, i.e., the general solution is obtained by Lyapunov’s first method.

But up to now no classes of differential equations have been singled out for which the series (3.7) will converge. Nor have such classes of differential equations been singled out for which there are other possibilities for the series (3.7) (when the coefficients of the series (3.7) have, with respect to \(t\), a period not equal to \(\omega\), or when they are nonperiodic, or when the first nonperiodic coefficient \(u^{(l)}\) has a form different from (3.8)), which we have noted.

If this system is canonical, then, as is known, under certain conditions similar to those in Siegel’s work, the construction method proposed by A. Kolmogorov and developed by V. Arnold [7] is possible. More,

¹ At least the matrix \(P(t)\) of the first approximation should be periodic.

the as yet undisclosed possibilities contained in the Krylov—Bogolyubov method, about which more is said in [3]. But, apparently, methods similar to the methods of the analytic theory noted by us have also not yet been sufficiently used.

Other methods of analytic theory have also not yet been sufficiently used, for example such as the methods developed in the works of the Yugoslav mathematician Pejović.

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Received by the editors
12 July 1967

Institute of Mathematics
Academy of Sciences of the BSSR

1. If the series \(\sum_{k=0}^{\infty}\varphi_k(x,y)c^k\) converges for \(|c|\le c_0\), where \(\varphi_k(x,y)\) are continuous and periodic with periods \(\omega\) and \(t\), then this series converges uniformly as well, since otherwise one could indicate \(x_1,x_2,\ldots\to x^*\), \(y_1,y_2,\ldots\to y^*\) such that \(\sum \varphi_k(x^*,y^*)\varepsilon^k\) diverges. ↩↩↩↩↩↩↩↩↩

2. Here the equalities (1.6) take place. ↩↩

3. We shall consider such systems. ↩