INTEGRAL CURVES OF A CERTAIN DIFFERENTIAL EQUATION

N. A. LUKASHEVICH

Consider the differential equation

\[ \frac{dy}{dx}= \frac{\displaystyle \sum_{i+j=0}^{2} a_{ij}x^i y^j} {\displaystyle \sum_{i+j=0}^{2} b_{ij}x^i y^j}. \tag{1} \]

Suppose that (1) has a singular point of center type. We shall assume that the latter is located at the origin. In [1] it is proved that a center for equation (1) can be expected only in the case when the roots of the characteristic equation, composed in order to determine the character of the singular point under study, are purely imaginary. By means of a nonsingular linear transformation, equation (1) can always be reduced to the form

\[ \frac{dy}{dx} = - \frac{x+ax^2+(2b+\alpha)xy+cy^2} {y+bx^2+(2c+\beta)xy+dy^2}. \tag{2} \]

According to [2], \((0,0)\) will be a center for equation (2) if at least one of the following six conditions is satisfied:

\[ \begin{aligned} &1)\ \alpha=\beta=0;\\ &2)\ a=c=\beta=0;\\ &3)\ ak^3-(3b+\alpha)k^2+(3c+\beta)k-d=0,\quad k=\frac{\alpha}{\beta}=\frac{b+d}{a+c};\\ &4)\ a+c=0,\quad b+d=0;\\ &5)\ b+d=\alpha=\beta+5(a+c)=ac+2a^2+d^2=0,\quad a+c\ne0;\\ &6)\ a_1+c_1=\beta_1=\alpha_1+5(b_1+d_1)=b_1d_1+2d_1^2+a_1^2=0,\quad b+d\ne0, \end{aligned} \tag{A} \]

where

\[ \begin{aligned} a_1={}&a\cos^3\varphi+(3b+\alpha)\cos^2\varphi\sin\varphi +(3c+\beta)\cos\varphi\sin^2\varphi+d\sin^3\varphi,\\ b_1={}&b\cos^3\varphi+(2c-a+\beta)\cos^2\varphi\sin\varphi\\ &\quad +(d-2b-\alpha)\cos\varphi\sin^2\varphi-c\sin^3\varphi,\\ c_1={}&c\cos^3\varphi+(d-2b-\alpha)\cos^2\varphi\sin\varphi\\ &\quad +(a-2c-\beta)\cos\varphi\sin^2\varphi+b\sin^3\varphi, \end{aligned} \tag{3} \]

\[ d_1=d\cos^3\varphi-(3c+\beta)\cos^2\varphi\sin\varphi+(3b+\alpha)\cos\varphi\sin^2\varphi-a\sin^3\varphi, \]

\[ \alpha_1=\alpha\cos\varphi-\beta\sin\varphi,\quad \beta_1=\alpha\sin\varphi+\beta\cos\varphi, \]

and in equation (2) one should put

\[ x=x_1\cos\varphi-y_1\sin\varphi,\quad y=x_1\sin\varphi+y_1\cos\varphi . \tag{4} \]

We note that the conditions \(A_3\) are equivalent to the conditions \(a_1=c_1=\beta_1=0\). Similarly, the conditions \(A_6\) are equivalent to the conditions \(A_5\). Thus, in order to determine the behavior of the integral curves as a whole for equation (2) when there is a center at \((0,0)\), it suffices to consider (2) under the conditions \(A_1,A_2,A_4\), and \(A_5\).

In what follows we agree to denote the \(\nu\)-th singular point located in the finite part of the plane by \(M_\nu\), where \(M_c,M_y,M_\phi,M_{\mathrm{ц}},M_{\mathrm{ос}у},M_{\mathrm{зс}у}\), respectively, mean that \(M_\nu\) is a saddle, node, focus, center, open saddle-node, closed saddle-node. In an analogous way we shall denote infinitely distant singular points \(N_\nu\), located on the equator of the Poincaré sphere.

A simply connected domain \(G\) containing only one singular point of center type and entirely filled with closed integral curves will be called a center domain.

1. Suppose that \(\alpha=\beta=0\). To simplify the subsequent calculations, in equation (2) we shall assume \(b=0\). The latter, as is easily seen from (3), does not restrict the generality of the investigation. From the system

\[ y(1+2cx+dy)=0,\quad x+ax^2+cy^2=0 \tag{5} \]

we see that the equation

\[ \frac{dy}{dx}=-\,\frac{x+ax^2+cy^2}{y+2cxy+dy^2} \tag{6} \]

has the single singular point \((0,0)\) only in the case when \(a=c=d=0\).

