On the Number of Limit Cycles in the Neighborhood of a Singular Point

K. S. Sibirskii

1. Consider the system of differential equations

\[ \left. \begin{aligned} -\frac{dx}{dt} &= b_{10}x+b_{01}y+\sum_{j+l\in A'} b_{jl}x^j y^l,\\ \frac{dy}{dt} &= c_{10}x+c_{01}y+\sum_{j+l\in A'} c_{jl}x^j y^l, \end{aligned} \right\} \tag{1} \]

in which \(A'=\{n_1,n_2,\ldots,n_s\}\) is a finite set of natural numbers distinct from one and unequal to one another, \(j\) and \(l\) are nonnegative integers, \(b_{jl}, c_{jl}\) are real numbers, and \(x,y\), and \(t\) are real variables.

We shall assume that the characteristic equation

\[ \left| \begin{array}{cc} -b_{10}-\Lambda & -b_{01}\\ c_{10} & c_{01}-\Lambda \end{array} \right|=0 \tag{2} \]

has imaginary roots, that is,

\[ (c_{01}-b_{10})^2+4(b_{10}c_{01}-c_{10}b_{01})<0. \tag{3} \]

In this case, as is known, the origin \(O\) of the phase plane \(XOY\) is, for system (1), a singular point of center or focus type.

Let \(E(A')\) denote the space of coefficients of the right-hand sides of system (1), under condition (3), with the Euclidean metric, and let \(E_c(A')\) \((E_f(A'))\) denote the set of points of \(E(A')\) to which there correspond systems (1) for which the point \(O\) is a center (focus). Then \(E(A')=E_c(A')\cup E_f(A')\). By \(S(O,\delta)\) and \(S(\Sigma,\varepsilon)\) we shall denote, respectively, the \(\delta\)-neighborhood of the point \(O\) of the plane \(XOY\) and the \(\varepsilon\)-neighborhood of the point \(\Sigma\) of the space \(E(A')\).

Following N. N. Bautin [1, 2], we shall say that, for a point \(\Sigma_0\in E(A')\), the origin \(O\) of the phase plane \(XOY\) has, relative to \(E(A')\), cyclicity of order \(k\) (or that, when the coefficients of system (1) corresponding to the point \(\Sigma_0\in E(A')\) are varied, \(k\) limit cycles are born from the singular point \(O\)), if the following two conditions are satisfied:

a) there exist numbers \(\varepsilon_0>0\) and \(\delta_0>0\) such that inside \(S(\Sigma_0,\varepsilon_0)\) there is not a single point to which there would correspond a system of the form (1) having inside \(S(O,\delta_0)\) more than \(k\) limit cycles;

b) whatever positive numbers \(\varepsilon<\varepsilon_0\) and \(\delta<\delta_0\) may be, there exists a point \(\Sigma\in S(\Sigma_0,\varepsilon)\) to which there corresponds a system of the form (1) having \(k\) limit cycles inside \(S(O,\delta)\).

Denote by \(N(A')\) the maximal order of cyclicity that the origin \(O\) can have for points \(\Sigma_0\) from \(E(A')\). In the above-cited works of N. N. Bautin it was shown that, for \(A'=\{2\}\), \(N(A')=3\).

The question of the number of limit cycles arising from a singular point under variation of the coefficients of system (1), subject to condition (3), was also considered by N. F. Otrokov [3, 4] and B. M. Peretyagin [5, 6]. In particular, N. F. Otrokov gave, in the form of a function of \(n\), a lower estimate for the number \(N(\{2,3,\ldots,n\})\), from which it follows that for \(n=3\) the number \(N(A')>3\).

In the present paper, using the method of paper [2], it is proved that if \(A'=\{3\}\) (the nonlinear additions are homogeneous polynomials of the third degree), then \(N(A')=5\).

This result shows the erroneousness of the conclusion of B. M. Peretyagin’s papers [5, 6], namely that for \(A'=\{n\}\) no more than \(n+1\) limit cycles can arise in some neighborhood of the origin.

As was established in the papers of I. G. Petrovskii and E. M. Landis [7, 8], for \(A'=\{2\}\) the maximal number of limit cycles of system (1) in the whole plane coincides with \(N(A')=3\). The question of whether system (1) for \(A'=\{3\}\) can have more than five limit cycles in the whole plane remains open.

2. Sufficient conditions for a center. Let \(A'=\{3\}\) and suppose that condition (3) is satisfied. Then the roots of equation (2) will be the numbers \(\Lambda_{1,2}=C\pm i\sqrt{-B}\), where \(2C\equiv c_{01}-b_{10}\), \(4B=(c_{01}-b_{10})^2+4(b_{10}c_{01}-c_{10}b_{01})\). In this case, by a nonsingular linear transformation and by dividing all coefficients of the right-hand sides by \(\sqrt{-B}\), system (1) can be reduced to the form

\[ \left. \begin{aligned} -\frac{dx}{dt} &= y-\lambda x+\sum_{j+l=3} B_{jl}x^j y^l,\\ \frac{dy}{dt} &= x+\lambda y+\sum_{j+l=3} C_{jl}x^j y^l, \end{aligned} \right\} \tag{4} \]

where \(\lambda=C/\sqrt{-B}\).

Performing a rotation of the coordinate axes through an angle \(\varphi\) satisfying the condition

\[ 2(B_{21}-C_{12})\cos 2\varphi+(B_{12}-3C_{03}+C_{21}-3B_{30})\sin 2\varphi=0, \]

one can ensure that, in the new system, the equality

\[ B_{21}=C_{12}. \tag{5} \]

is satisfied. Such a system can be written in the form

\[ \frac{dx}{dt}=-P(x,y), \qquad \frac{dy}{dt}=Q(x,y), \tag{6} \]

where

\[ P(x,y)=y-\lambda x+(\omega+\theta-a)x^3+(\eta-3\mu)x^2y+(3\omega-3\theta+ \]

\[ +\,2a-\xi)xy^2+(\mu-\nu)y^3, \]

\[ Q(x,y)=x+\lambda y+(\mu+\nu)x^3+(3\omega+3\theta+2a)x^2y+ \]

\[ +\,(\eta-3\mu)xy^2+(\omega-\theta-a)y^3. \]

It is easy to see that the point \(O\) always has, for system (6), with respect to the space \(\widetilde E\) of the coefficients \(\lambda, \xi, \nu, \omega, \eta, \mu, \theta, a\), the same cyclicity as for system (1) when \(A'=\{3\}\) and condition (3) with respect to \(E(\{3\})\).