Let one of the following conditions hold:

\[ \begin{gathered} 1)\ a\ne0,\ d=c=0;\quad 2)\ d=0,\ a=2c\ne0;\\ 3)\ ad\ne0,\ a=2c;\quad 4)\ a=c=0,\ d\ne0, \end{gathered} \tag{7} \]

i.e., equation (6) has only two singular points \(M_0(0,0)\) and \(M_1\). The characteristic equation composed for determining the character of the singular point \(M_1\) in cases 1, 3, and 4 of conditions (7) has the form \(\lambda^2=p^2\) and \(\lambda^2=0\) in case 2 \((p\ne0\) is a real number). According to the results of [3], \(M_1\) is a saddle if \(a=2c\ne0,\ d=0\). Thus, if equation (6) has only two singular points \(M_\nu\), then one of them is in all cases a saddle.

To investigate the behavior of the integral curves in the infinitely distant part of the plane, it is necessary to determine the possible number and character of infinitely distant singular points. To this end we put [4]

\[ x=\frac{1}{z},\quad y=\frac{\tau}{z}. \tag{8} \]

In the variables \(z,\tau\), equation (6) has the form

\[ \frac{d\tau}{dz}= \frac{z(1+\tau^2)+a+3c\tau^2+d\tau^3} {z(2c\tau+d\tau^2+\tau z)} . \tag{9} \]

The transformation (8) makes it possible to investigate neighborhoods of all points of the equator of the Poincaré sphere, except those situated in the direction of the \(OY\) axis. To study neighborhoods of the latter, one carries out the transformation

\[ x=\frac{\xi}{z},\quad y=\frac{1}{z}. \tag{10} \]

In the variables \(\xi, z\), equation (6) has the form

\[ \frac{d\xi}{dz}= \frac{z(1+\xi^2)+d+3c\xi+a\xi^3} {z(c+a\xi^2+\xi z)}. \tag{11} \]

When conditions (7) are satisfied, in the infinitely distant part of the plane there is only one singular point \(N_{1y}\).

According to the results of [5], the center domain for equation (1) is always open. Consequently, the behavior of the integral curves as a whole for equation (6), which has only two singular points \(M_\nu\), corresponds to Fig. 2, ж1.

Suppose that equation (6) has only three singular points.

In this case the coefficients of the equation must satisfy one of the following relations:

\[ \begin{gathered} 1)\ a=0,\ c\ne0;\quad 2)\ adc\ne0,\ ad^2+4c^3=0;\\ 3)\ adc\ne0,\ ad^2+4c^3\ne0;\ d^2+8c^2-4ac=0. \end{gathered} \tag{12} \]

Let \(a=0,\ c\ne0\). The characteristic equation, composed for determining the character of the singular points \(M_1\) and \(M_2\), distinct from \(M_0(0,0)\), has the form

\[ \lambda^2=\pm \sqrt{d^2+8c^2} \left( \frac{d\pm\sqrt{d^2+8c^2}}{4c^2} \right). \tag{13} \]

From (13) it is easy to conclude that \(M_1\) and \(M_2\) are saddles. In the infinitely distant part of the plane we have \(N_1(\tau=z=0)\) and \(N_2\left(\xi=-\dfrac{d}{3c},\ z=0\right)\).

According to the results of [3], one of them is \(N_{3cy}\), and the other is \(N_y\).

Let us determine the behavior of the separatrices of the saddles \(M_c\). The general integral of equation (6) has the form

\[ \Phi(x,y)=\frac{x^2+y^2}{2}+\frac{ax^3+dy^3}{3}+cxy^2=C. \tag{14} \]

Taking (5) into account, it is not difficult to verify that the equations of the separatrices of the saddles have the form

\[ \Phi(x,y)=\frac{1}{6}\left(x_i^2+y_i^2\right), \tag{15} \]

where \(x_i, y_i\) are the coordinates of the singular points \(M_1\) and \(M_2\). Since \(x_1^2+y_1^2=x_2^2+y_2^2\) only in the case when \(d=0\), the behavior of the integral curves as a whole for equation (6), which has only three

special points \(M_\nu\), corresponds to Fig. 1, a, if \(d=0\), and to Fig. 1, b, if \(d\ne 0\).