Let us denote an arbitrary point of the space \(\widetilde E\) by \(\sigma\), the images of the sets \(E_c(\{3\})\) and \(E_f(\{3\})\) in \(\widetilde E\) respectively by \(\widetilde E_c\) and \(\widetilde E_f\), and the expression \(4(\mu^2+\theta^2)-a^2\) by \(x\).

Lemma 1. In each of the following three cases:

\[ \left. \begin{array}{ll} 1) & \lambda=\xi=a=0;\\ 2) & \lambda=\xi=\nu=\theta=0;\\ 3) & \lambda=\xi=\nu=\omega=\eta=4(\mu^2+\theta^2)-a^2=0 \end{array} \right\} \tag{7} \]

system (6) has a center at the origin.

Proof. When conditions 1) are fulfilled, the existence of a center is ensured by the identity

\[ \frac{\partial P}{\partial x}\equiv \frac{\partial Q}{\partial y}. \]

In case 2) the identities hold

\[ P(-y,-x)\equiv -Q(x,y), \quad Q(-y,-x)\equiv -P(x,y), \]

by virtue of which the straight line \(y=-x\) is an axis of symmetry of the field of directions defined by system (6) in the phase plane. Suppose now that conditions 3) are satisfied and \(\mu\theta\ne 0\). In this case, as shown by K. E. Malkin [9], system (6) has an A. M. Lyapunov integral of the form \(F^2 f^{-3}=C\), where \(F\) and \(f\) are, respectively, polynomials of the sixth and fourth degrees. By direct differentiation it is not difficult to verify that one may take

\[ F=1-\frac{3a}{32\mu^3}\left(bu^2+cv^2-\frac{bu^3v}{2\mu}+\frac{b^3u^6}{96\mu^4}\right), \]

\[ f=1-\frac{uv}{2\mu}+\frac{b^2u^4}{64\mu^4}, \]

where

\[ b\equiv a-2\theta,\quad c\equiv a+2\theta,\quad u\equiv cx-2\mu y,\quad v\equiv bx-2\mu y. \tag{8} \]

At the same time, using the equality \(bc=4\mu^2\), we obtain

\[ \frac{dv}{du}= \frac{8ab\mu u-32\mu^3v+4b^2u^3-7abu^2v+3acv^3} {32\mu^3u-8ac\mu v-abu^3+acuv^2}, \]

after which, making the substitution (8) in this equation, we arrive at the equation

\[ \frac{dy}{dx} = -\frac{x+\mu x^3+(2a+3\theta)x^2y-3\mu xy^2-(a+\theta)y^3} {y+(\theta-a)x^3-3\mu x^2y+(2a-3\theta)xy^2+\mu y^3}, \]

equivalent to system (6) with condition 3).

When \(\mu\theta=0\) in case 3), the existence of a center follows from the closedness of the set \(\widetilde E_c\). Lemma 1 is proved. We have not used here the center conditions of M. I. Al’mukhamedov [10] and N. A. Sakharnikov [11], since they do not contain all cases of a center [9].

3. The succession function. Eliminating \(t\) from system (6) and passing to polar coordinates, we arrive at the equation

\[ \frac{d\rho}{d\varphi}=\frac{\lambda\rho+a\rho^3}{1+\beta\rho^2}, \tag{9} \]

where

\[ \alpha \equiv (a-\theta-\omega)\cos^{4}\varphi+(4\mu+\nu-\eta)\cos^{3}\varphi\sin\varphi+(6\theta+\xi)\cos^{2}\varphi\sin^{2}\varphi+ \]
\[ +(\eta-4\mu+\nu)\cos\varphi\sin^{3}\varphi+(\omega-\theta-a)\sin^{4}\varphi, \tag{10} \]

\[ \beta \equiv (\mu+\nu)\cos^{4}\varphi+(4\omega+4\theta+a)\cos^{3}\varphi\sin\varphi+(2\eta-6\mu)\cos^{2}\varphi\sin^{2}\varphi+ \]
\[ +(4\omega-4\theta+a-\xi)\cos\varphi\sin^{3}\varphi+(\mu-\nu)\sin^{4}\varphi . \tag{11} \]

By means of the substitution \(r=\rho^{2}\), one can pass from equation (9) to the equation

\[ \frac{dr}{d\varphi}=2\,\frac{\lambda r+\alpha r^{2}}{1+\beta r}. \tag{12} \]

The solution \(r=r(\varphi,\sigma)\) of this equation satisfying the initial condition \(r(0,\sigma)=r_{0}>0\) can be represented as a series in powers of \(r_{0}\),

\[ r(\varphi,\sigma)=r_{0}u_{0}(\varphi,\sigma)+r_{0}^{2}u_{1}(\varphi,\sigma)+r_{0}^{3}u_{2}(\varphi,\sigma)+\cdots, \tag{13} \]

in which

\[ u_{0}(0,\sigma)=1,\qquad u_{j}(0,\sigma)=0\quad (j=2,3,\ldots). \tag{14} \]

At the same time, let us note that, whatever \(\varepsilon>0\) and \(\sigma_{0}\in \widetilde{E}\) may be, there is a positive number \(R=R(\varepsilon,\sigma_{0})\) such that the series (13) converges for all \(\varphi\in[0,2\pi]\), \(\sigma\in S(\sigma_{0},\varepsilon)\), and \(r_{0}\in(0,R)\).

From (13), for \(\varphi=2\pi\) we obtain the so-called successor function

\[ r(2\pi,\sigma)-r_{0}=r_{0}\,[u_{0}(2\pi,\sigma)-1]+r_{0}^{2}u_{1}(2\pi,\sigma)+ \]
\[ +r_{0}^{3}u_{2}(2\pi,\sigma)+\cdots, \tag{15} \]

whose positive roots correspond, obviously, to closed integral curves of system (6).