Let \(ac\ne 0,\ ad^2+4c^3=0\). We have \(x_1=-\dfrac{1}{a},\ y_1=0\)—the coordinates of the singular point \(M_1\), and \(x_2=-\dfrac{c}{d^2+4a^2},\ y_2=-\dfrac{d^2+2c^2}{d(d^2+4a^2)}\)—the coordinates of the point \(M_2\). It is easy to show that \(M_1\) and \(M_2\) are \(M_c\). Put \(d^2=kc^2\). The condition \(x_1^2+y_1^2=x_2^2+y_2^2\) takes the form

\[ k^5+8k^4+16k^3-16k^2-80k-64=0. \tag{16} \]

Equation (16) has only one positive root, \(k=2\). Let us determine the possible number and type of singular points \(N_\nu\). Since the discriminant of the equation

\[ a\xi^3+3c\xi+d=0 \tag{17} \]

is equal to zero, in the infinitely distant part of the plane there are only two singular points
\(N_1\left(\xi=-\dfrac{d}{c},\ z=0\right)\) and
\(N_2\left(\xi=\dfrac{d}{2c},\ z=0\right)\), with \(N_{1y}\) and \(N_{23:y}\). Therefore, the behavior of the integral curves as a whole corresponds to Fig. 1, a, if \(k=2\), and to Fig. 1, b, if \(k\ne 2\).

Fig. 1

Finally, suppose that the third condition (12) holds. The point \(M_1\) remains a saddle. As for the other singular point, according to the results of [3], it is a two-separatrix saddle. Since the discriminant of equation (17) is less than zero, there is only one point \(N_1(\xi_0,0)\), for which the roots of the characteristic equation are

\[ \mu_1=a\xi_0^2+c,\qquad \mu_2=3\mu_1. \]

We shall show that \(N_1\) is a node. Indeed, if \(\mu_1=0\), then \(\xi_0^2=-\dfrac{c}{a}\). Since \(\xi_0\) is a root of equation (17), it must be that \(2\xi_0 c+d=0\), which is impossible, because \(ad^2+4c^3\ne 0\).

The qualitative picture of the behavior of the integral curves as a whole corresponds to Fig. 1, c.

Suppose that (6) has four singular points \(M_\nu\). Clearly, all of them are simple.

Three cases are possible:

\[ \begin{gathered} 1)\ acd\ne 0,\quad ad^2+4c^3\ne 0;\quad 8c^2+d^2-4ac>0;\\ 2)\ ad\ne 0,\quad c=0;\quad 3)\ ac\ne 0,\quad d=0,\quad c(2c-a)>0. \end{gathered} \tag{18} \]

Let the first of the conditions (18) hold. To simplify the exposition, put

\[ x_1=cx,\qquad y_1=cy,\qquad a=lc,\qquad d=hc. \tag{19} \]

Equation (6) takes the form

\[ \frac{dy_1}{dx_1}=-\frac{x_1+lx_1^2+y_1^2}{y_1(1+2x_1+hy_1)},\quad hl\ne0,\quad l\ne2. \tag{20} \]

The singular points will be \(M_0(0,0)\), \(M_1\left(-\frac{1}{l},0\right)\),

\[ M_2\left[ \frac{-(h^2+4)+h\sqrt{h^2-4l+8}}{2(h^2+4)},\, -\frac{\sqrt{h^2-4l+8}}{h(h^2+8)} \right], \]

\[ M_3\left[ -\frac{(h^2+4)+h\sqrt{h^2-4l+8}}{2(h^2+4)},\, \frac{\sqrt{h^2-4l+8}}{h(h^2+4)} \right]. \]

It is easy to see that \(M_{1\text{ц}}\), if \(0<l<2\), and \(M_{1\text{с}}\), if \(0>l>2\). The characteristic equation constructed to determine the character of \(M_2\) and \(M_3\) has the form
\(\lambda^2=1+2(l-1)x_i,\ i=2,3\) (\(x_i\) is the abscissa of the point \(M_2\) or \(M_3\)).

It is not difficult to verify that

a) if \(l<0,\ lh^2+4>0\) or \(0<l<2\), then \(M_{2\text{с}}\) and \(M_{3\text{с}}\);

b) if \(l<0,\ lh^2+4>0\) or \(l>2\), then \(M_{2\text{с}}\), and \(M_{3\text{ц}}\).

Moreover, \(x_2^2+y_2^2\ne x_3^2+y_3^2\) for all values of the parameters \(l\) and \(h\) admissible in the present case.

Consider the equations

\[ f(x)\equiv(4+lh^2)x^2+(4+h^2)x+1=0 \tag{21} \]

and

\[ g(x)\equiv(4+lh^2)x^2+4x+\frac{l^2-h^2}{l^2}=0, \tag{22} \]

where \(\Phi(x_1,y_1)=f(x_1)\), \(\Phi(x_i,y_i)=g(x_i)\) \((i=2,3)\). If one of the roots of equation (21) is a root of equation (22), then the separatrices of \(M_{1\text{с}}\) will be separatrices of \(M_{2\text{с}}\) or \(M_{3\text{с}}\), depending on whether \(x_1=x_2\) or \(x_1=x_3\). If, however, neither of the two roots of equation (21) is a root of equation (22), then the separatrices of \(M_{1\text{с}}\) will not be separatrices of \(M_{2\text{с}}\) and \(M_{3\text{с}}\).