Substituting the series (13) into equation (12), we obtain, for determining the functions \(u_{j}(\varphi,\sigma)\), the following recurrence relations:

\[ u_{0}'=2\lambda u_{0}, \tag{16} \]

\[ u_{j}'-2\lambda u_{j}=R_{j}, \tag{17} \]

where

\[ R_{j}=\sum_{l=0}^{j-1}u_{l}\bigl(2\alpha u_{j-1-l}-\beta u_{j-1-l}'\bigr)\quad (j=1,2,3,\ldots). \tag{18} \]

From the linear equations (16) and (17), taking into account the initial conditions (14), we find

\[ u_{0}(\varphi,\sigma)=e^{2\lambda\varphi}, \tag{19} \]

\[ u_{j}(\varphi,\sigma)=e^{2\lambda\varphi}\int_{0}^{\varphi}e^{-2\lambda\varphi}R_{j}\,d\varphi\quad (j=1,2,3,\ldots). \tag{20} \]

Lemma 2. The coefficients \(u_{j}(2\pi,\sigma)\) \((j=0,1,2,\ldots)\) of the successor function are entire functions of the coordinates of the point \(\sigma\in\widetilde{E}\), which, for \(\lambda=0\), turn into homogeneous polynomials of degree \(j\) with respect to the remaining coordinates of the point \(\sigma\), and \(u_{0}(2\pi,\sigma)=\exp(4\pi\lambda)\).

Proof. The equality \(u_{0}(2\pi,\sigma)=\exp(4\pi\lambda)\) follows directly from (19). Now it remains only to show that the functions

\(u_j(\varphi,\sigma)\) are homogeneous polynomials of degree \(j\) in the totality of all coordinates of the point \(\sigma \in \tilde E\), except \(\lambda\), and the coefficients of these polynomials are integer functions of \(\varphi\) and \(\lambda\). We shall apply here the method of complete mathematical induction with respect to \(j\).

From (19) it is seen that the function \(u_0\) is an integer function of \(\varphi\) and \(\lambda\). Suppose now that the functions \(u_l(l=0,1,2,\ldots,j-1)\) are homogeneous polynomials of degree \(l\) in the totality of all coordinates of the point \(\sigma\), except \(\lambda\), with coefficients in the form of integer functions of \(\varphi\) and \(\lambda\). Then the derivatives \(u_l'(l=0,1,\ldots,j-1)\) with respect to \(\varphi\) also have the same property. Formulas (18) and (20), taking into account (10) and (11), show that in this case the function \(u_j(\varphi,\sigma)\), as well as the function \(R_j\), is a homogeneous polynomial of degree \(j\) in the totality of all coordinates of the point \(\sigma\), except \(\lambda\), and its coefficients are integer functions of \(\varphi\) and \(\lambda\).

The assertion of Lemma 2 follows immediately from this.

4. The main lemma. We have

Lemma 3. The coefficients of the sequence function \(u_j(2\pi,\sigma)\), for all \(j>0\), have the form

\[ u_j(2\pi,\sigma)=a^2\theta_\chi K_j^{(6)}+a^2\theta_\eta K_j^{(5)}+a\theta_\omega K_j^{(4)}+a\nu K_j^{(3)}+\xi K_j^{(2)}+\lambda K_j^{(1)}, \tag{21} \]

where \(K_j^{(l)}\) are integer functions of the coordinates of the point \(\sigma \in \tilde E\), independent of the first \(l-1\) coordinates, \(K_j^{(j+l)}=0\) for \(l>1\), and \(K_j^{(j+1)}\) are nonzero real numbers.

Proof. On the basis of Lemma 1, when each of the three series of conditions (7) is fulfilled, all coefficients \(u_j(2\pi,\sigma)\) \((j>0)\) must vanish. Taking Lemma 2 into account and denoting by \(\lambda K_j^{(1)}\) the sum of all terms in \(u_j(2\pi,\sigma)\) containing \(\lambda\), we write \(u_j(2\pi,\sigma)\) in the form

\[ u_j(2\pi,\sigma)=K_j^{*(1)}+\lambda K_j^{(1)}. \]

The sum of the terms in \(K_j^{*(1)}\) containing \(\xi\) we denote by \(\xi K_j^{(2)}\). Then we may write

\[ u_j(2\pi,\sigma)=K_j^{*(2)}+\xi K_j^{(2)}+\lambda K_j^{(1)}. \]

From the first series of conditions (7) it follows that the function \(K_j^{*(2)}\) contains the factor \(a\). Denote by \(a\nu K_j^{(3)}\) the sum of all terms in \(K_j^{*(2)}\) which contain \(\nu\). Then the function \(u_j(2\pi,\sigma)\) can be represented in the form

\[ u_j(2\pi,\sigma)=K_j^{*(3)}+a\nu K_j^{(3)}+\xi K_j^{(2)}+\lambda K_j^{(1)}. \]

On the basis of the second series of conditions (7), the function \(K_j^{*(3)}\), in addition to the factor \(a\), must also contain the factor \(\theta\). Denote by \(a\theta_\omega K_j^{(4)}\) the sum of all terms in \(K_j^{*(3)}\) which contain \(\omega\). In this case we may write

\[ u_j(2\pi,\sigma)=K_j^{*(4)}+a\theta_\omega K_j^{(4)}+a\nu K_j^{(3)}+\xi K_j^{(2)}+\lambda K_j^{(1)}. \]

Denoting by \(a\theta_\eta \tilde K_j^{(5)}\) the sum of all terms in \(K_j^{*(4)}\) which contain \(\eta\), we write \(u_j(2\pi,\sigma)\) in the form

\[ u_j(2\pi,\sigma)=a\theta K_j^{*(5)}+a\theta_\eta \tilde K_j^{(5)}+a\theta_\omega K_j^{(4)}+a\nu K_j^{(3)}+\xi K_j^{(2)}+\lambda K_j^{(1)}. \]

Here \(K_j^{*(5)}\) is a polynomial depending only on three variables \(a\), \(\mu\), and \(\theta\). The third series of conditions (7) shows that this polynomial must vanish for

\[ \chi \equiv 4(\mu^2+\theta^2)-a^2=0. \]

Considering

\(K_j^{*(5)}\) as a polynomial in \(\mu\), we conclude, by Bezout’s theorem, that it is divisible both by \(2\mu+\sqrt{a^2-4\theta^2}\) and by \(2\mu-\sqrt{a^2-4\theta^2}\), and therefore it is divisible by \(\chi\). Let \(K_j^{*(5)}=\chi \widetilde K_j^{(6)}\). Then, for all \(j>0\),

\[ u_j(2\pi,\sigma)=a\theta\chi \widetilde K_j^{(6)}+a\theta\eta K_{\eta j}^{(5)}+a\theta\omega K_j^{(4)}+ \]
\[ {}+a\nu K_j^{(3)}+\xi K_j^{(2)}+\lambda K_j^{(1)}. \tag{22} \]

To obtain formula (21), it now remains only to show that \(\widetilde K_j^{(6)}\) and \(\widetilde K_j^{(5)}\) contain the factor \(a\). For this purpose it is sufficient to restrict ourselves to the case

\[ \lambda=\xi=\nu=\omega=0 \tag{23} \]

and to show that, in this case, the expansion of \(r(2\pi,\sigma)-r_0\) in powers of \(a\) contains no terms below the second degree. Assuming the conditions (23) to be fulfilled and taking into account equalities (22) and (15), we write the difference \(r(2\pi,\sigma)-r_0\) in the form

\[ r(2\pi,\sigma)-r_0=a\psi(r_0,a). \tag{24} \]

Here \(\theta\) and \(\mu\) are regarded as fixed.

We shall prove that the expansion of the function \(\psi(r_0,a)\) in powers of \(a\) contains no terms below the first degree, that is, that

\[ \psi(r_0,0)=0 \tag{25} \]

for sufficiently small \(r_0\).

Under the conditions (23), system (6) can be written in the form

\[ \frac{dx}{dt}=-H_y'(x,y)+ap(x,y),\qquad \frac{dy}{dt}=H_x'(x,y)+aq(x,y), \tag{26} \]

where

\[ H(x,y)\equiv \frac{1}{2}(x^2+y^2)+\frac{1}{4}\mu x^4+\theta x^3y+ \]
\[ {}+\frac{1}{2}(\eta-3\mu)x^2y^2-\theta xy^3+\frac{1}{4}\mu y^4, \tag{27} \]

\[ p(x,y)\equiv x^3-2xy^2,\qquad q(x,y)\equiv 2x^2y-y^3. \tag{28} \]

For sufficiently small \(h\), consider the family of closed curves \(C_h\) with equation

\[ H(x,y)=h. \tag{29} \]

Let \(C_{h_0}\) be one of these curves. Introduce in its neighborhood new coordinates by the relations

\[ H(x,y)=h_0+\zeta,\qquad M(x,y;s)=0, \tag{30} \]

the latter of which, for each \(s\), determines a contact-free arc coinciding, for \(s=0\), with the segment of the half-line \(\varphi=0\), while \(s\) is the cyclic coordinate.

Let from (30) we have

\[ x=f_1(s,\zeta),\qquad y=f_2(s,\zeta). \tag{31} \]

Then

\[ x=f_1(s,0),\qquad y=f_2(s,0) \tag{32} \]

is the equation of the curve \(C_{h_0}\), and the identities

\[ H(f_1(s,\zeta),f_2(s,\zeta))\equiv h_0+\zeta,\qquad M(f_1(s,\zeta),f_2(s,\zeta),s)\equiv 0. \tag{33} \]

hold.

Passing in system (26) to the new coordinates (31). From (30), taking (26) into account, we have

\[ \frac{d\zeta}{dt}=a(pH'_x+qH'_y), \]

\[ \frac{ds}{dt}=\bigl[M'_x(H'_y-ap)-M'_y(H'_x+aq)\bigr](M'_s)^{-1}. \]

Hence, using the identities

\[ H'_x f'_{1s}+H'_y f'_{2s}=0,\qquad M'_x f'_{1s}+M'_y f'_{2s}=-M'_s, \]

\[ H'_x f'_{1\zeta}+H'_y f'_{2\zeta}=1,\qquad M'_x f'_{1\zeta}+M'_y f'_{2\zeta}=0, \]

obtained by differentiating (33), we arrive at the equation

\[ \frac{d\zeta}{ds} = -\frac{a(f'_{2s}p-f'_{1s}q)} {1-a(f'_{2\zeta}p-f'_{1\zeta}q)}. \tag{34} \]

Expanding the right-hand side of this equation in a series in powers of \(a\) and \(\zeta\), we obtain

\[ \frac{d\zeta}{ds} = (pf'_{2s}-qf'_{1s})_{\zeta=0}\,a + R^*(\zeta,s,a), \tag{35} \]

where \(R^*\) contains only terms of order higher than the first with respect to the pair of variables \(a\) and \(\zeta\).

We shall seek a solution of equation (35) in the form of a series in powers of \(a\) and of the initial value \(\zeta_0=\zeta(0)\):

\[ \zeta = g_{10}(s)\zeta_0 + g_{01}(s)a + g_{20}(s)\zeta_0^2 + g_{11}(s)\zeta_0 a + g_{02}(s)a^2+\cdots . \tag{36} \]

Substituting this series into (35), we find that

\[ g'_{01}(s)=(pf'_{2s}-qf'_{1s})_{\zeta=0}, \]

whence

\[ g_{01}(s)=\int_0^s (pf'_{2s}-qf'_{1s})_{\zeta=0}\,ds. \]

Let \(\tau\) be the increment of the cyclic coordinate \(s\) along the curve \(C_{h_0}\). Then, taking (32) into account,

\[ g_{01}(\tau) = \int_{C_{h_0}}(pf'_{2s}-qf'_{1s})\,ds = \int_{C_{h_0}}p\,dy-q\,dx. \tag{37} \]

Putting \(s=\tau\) in (36), we obtain, for the differential equation (34), the succession function

\[ \zeta(\tau,\sigma)-\zeta_0 = g_{01}(\tau)a + g_{11}(\tau)\zeta_0 a + g_{02}(\tau)a^2+\cdots . \tag{38} \]

The equalities \(g_{10}(\tau)=1,\ g_{20}(\tau)=g_{30}(\tau)=\cdots=0\) are satisfied because \(\zeta=\zeta_0\) is a solution of equation (34) when \(a=0\).