The case in which both roots of equation (21) are roots of equation (22) is excluded, since \(h^2-4l+8>0\). We have

\[ R(f,g)=\frac{h^4}{l^4}(l-2)^3[(l-2)(l+1)^2-k^2l], \tag{23} \]

where \(R(f,g)\) is the resultant of equations (21) and (22).

For admissible values of the parameters \(l\) and \(h\), \(R(f,g)=0\) only in the case when

\[ h^2=\frac{(l-2)(l+1)^2}{l}. \tag{24} \]

Suppose that condition (24) is satisfied. It is easy to see that
\(M_2\left[-\frac{1}{l(l-1)},-\frac{1}{(l+2)(l-1)}\right]\). Since
\(x_2=-\frac{1}{l(l-1)}\) is a root of equation (22), the separatrices of \(M_{1\text{с}}\) will simultaneously be separatrices of \(M_{2\text{с}}\). Since \(x_2\ne x_3\), the separatrices of \(M_{1\text{с}}\) are not simultaneously separatrices of \(M_{3\text{с}}\).

Obviously, if \(\delta=-27l^{-3}(lh^2+4)>0\), then there are three points \(N_\nu\); if \(\delta<0\), then there is only one, \(N_1\).

Let \(\delta>0\). All \(N_\nu\) \((\nu=1,2,3)\) are simple. According to Poincaré’s theorem [4] on the complete distribution of singular points, we must have

\[ N+F+F'+N'=C+C'+1, \tag{25} \]

where \(N\), \(F\), and \(C\) are, respectively, the numbers of nodes, foci, and saddles situated in the finite part of the plane, and \(N'\), \(F'\), and \(C'\) are those in the infinitely distant part of the plane.

From (25) we see that if \(M_{i\mathrm{c}}\) \((i=1,2,3)\), then \(N_{i\mathrm{u}}\) \((i=1,2,3)\). In an analogous way it is easy to verify that if in the finite part of the plane there are two singular points of center type, then \(\delta<0\), and in the infinitely distant part of the plane there is only one singular point—a node.

The qualitative picture of the behavior of the integral curves as a whole is equivalent to the picture shown in Fig. 1, g, if \(lh^2\ne (l-2)(l+1)^2\) and \(M_{i\mathrm{c}}(i=1,2,3)\); to Fig. 1, d, if \(lh^2=(l-2)(l+1)^2\) and \(M_{i\mathrm{c}}(i=1,2,3)\); to Fig. 1, e, if \(M_{1\mathrm{u}}, M_{2\mathrm{c}}, M_{3\mathrm{c}}\), or \(M_{1\mathrm{c}}, M_{2\mathrm{c}}, M_{3\mathrm{u}}\), and \(lh^2\ne (l-2)(l+1)^2\); and to Fig. 1, zh, if \(M_{1\mathrm{u}}, M_{2\mathrm{c}}, M_{3\mathrm{c}}\), or \(M_{1\mathrm{c}}, M_{2\mathrm{c}}, M_{3\mathrm{u}}\), and \(lh^2=(l-2)(l+1)^2\).

Let \(ad\ne 0\), \(c=0\). The singular points will be \(M_0(0,0)\), \(M_1\left(-\dfrac{1}{a},0\right)\),

\[ M_2\left(0,-\frac{1}{d}\right) \quad\text{and}\quad M_3\left(-\frac{1}{a},-\frac{1}{d}\right). \]

It is easy to verify that \(M_{1\mathrm{c}}\) and \(M_{2\mathrm{c}}, M_{3\mathrm{u}}\); moreover, if \(a\ne d\), then \(x_1^2+y_1^2\ne x_2^2+y_2^2\), and the qualitative picture of the behavior of the integral curves as a whole is equivalent to Fig. 1, e, if \(a\ne d\), and to Fig. 1, zh, if \(a=d\).

Let \(ac\ne 0\), \(d=0\), \(c(2c-a)>0\). We have \(M_0(0,0)\), \(M_1\left(-\dfrac{1}{a},0\right)\),

\[ M_2\left(-\frac{1}{2c},-\frac{1}{2}\sqrt{\frac{2c-a}{c^3}}\right), \quad M_3\left(-\frac{1}{2c},\frac{1}{2}\sqrt{\frac{2c-a}{c^3}}\right), \]

with \(M_{2\mathrm{c}}\) and \(M_{3\mathrm{c}}\). As for \(M_1\), it is \(M_{1\mathrm{c}}\) if \(ac<0\), and \(M_{1\mathrm{u}}\) if \(ac>0\). The equation of the separatrices \(M_{2\mathrm{c}}\) and \(M_{3\mathrm{c}}\) is

\[ (2cx+1)\left[\frac{a}{c}x^2+3y^2+\frac{3c-a}{2c^2}x-\frac{3c-a}{4c^3}\right]=0. \]

It is not difficult to verify that in the infinitely distant part of the plane there are \(N_{i\mathrm{u}}(i=1,2,3)\) if \(ac<0\), and \(N_{1\mathrm{u}}\) if \(ac>0\). Thus, the qualitative picture of the behavior of the integral curves as a whole is equivalent to Fig. 1, g, if \(ac<0\), and to Fig. 1, zh, if \(ac>0\).