Let \(H(\sqrt{r_0^*},0)=h_0,\ H(\sqrt{\bar r_0},0)=h_0+\zeta_0\) and \(H(\sqrt{\bar r},0)=h_0+\zeta\), where \(r=r(2\pi,\sigma)\) is expressed by formula (24), and \(\zeta=\zeta(\tau,\sigma)\) by formula (38). Then from (27) we have

\[ \frac12 r_0^*+\frac14\mu r_0^{*2}=h_0, \tag{39} \]

\[ \frac{1}{2}r_0+\frac{1}{4}\mu r_0^2=h_0+\zeta_0, \tag{40} \]

\[ \frac{1}{2}r+\frac{1}{4}\mu r^2=h_0+\zeta . \tag{41} \]

From (39) and (40) we find

\[ \zeta_0=\frac{1}{2}r_0-\frac{1}{2}r_0^*+\frac{1}{4}\mu r_0^2-\frac{1}{4}\mu r_0^{*2}, \]

whence

\[ \zeta_0=\left(\frac{1}{2}+\frac{1}{2}\mu r_0^*\right)(r_0-r_0^*)+\frac{1}{4}\mu (r_0-r_0^*)^2 . \tag{42} \]

Similarly, from (40) and (41),

\[ \zeta-\zeta_0=\left(\frac{1}{2}+\frac{1}{2}\mu r_0\right)(r-r_0)+\frac{1}{4}\mu (r-r_0)^2, \]

whence

\[ \zeta-\zeta_0=\left[\frac{1}{2}+\frac{1}{2}\mu r_0^*+\frac{1}{2}\mu (r_0-r_0^*)\right](r-r_0)+\frac{1}{4}\mu (r-r_0)^2 . \tag{43} \]

Expanding the difference (24) in powers of \(a\) and \(r_0-r_0^*\), we obtain

\[ r-r_0=a\left[\psi(r_0^*,0)+\psi'_{r_0^*}(r_0^*,0)(r_0-r_0^*)+a\psi'_a(r_0^*,0)+\cdots\right]. \]

Substituting the right-hand side of this equality into (43) in place of \(r-r_0\), we find that the coefficient of \(a(r_0-r_0^*)^0\) in the expansion of \(\zeta-\zeta_0\) in powers of \(a\) and \(r_0-r_0^*\) is equal to \(\left(\frac{1}{2}+\frac{1}{2}\mu r_0^*\right)\psi(r_0^*,0)\). Substituting (42) into (38), we obtain that this same coefficient is equal to \(g_{01}(\tau)\). Thus, in order to prove the equality (25), it remains only to prove that \(g_{01}(\tau)=0\), independently of the choice of the curve \(C_{h_0}\), which is already not difficult to establish.

Indeed, taking (28) into account, from (37) we obtain

\[ g_{01}(\tau)=\int_{C_{h_0}}(x^3-2xy^2)\,dy+(y^3-2x^2y)\,dx . \]

In this integral, replacing \(x\) by \(y\), and \(y\) by \(-x\), entails a change of sign of the integrand. At the same time, from the equation of the curve \(C_{h_0}\) \((H(x,y)=h_0)\) it is clear (see (27)) that, independently of the choice of \(h_0\), if a point with coordinates \((x,y)\) belongs to this curve, then the point \((y,-x)\) also lies on \(C_{h_0}\). Hence it follows at once that \(g_{01}(\tau)=0\), independently of the choice of the curve \(C_{h_0}\) from the family (29). Thus it is proved that \(u_j(2\pi,\sigma)\), for all \(j>0\), have the form (21).

The fact that \(K_j^{(j+l)}=0\) for all \(l>1\), while \(K_j^{(j+1)}\) are constants, follows directly from Lemma 2.

To complete the proof of Lemma 3, it remains to carry out

\(^1\) We note that in [1, 2] it was likewise unnecessary to compute \(c_{02}(\tau)\), since the requirement that \(\theta_k^{***}\) contain \(\mu\) as a factor is equivalent to the requirement that the expansion of \(\rho-\rho_0\) in powers of \(\mu\) contain no terms below the second degree, and the latter is equivalent to the fulfillment of the equality \(c_{01}(\tau)=0\), independently of the choice of the curve \(C_{h_0}\).

1. Calculation of the constants \(K_j^{(j+1)}\) \((j=1,2,3,4,5)\). We shall prove that these constants are different from zero. Here, obviously, one may assume \(\lambda=0\). Then from (20) and (18) we have

\[ u_0(\varphi,\sigma)=1, \]

\[ u_j(\varphi,\sigma)=\int_0^\varphi \sum_{l=0}^{j-1} u_l\left(2\alpha u_{j-1-l}-\beta u'_{j-1-l}\right)\,d\varphi \quad (j\geqslant 1). \tag{44} \]

From the recurrence relations (44), using the formula for integration by parts, it follows that

\[ u_1(2\pi,\sigma)=2\int_0^{2\pi}\alpha\,d\varphi; \tag{45} \]

if \(u_1(2\pi,\sigma)=0\), then

\[ u_2(2\pi,\sigma)=-2\int_0^{2\pi}\alpha\beta\,d\varphi; \tag{46} \]

and if \(u_1(2\pi,\sigma)=u_2(2\pi,\sigma)=0\), then

\[ u_3(2\pi,\sigma)=-2\int_0^{2\pi}\alpha\beta\gamma\,d\varphi, \tag{47} \]

\[ u_4(2\pi,\sigma)=-2\int_0^{2\pi}\alpha\beta\gamma^2\,d\varphi, \tag{48} \]

\[ u_5(2\pi,\sigma) = -2\int_0^{2\pi} \left( \alpha\beta\gamma^3 - 2\alpha\beta\gamma\int_0^\varphi \alpha\beta\,d\varphi \right)d\varphi, \tag{49} \]

where

\[ \gamma\equiv 2\int_0^\varphi \alpha\,d\varphi-\beta. \]

Lemma 3 permits us to assert that \(K_1^{(2)}=u_1(2\pi,\sigma)\), if

\[ \lambda=\nu=\omega=\eta=\mu=\theta=a=0,\quad \xi=1; \tag{50} \]

\(K_2^{(3)}=u_2(2\pi,\sigma)\), if

\[ \lambda=\xi=\omega=\eta=\mu=\theta=0,\quad \nu=a=1; \tag{51} \]

\(K_3^{(4)}=u_3(2\pi,\sigma)\), if

\[ \lambda=\xi=\nu=\eta=\mu=0,\quad \omega=\theta=a=1; \tag{52} \]

\(K_4^{(5)}=u_4(2\pi,\sigma)\), if

\[ \lambda=\xi=\nu=\omega=\mu=0,\quad \eta=\theta=a=1; \tag{53} \]

\[ K_5^{(6)}=\frac{1}{3}\,u_5(2\pi,\sigma), \]

if

\[ \lambda=\xi=\nu=\omega=\eta=\mu=0,\quad \theta=a=1. \tag{54} \]

Since for \(\lambda=\xi=0\), \(u_1(2\pi,\sigma)=0\), and for \(\lambda=\xi=\nu=0\), \(u_2(2\pi,\sigma)=0\), to calculate \(K_j^{(j+1)}\) one may use formulas (45–49).