1. Suppose that condition \(A_2\) is fulfilled. Without restricting the generality of the investigation, equation (2) in this case can always be reduced to the form

\[ \frac{dy}{dx}=-\frac{x+mx^2+ny^2}{y(1+lx)}. \tag{26} \]

Transformation (8) brings equation (26) to the form

\[ \frac{d\tau}{dz} = \frac{\tau z^2+z+m+(n+l)\tau^2}{\tau z^2+l\tau z}. \tag{27} \]

Accordingly, transformation (10) gives

\[ \frac{d\xi}{dz} = \frac{z\xi^2+z+m\xi^3+(l+n)\xi}{z^2\xi+m\xi^2z+nz}. \tag{28} \]

The roots of the characteristic equation compiled to determine the type of the singular point \(N_1(\xi = z = 0)\) are \(\mu_1 = n,\ \mu_2 = l+n\).

According to the results of [3], the type of the singular point \(N_2(\tau = z = 0)\) is determined by the coefficients of the expansions into series of the functions

\[ f(\tau) = -l(l+n)\tau^3 + (l+n)^2\tau^5 - l(n+l)^2\tau^6 + \ldots, \]

\[ \varphi(\tau) = (3l+2n)\tau - 2(l+n)\tau^3 + \ldots \]

In what follows, when clarifying the behavior of the integral curves as a whole, one must take into account the fact of symmetry of the field of directions of equation (26).

Fig. 2

Suppose that (26) has only one singular point \(M_0(0,0)\), i.e. \(m=0,\ nl<0\).

The case \(n=l=0\) is trivial.

Let \(n=0,\ l\ne 0\). Then \(N_1(\xi=z=0)\) is a saddle if \(l<0\), and a node if \(l>0\); \(N_2(\tau=z=0)\) is a closed saddle-node. The general integral of equation (26) for \(m=n=0,\ l\ne 0\) has the form

\[ \frac{y^2}{2}+\frac{x}{l}-\frac{1}{l^2}\ln|lx+1|=C. \]

The qualitative picture of the behavior of the integral curves as a whole is equivalent to the picture shown in Fig. 2, в, if \(l<0\), and Fig. 2, б, if \(l>0\).

Let \(n\ne 0,\ l=0\). In this case \(N_1\) is a dicritical node, \(N_2\) is a saddle. The general integral has the form

\[ y^2=-\frac{2}{n^2}(nx-1)+Ce^{-nx}. \]

The arrangement of the integral curves as a whole is given in Fig. 2, а.

Let \(ln<0,\ l+n\ne 0\). Then \(N_{1c}, N_{2zy}\), if \(n(l+n)<0\), and \(N_{1y}, N_{2c}\), if \(n(l+n)>0\). The qualitative picture of the behavior of the integral curves as a whole is equivalent to the picture shown in Fig. 2, в, if \(N_{1c}\), and in Fig. 2, б, if \(N_{1y}\).

Let \(n+l=0,\ l\ne0\). In this case the equator \(z=0\) of the Poincaré sphere consists entirely of singular points of the two-separatrix saddle type. The qualitative picture of the behavior of the integral curves as a whole is equivalent to the picture shown in Fig. 2, г.

Let, finally, \(3l+2n=0\). Then \(N_{1y}\), and \(N_{2c}\), and the qualitative picture of the behavior of the integral curves as a whole is equivalent to the picture shown in Fig. 2, a.

Suppose that equation (26) has two singular points \(M_0(0,0)\) and \(M_1\left(-\dfrac{1}{m},0\right)\). The latter is possible if \(m\ne0,\ n(l-m)=0,\ n^2+(l-m)^2\ne0\).

Let \(n(l-m)<0\). It is easy to see that \(M_{1c}\) if \(m(m-l)>0\), and \(M_{1ц}\) if \(m(m-l)<0\). In addition, if \(m(l+n)<0\), then there are three singular points \(N_i\); whereas if \(m(l+n)>0\) or \(l+n=0\), then there is only one singular point. Direct calculations show that two of the \(N_i\) are nodes and the third is a saddle if \(m(l+n)<0\) and \(m(m-l)>0\), and, conversely, two of the three singular points \(N_i\) are saddles and the third is a node if \(m(l+n)<0\) and \(m(m-l)<0\).