Under the conditions (50), from (10) we have

\[ \alpha=\cos^2\varphi\sin^2\varphi. \]

Then from (45) we obtain

\[ K_1^{(2)}=u_1(2\pi,\sigma)=2\int_0^{2\pi}\cos^2\varphi\sin^2\varphi\,d\varphi=\frac{\pi}{2}. \]

Under conditions (51), from (10) and (11) we find

\[ \alpha=\beta=\cos^4\varphi+\cos^3\varphi\sin\varphi+\cos\varphi\sin^3\varphi-\sin^4\varphi. \]

Then from (46) we have

\[ K_2^{(3)}=u_2(2\pi,\sigma)=-2\int_0^{2\pi}\alpha\beta\,d\varphi=-\frac{5\pi}{2}. \]

When conditions (52) are satisfied, from (10) and (11) we find

\[ \alpha=-\cos^4\varphi+6\cos^2\varphi\sin^2\varphi-\sin^4\varphi, \]

\[ \beta=9\cos^3\varphi\sin\varphi+\cos\varphi\sin^3\varphi. \]

Then \(\gamma=-11\cos^3\varphi\sin\varphi+\cos\varphi\sin^3\varphi\), and from (47) we have

\[ K_3^{(4)}=u_3(2\pi,\sigma)=-2\int_0^{2\pi}\alpha\beta\gamma\,d\varphi=\frac{25\pi}{4}. \]

In the case (53), from (10) and (11) it follows that

\[ \alpha=-\cos^3\varphi\sin\varphi+6\cos^2\varphi\sin^2\varphi+\cos\varphi\sin^3\varphi-2\sin^4\varphi, \]

\[ \beta=5\cos^3\varphi\sin\varphi+2\cos^2\varphi\sin^2\varphi-3\cos\varphi\sin^3\varphi. \]

Then, taking into account that

\[ 2\int_0^{\varphi}(6\cos^2\varphi\sin^2\varphi-2\sin^4\varphi)\,d\varphi=4\cos\varphi\sin^3\varphi, \]

we obtain

\[ \gamma=-\frac{1}{2}\cos^4\varphi-5\cos^3\varphi\sin\varphi-2\cos^2\varphi\sin^2\varphi+ \]

\[ +7\cos\varphi\sin^3\varphi+\frac{1}{2}\sin^4\varphi, \]

and from (48) we find

\[ K_4^{(5)}=u_4(2\pi,\sigma)=-2\int_0^{2\pi}\alpha\beta\gamma^2\,d\varphi=\frac{5\pi}{16}. \]

If conditions (54) are satisfied, then

\[ \alpha=6\cos^2\varphi\sin^2\varphi-2\sin^4\varphi, \]

\[ \beta=5\cos^3\varphi\sin\varphi-3\cos\varphi\sin^3\varphi. \]

In this case, on the basis of (55),

\[ \gamma=-5\cos^3\varphi\sin\varphi+7\cos\varphi\sin^3\varphi, \]

\[ \int_0^\varphi \alpha\beta\,d\varphi = 8\sin^8\varphi-\frac{44}{3}\sin^6\varphi+\frac{15}{2}\sin^4\varphi \]

and from (49) we obtain that

\[ K_5^{(6)}=\frac{1}{3}u_5(2\pi,\sigma) =-\frac{2}{3}\int_0^{2\pi}\left(\alpha\beta\gamma^3-2\alpha\beta\gamma\int_0^\varphi \alpha\beta\,d\varphi\right)d\varphi =-\frac{5\pi}{48}. \]

This completes the proof of Lemma 3.

Since, in order for the origin to be a center of system (6), the equalities \(u_j(2\pi,\sigma)=0\) \((j=1,2,3,4,5)\), \(u_0(2\pi,\sigma)=1\), must in particular hold, it follows directly from Lemmas 1 and 3 that

Corollary 1. For the origin to be a center of system (6), it is necessary and sufficient that at least one of the three series of conditions (7) be satisfied.

This result agrees with the result of K. E. Malkin [9]. Passing from system (6) back to system (1), one could express the conditions for the origin to be a center of system (1) directly in terms of its coefficients \(b_{jl}, c_{jl}\); however, in the present paper we shall not do this.

6. Main theorem. On the basis of Lemma 3, the displacement function (15) can be written as follows:

\[ r(2\pi,\sigma)-r_0=\Phi_0(r_0,\sigma)\,r_0, \]

where

\[ \begin{aligned} \Phi_0(r_0,\sigma)\equiv{}&4\pi\lambda(1+\lambda\Phi^*+r_0\Psi^*)+K_1^{(2)}\xi(1+r_0\Psi_1)r_0+\\ &+K_2^{(3)}a\nu(1+r_0\Psi_2)r_0^2+K_3^{(4)}a\theta_\omega(1+r_0\Psi_3)r_0^3+\\ &+K_4^{(5)}a^2\theta_\eta(1+r_0\Psi_4)r_0^4+K_5^{(6)}a^2\theta_\chi(1+r_0\Psi_5)r_0^5, \end{aligned} \tag{56} \]

\(\Psi^*, \Psi_j\) are series in powers of \(r_0\) with coefficients in the form of integral functions of the last \(8-j\) coordinates of the point \(\sigma\), and

\[ 4\pi\lambda(1+\lambda\Phi^*)\equiv e^{4\pi\lambda}-1. \tag{57} \]

As is easy to see, the following holds.

Lemma 4. Suppose that in the domain

\[ r_0\in(0,\delta_0),\quad \sigma\in S(\sigma_0,\varepsilon_0) \tag{58} \]

the function \(\Phi(r_0,\sigma)\) expands in a series in powers of \(r_0\) with coefficients in the form of integral functions of the coordinates of the point \(\sigma\), and that the term independent of \(r_0\) at the point \(\sigma_0\) is not equal to zero. Then, by decreasing \(\delta_0\) and \(\varepsilon_0\), one can arrange that throughout the entire domain (58) the sign of the function \(\Phi(r_0,\sigma)\) coincides with the sign of its free term at the point \(\sigma_0\).

Corollary 2. For any point \(\sigma_0\in \widetilde E\), whose first coordinate \(\lambda_0\ne 0\), the origin of the phase plane \(XOY\) has cyclicity 0 relative to \(\widetilde E\).