Thus, the qualitative picture of the behavior of the integral curves as a whole is equivalent to the picture shown in Fig. 2, д, if \(m(l+n)<0\) and \(m(m-l)>0\), and to Fig. 2, е, if \(m(l+n)<0\) and \(m(m-l)<0\).

If \(m(l+n)>0\) or \(l+n=0\), then the unique singular point at infinity will be a node if \(M_{1c}\), and a saddle if \(M_{1ц}\). The qualitative picture of the arrangement of the integral curves as a whole is equivalent to the picture shown in Fig. 2, ж, if \(M_{1c}\), and Fig. 2, и, if \(M_{1ц}\).

Let \(n=0,\ l-m\ne0\). In the infinitely distant part of the plane there are three singular points \(N_i\ (i=1,2,3)\) if \(lm<0\), and one if \(lm>0\) or \(l=0\).

Suppose that \(lm<0\). Then, obviously, \(M_{1c}\). Consequently, \(N_{1c}\), and \(N_{2y}\), and \(N_{3y}\). The arrangement of the integral curves as a whole is given in Fig. 2, д.

In the case \(lm>0\), \(M_{1c}\) if \(m-l<0\), and \(M_{1ц}\) if \(m-l>0\). Consequently, the qualitative picture of the behavior of the integral curves as a whole is equivalent to the picture shown in Fig. 2, ж, if \(M_{1c}\), and Fig. 2, и, if \(M_{1ц}\).

Finally, if \(l=n=0\), then \(M_{1c}\), and \(N_{1y}\). The arrangement of the integral curves as a whole is given in Fig. 2, ж.

Let \(l=m\ne0,\ n\ne0\). Put \(x=-\dfrac{1}{m}+u\). In the variables \(u,y\), equation (26) has the form

\[ \frac{dy}{du}=\frac{u-mu^2-ny^2}{muy}. \tag{29} \]

According to the results of [3], \(M_{1c}\) if \(mn>0\), and \(M_{1эсу}\) if \(mn<0\); moreover, if \(mn>0\), then \(N_{1y}\). There are no other singular points at infinity. Consequently, the qualitative picture of the behavior of the integral curves as a whole is equivalent to the picture shown in Fig. 2, ж.

If \(mn<0,\ m(m+n)>0\), then in the infinitely distant part of the plane there is only one singular point \(N_{1c}\). If, however, \(mn<0,\ m(m+n)<0\), then we have three singular points at infinity \(N_v\), with \(N_{1y}\), and the other two being saddles. The arrangement of the integral curves as a whole is given in Fig. 2, з, if \(mn<0,\ m(m+n)>0\), and Fig. 2, к, if \(mn<0,\ m(m+n)<0\).

Finally, if \(m+n=0\), then we have arrangement 2, з.

Suppose that equation (26) has three singular points \(M_i\). The latter is possible when the conditions \(m=0,\ ln>0\) are satisfied, and the singular points distinct from \(M_0(0,0)\) are saddles. In the infinitely distant part of the plane there are two singular points \(N_{1y}\) and \(N_{2scy}\). The qualitative picture of the behavior of the integral curves as a whole is equivalent to the picture shown in Fig. 2, л.

Suppose that equation (26) has four singular points \(M_i\), i.e. \(mn\ne 0,\ n(l-m)>0\).

The singular points will be

\[ M_0(0,0),\quad M_1\left(-\frac{1}{m},0\right),\quad M_2\left(-\frac{1}{l},\frac{1}{l}\sqrt{\frac{l-m}{n}}\right) \]

and

\[ M_3\left(-\frac{1}{l},-\frac{1}{l}\sqrt{\frac{l-m}{n}}\right). \]

It is easy to verify that \(M_{2c}\) and \(M_{3c}\), if \(ln>0\), and \(M_{2y}\), \(M_{3y}\), if \(ln<0\). In this case \(M_{1c}\), if \(ln<0\), and \(M_{1c}\) or \(M_{1u}\), if \(ln>0\).

Suppose that \(M_{1u}\), i.e. \(m(m-l)<0\). If \(m>0\), then also \(l>0\), with \(l>m\). From \(n(l-m)>0\) it follows that \(n>0\). In this case \(m(l+n)>0\). If \(m<0\), then \(l<0\) and \(|l|>|n|\). From \(n(l-m)>0\) it follows that \(n<0\). Thus, in the infinitely distant part of the plane there is only one singular point \(N_{1y}\). The arrangement of the integral curves as a whole is given in Fig. 1, з.

Suppose that \(M_{1c}\). If \(nl>0\), then there are three singular points \(N_i\ (i=1,2,3)\), and all of them are nodes. It is not difficult to verify that

\[ y^2=-\frac{m}{l^2(n+l)}(1+lx)^2-\frac{4m-2l}{l^2(2n+l)}\times \]

\[ \times(1+lx)-\frac{m-l}{cl^2}+C|1+lx|^{-\frac{2n}{l}} \tag{30} \]

is the general solution of equation (26) under the condition that \(nl(n+l)\times(2n+l)\ne0\). The equation of the separatrices of the saddles \(M_2\) and \(M_3\) has the form

\[ |1+lx|^{\frac{2n}{l}}\left[ y^2+\frac{m}{l^2(n+l)}(1+lx)^2 -\frac{4m-2l}{l^2(2n+l)}(1+lx) +\frac{m-l}{nl^2} \right]=0. \tag{31} \]

Let us determine whether \(M_{1c}\) can lie on the curve (31). This is possible if

\[ (m+n)(m-l)=0. \tag{32} \]

Since \(m-l\ne0\), from (32) we have \(m+n=0\), and, consequently, the curve

\[ y^2+\frac{m}{l^2(n+l)}(1+lx)^2 -\frac{4m-2l}{l^2(2n+l)}(1+lx) +\frac{m-l}{nl^2}=0 \]

splits into two straight lines. The qualitative picture of the behavior of the integral curves as a whole is equivalent to the picture shown in Fig. 2, м, if \(n+m=0\), and Fig. 2, п, if \(m+n\ne0\).

Let \(ln<0\). In the infinitely distant part of the plane there are three singular points \(N_i(i=1,2,3)\), if \(m(l+n)<0\), and one, if \(m(l+n)>0\) or \(l+n=0\). According to Poincaré’s theorem [4] on the complete distribution of singular points, we have \(N_{1y}\) and \(N_{2c}, N_{3c}\), if \(m(l+n)<0\). Accordingly, the qualitative picture of the behavior of the integral curves is equivalent to the picture shown in Fig. 2, о. If \(m(l+n)>0\), then we have \(N_{1c}\). The arrangement of the integral curves as a whole is given in Fig. 2, п.

If \(l+n=0\), then from \(n(l-m)>0\) we infer \(mn<0\). Put \(z=u+n\xi\). In the variables \(\xi,u\), equation (28) has the form

\[ \frac{d\xi}{du}= \frac{u+n\xi+\xi^2u+(m+n)\xi^3}{\xi u^2+(m+n)u\xi^2}. \tag{33} \]

According to the results of [6], we must conclude that \(N_{1c}\) if \(mn<0\). The arrangement of the integral curves as a whole coincides with Fig. 2, p.

Let \(n=0,\ m=l\). Obviously, the straight line \(1+lx=0\) consists entirely of singular points of the type two-separatrix saddle. The arrangement of the integral curves as a whole is given in Fig. 2, p.

1. Suppose that the conditions \(A_4\) are fulfilled. Equation (2) has the form

\[ \frac{dy}{dx}= -\frac{x+a(x^2-y^2)+\gamma xy}{y+b(x^2-y^2)+\varepsilon xy}, \qquad \gamma=2b+\alpha,\quad \varepsilon=\beta-2a. \tag{34} \]

By means of transformation (4), equation (34) can always be brought to the form

\[ \frac{dy_1}{dx_1}= -\frac{x_1+a_1(x_1^2-y_1^2)+\gamma_1x_1y_1}{y_1(1+\varepsilon_1x_1)}, \qquad \gamma_1=2b_1+a_1,\quad \varepsilon_1=\beta_1-2a_1, \tag{35} \]

where the coefficients \(a_1,\gamma_1\) and \(\varepsilon_1\) are determined according to (3). In what follows we shall assume \(\gamma_1\ne0\) (\(\gamma_1=0\) is a particular case of equation (26)).

The number and type of infinitely distant singular points of equation (35) is not difficult to establish by considering the equations

\[ \frac{d\tau}{dz}= \frac{z(1+\tau^2)+(\varepsilon_1-a_1)\tau^2+\gamma_1\tau+a_1}{z(\tau z+\varepsilon_1\tau)} \qquad \left(x_1=\frac{1}{z},\ y_1=\frac{\tau}{z}\right) \tag{36} \]

and

\[ \frac{d\xi}{dz}= \frac{z(1+\xi^2)+a_1\xi^3+\gamma_1\xi^2+(\varepsilon_1-a_1)\xi}{z(z\xi+a_1\xi^2+\gamma_1\xi-a_1)} \qquad \left(x_1=\frac{\xi}{z},\ y_1=\frac{1}{z}\right). \tag{37} \]

If

\[ \Delta \equiv \gamma_1^2-4a_1(\varepsilon_1-a_1)<0, \]

there is only one point \(N_1(\xi=z=0)\); if \(\Delta=0\), there are two points \(N_1\) and \(N_2\)

\[ \left(\tau_0=-\frac{\gamma_1}{2(\varepsilon_1-a_1)},\ z=0\right) \]

when \(\varepsilon_1\ne a_1\), or one \(N_2\)

\[ \left(-\frac{a_1}{\gamma_1},\ z=0\right) \]

when \(\varepsilon_1=a_1\).