Indeed, when \(\lambda_0\ne 0\), on the basis of Lemma 4 the domain (58) can be chosen so that in it

\[ \operatorname{sgn}\Phi_0(r_0,\sigma)=\operatorname{sgn}4\pi\lambda_0(1+\lambda_0\Phi^*). \]

In this case it follows from (57) that in the domain (58)

\[ \operatorname{sgn}\Phi_0(r_0,\sigma)=\operatorname{sgn}\lambda_0 . \]

Thus, in (58) the function (56) does not vanish and, consequently, whatever the point \(\sigma\in S(\sigma_0,\varepsilon_0)\), in the domain \(S(0,\sqrt{\tilde\delta_0})\) there is not a single limit cycle of system (6). The cyclicity of the point \(O\) is in this case equal to zero.

Theorem 1. In \(\widetilde E\) there are no points for which the origin of coordinates of the phase plane \(XOY\) would have, with respect to \(\widetilde E\), cyclicity order greater than five.

Proof. Consider an arbitrary point \(\sigma_0\in \widetilde E\) with coordinates \(\lambda_0,\xi_0,\nu_0,\omega_0,\eta_0,\mu_0,\theta_0,a_0\). As Corollary 2 shows, it suffices to restrict ourselves to the case when \(\lambda_0=0\). In this case one may always assume that in the domain (58) the function \((1+\lambda\Phi^*)^{-1}\) is an integral function of \(\lambda\), and the function \(\Phi_0(r_0,\sigma)\) can be written as follows:

\[ \Phi_0(r_0,\sigma)=\sum_{j=0}^{5} Q_j(\sigma)Z_j^{(0)}(r_0,\sigma)r_0^j, \tag{59} \]

where

\[ Q_0\equiv e^{4\pi\lambda}-1,\qquad Q_1\equiv K_1^{(2)}\xi,\qquad Q_2\equiv K_2^{(3)}a\nu, \]

\[ Q_3\equiv K_3^{(4)}a\theta\omega,\qquad Q_4\equiv K_4^{(5)}a^2\theta\eta,\qquad Q_5\equiv K_5^{(6)}a^2\theta\mu, \]

and \(Z_j^{(0)}(r_0,\sigma)\equiv 1+r_0\Psi_j\) are series in powers of \(r_0\) with coefficients in the form of integral functions of the last \(8-j\) coordinates of the point \(\sigma\), and with constant terms, independent of \(r_0\), equal to 1.

Along with the function (59), consider in the domain (58) the functions

\[ \Phi_l(r_0,\sigma)=\frac{d}{dr_0} \left\{[Z_{l-1}^{(l-1)}(r_0,\sigma)]^{-1}\Phi_{l-1}(r_0,\sigma)\right\}= \]

\[ =\sum_{j=l}^{5} n_j^{(l)}Q_j(\sigma)Z_j^{(l)}r_0^{j-l} \qquad (l=1,2,3,4,5). \tag{60} \]

Here \(n_j^{(l)}\) are natural numbers, and if the domain (58) is chosen so that in it

\[ |Z_l^{(l)}-1|<\frac12 \qquad (l=0,1,2,3,4,5), \tag{61} \]

then the \(Z_j^{(l)}\) may be regarded as series in powers of \(r_0\) with coefficients in the form of integral functions of the coordinates of the point \(\sigma\) and with constant terms, independent of \(r_0\), equal to 1.

If at the point \(\sigma\in S(\sigma_0,\varepsilon_0)\) all \(Q_j\) \((j=0,1,2,3,4,5)\) are equal to zero, then in \((0,\tilde\delta_0)\) the function (59) is identically equal to zero, and all integral curves of system (6) situated in \(S(0,\sqrt{\tilde\delta_0})\) are closed. Then in this domain there are no limit cycles.

Suppose now that the expressions

\[ Q_0,\ Q_1,\ Q_2,\ Q_3,\ Q_4,\ Q_5 \tag{62} \]

at the point \(\sigma\) are not all equal to zero, and in the row (62) the last expression different from zero is \(Q_l\) \((0\le l\le 5)\). Then, by virtue of (61), the function \(\Phi_l(r_0,\sigma)\) does not vanish for any \(r_0\in(0,\tilde\delta_0)\).

In this case, obviously, in \((0,\delta_0)\) the function \(\Phi_{l-1}(r_0,\sigma)\) has at most one zero, the function \(\Phi_{l-2}(r_0,\sigma)\) has at most two zeros, and so on, while the function \(\Phi_0(r_0,\sigma)\) has at most \(l\) zeros. Hence the number of limit cycles of system (6) contained in the domain \(S(0,\sqrt{\bar\delta_0})\) cannot exceed \(l\) \((l\leq 5)\).

Thus, whatever the point \(\sigma\in S(\sigma_0,\varepsilon_0)\), the system (6) corresponding to it has in \(S(0,\sqrt{\bar\delta_0})\) at most five limit cycles. Consequently, in \(\widetilde E\) there are no points for which the order of cyclicity of the origin of the phase plane \(XOY\) with respect to \(\widetilde E\) would be greater than five. Theorem 1 is proved.

7. The number of limit cycles. We have

Theorem 2. Whatever the number \(k=0,1,2,3,4,5\), in \(\widetilde E\) there exists a point for which the origin \(O\) of the phase plane \(XOY\) has, with respect to \(\widetilde E\), cyclicity of order \(k\).

Proof. Taking into account Corollary 2, it is sufficient to restrict ourselves to the cases where \(k>0\).

Fix a natural number \(k\) satisfying the conditions \(1\leq k\leq 5\), and consider a point \(\sigma_0\) at which the first \(k\) expressions (62) are equal to zero, while \(Q_k\neq 0\). In this case the first \(k\) coordinates of the point \(\sigma_0\) are equal to zero. We shall show that for \(\sigma_0\) the origin of the phase plane \(XOY\) has, with respect to \(\widetilde E\), cyclicity of order \(k\). On the basis of Lemma 4 and the condition \(Q_k\neq 0\), the domain (58) can be chosen so that in it the function \(\Phi_k(r_0,\sigma)\) does not vanish. Then for any point \(\sigma\in S(\sigma_0,\varepsilon_0)\) the function \(\Phi_0(r_0,\sigma)\) has in \((0,\delta_0)\) at most \(k\) zeros.

On the other hand, whatever \(\varepsilon>0\) and \(\delta>0\) such that \(\varepsilon<\varepsilon_0,\ \delta<\delta_0\), by changing the first \(k\) coordinates of the point \(\sigma_0\) one can always obtain a point \(\sigma\in S(\sigma_0,\varepsilon)\) for which the function \(\Phi_0(r_0,\sigma)\) has \(k\) zeros in \((0,\delta)\). Indeed, choose in \(S(\sigma_0,\varepsilon)\) an arbitrary point \(\sigma\) for which the last \(8-k\) coordinates coincide with the coordinates of the point \(\sigma_0\), and the signs of the first \(k\) coordinates are such that

\[ \operatorname{sgn} Q_0=-\operatorname{sgn} Q_1=\operatorname{sgn} Q_2=\cdots=(-1)^k\operatorname{sgn} Q_k . \tag{63} \]

We now find \(\delta_1>0\) such that \(\delta_1<\delta\) and all integral curves of system (6) with initial conditions \(r_i<\delta_1\) do not leave, as \(\varphi\) varies from \(0\) to \(2\pi\), the set \(S(0,\sqrt{\delta})\), and then choose successively the points

\[ 0<r_{k+1}<r_k<\cdots<r_2<r_1<\delta_1 \tag{64} \]

and fix the first \(k\) coordinates of the point \(\sigma\). If the points \(r_g\) \((g=1,2,\ldots,l)\) have already been chosen and the last \(l\) of the first \(k\) coordinates of the point \(\sigma\) have been fixed, then the point \(r_{l+1}\) is chosen so that

\[ \operatorname{sgn}\sum_{j=k-l}^{5} Q_j(\sigma) Z_j^{(0)}(r_{l+1},\sigma) r_{l+1}^{j+1} =\operatorname{sgn} Q_{k-l}(\sigma). \tag{65} \]

Such a choice can be made, since for \(k-l\leq j\leq 5\) the expressions \(Q_j(\sigma)\) and \(Z_j^{(0)}(r,\sigma)\) do not depend on the first (as yet not fixed) \(k-l\) coordinates of the point \(\sigma\). After this we fix the \((k-l)\)-th coordinate of the point \(\sigma\) so that at the already chosen points \(r_g\) one has

\[ \left|Q_{k-l-1}(\sigma) Z_{k-l-1}^{(0)}(r_g,\sigma) r_g^{k-l}\right| < \frac{1}{k-g+1} \left| \sum_{j=k-g+1}^{5} Q_j(\sigma) Z_j^{(0)}(r_g,\sigma) r_g^j \right| \quad (g=1,2,\ldots,l+1). \tag{66} \]

The latter is easy to do, since in (66) the right-hand sides do not depend on the fixed coordinate, while in the left-hand sides of (66) this coordinate enters into \(Q_{k-l-1}(\sigma)\) as a factor.

We continue the process of successively choosing a point from \((0,\hat\delta_1)\) and the value of the coordinate of the point \(\sigma\) until all the points (64) have been chosen and the values of all coordinates of the point \(\sigma \in S(\sigma_0,\varepsilon)\) have been fixed. As a result, by (59), we obtain

\[ \operatorname{sgn}\Phi_0(r_g,\sigma)= \]

\[ =\operatorname{sgn}\left[ \sum_{j=0}^{k-g} Q_j(\sigma) Z_j^{(0)}(r_g,\sigma) r_g^j + \sum_{j=k-g+1}^{5} Q_j(\sigma) Z_j^{(0)}(r_g,\sigma) r_g^j \right]. \]

Here, on the basis of (66), the absolute value of each of the \(k-g+1\) terms of the first sum is less than \(\dfrac{1}{k-g+1}\) of the absolute value of the second sum. Therefore

\[ \operatorname{sgn}\Phi_0(r_g,\sigma) = \operatorname{sgn}\sum_{j=k-g+1}^{5} Q_j(\sigma) Z_j^{(0)}(r_g,\sigma) r_g^j, \]

whence, by (65),

\[ \operatorname{sgn}\Phi_0(r_g,\sigma) = \operatorname{sgn}Q_{k-g+1}(\sigma) \qquad (g=1,2,\ldots,k+1). \]

Taking (63) into account, it follows that the function \(\Phi_0(r_0,\sigma)\) changes sign \(k\) times in the interval \((0,\hat\delta_1)\), and, consequently, it has \(k\) zeros in this interval. Then in the domain \(S(0,\sqrt{\hat\delta})\) the system (6) has exactly \(k\) limit cycles, and for the point \(\sigma_0\) the origin \(O\) has, relative to \(\tilde E\), cyclicity order \(k\). In particular, for a point \(\sigma_0 \in \tilde E\) the origin \(O\) has, relative to \(\tilde E\), cyclicity of order 5 if at the point \(\sigma_0\) the first five expressions (62) vanish, and \(Q_5 \ne 0\), that is,
\(\lambda_0=\xi_0=\nu_0=\omega_0=\eta_0=0\), and \(\alpha_0\beta_0\chi_0 \ne 0\).

From Theorems 1 and 2 it follows that

Theorem 3. \(N(\{3\})=5\).

References

1. N. N. Bautin, DAN SSSR, 24, no. 7, 1939, pp. 668—671.

2. N. N. Bautin, On the number of limit cycles appearing under variation of the coefficients from an equilibrium state of focus or center type. Matematicheskii sbornik, 30, no. 1, 1952, pp. 181—196.

3. N. F. Otrokov, DAN SSSR, 43, no. 3, 1944, pp. 102—103.

4. N. F. Otrokov, On the number of limit cycles of a differential equation in a neighborhood of a singular point. Matematicheskii sbornik, 34, issue 1, 1954, pp. 127—144.

5. B. M. Peretyagin, DAN SSSR, 114, no. 1, 1957, pp. 29—32.

6. B. M. Peretyagin, Uchenye zapiski Smolenskogo pedagogicheskogo instituta, issue 10, 1962, pp. 67—88.

7. I. G. Petrovskii, E. M. Landis, DAN SSSR, 102, no. 1, 1955, pp. 29—32.

8. I. G. Petrovskii, E. M. Landis, Matematicheskii sbornik, 37, issue 2, 1955, pp. 209—250.

9. K. E. Malkin, Volzhskii matematicheskii sbornik, issue 2, 1964, pp. 87—91.

10. M. I. Al’mukhamedov, Kazan, Izvestiya fiziko-matematicheskogo obshchestva, 9, series 3, 1937, pp. 105—121.

11. N. A. Sakharnikov, PMM, 14, issue 6, 1950, pp. 651—658.

Received by the editors
October 5, 1964

Institute of Mathematics
of the Computing Center of the Academy of Sciences of the Moldavian SSR