Finally, if \(\Delta>0\), then in the infinitely distant part of the plane there are three singular points \(N_i\ (i=1,2,3)\).

Let us determine their character:

1) \(\Delta<0\): \(N_{1y}\), if \(a_1(\varepsilon_1-a_1)>0\), and \(N_{1c}\), if \(a_1(\varepsilon_1-a_1)<0\);

2) \(\Delta=0,\ \varepsilon_1\ne a_1\): the character of \(N_1\) is the same as for the case \(\Delta<0\), while \(N_{2ocy}\);

3) \(\Delta=0,\ \varepsilon_1=a_1\): \(N_{1ocy},\ N_{2c}\), since \(\varepsilon_1a_1>0\);

4) \(\Delta>0\): all \(N_i\ (i=1,2,3)\) are simple. We shall determine their possible character below.

It is easy to see that equation (35) always has at least two singular points \(M_i\).

Suppose that (35) has only two singular points \(M_0(0,0)\) and \(M_1\). The coefficients \(a_1\), \(\varepsilon_1\), and \(\gamma_1\) must satisfy one of the conditions:

\[ \text{1) } a_1=0,\ \varepsilon_1\ne 0;\quad \text{2) } a_1\ne 0,\ \varepsilon_1=0;\quad \text{3) } a_1\varepsilon_1\ne 0,\ \gamma_1^2+4a_1(a_1-\varepsilon_1)<0. \tag{38} \]

Let \(a_1=0,\ \varepsilon_1\ne 0\). We have

\[ M_{1c}\left(-\frac{1}{\varepsilon_1},\ -\frac{1}{\gamma_1}\right). \]

Fig. 3

In the infinitely distant part of the plane there are three singular points \(N_{1y}\), \(N_{2y}\), \(N_{3c}\). The general integral of the equation has the form

\[ Ce^{\frac{y_1}{\gamma_1}+\frac{x_1}{\varepsilon_1}} = |1+\gamma_1 y_1|^{\frac{1}{\gamma_1^2}} |1+\varepsilon_1 x_1|^{\frac{1}{\varepsilon_1^2}}. \]

The qualitative picture of the behavior of the integral curves is, as a whole, equivalent to the picture shown in Fig. 3, a.

Let \(a_1\ne 0,\ \varepsilon_1=0\). We have

\[ M_{1c}\left(-\frac{1}{a_1},\,0\right). \]

The general integral has the form

\[ y_1^2=x_1^2+\frac{2}{a_1}x_1+\frac{1}{a_1^2}+Ce^{-2a_1x_1}. \]

In the infinitely distant part of the plane there are three singular points, two nodes and a saddle. The arrangement of the integral curves as a whole is given in Fig. 3, a.

Let \(a_1\varepsilon_1\ne 0,\ \gamma_1^2+4a_1(a_1-\varepsilon_1)<0\). The characteristic equation, constructed to determine the character of the singular point

\[ M_1\left(-\frac{1}{a_1},\,0\right), \]

has the form

\[ \lambda^2-\frac{\gamma_1}{a_1}\lambda+\frac{\varepsilon_1}{a_1}-1=0. \tag{39} \]

Since \(\Delta<0\), \(M_{1\phi}\). We shall show that \(M_{1\phi}\) cannot be surrounded by limit cycles. For this purpose, consider equation (35) under the assumption that \(\gamma_1=0\), i.e.

\[ \frac{dy_1}{dx_1} = -\frac{x_1+a_1(x_1^2-y_1^2)} {y_1(1+\varepsilon_1 x_1)}, \qquad a_1(a_1-\varepsilon_1)<0. \tag{40} \]

Equation (40) has two singular points \(M^*(0,0)\) and

\[ M^{**}\left(-\frac{1}{a_1},\,0\right), \]

and both are of center type. In the infinitely distant part of the plane there is only one singular point—a saddle. Consequently, the two half-planes separated by the straight line

\[ x_1=-\frac{1}{\varepsilon_1} \]

are entirely filled with non-...

an interrupted family of closed curves enclosing, respectively, the points \((0,0)\) and \(\left(-\dfrac{1}{a_1},0\right)\). As a Poincaré topographic system we take the family of curves \(F(x_1,y_1)=C\) \((F(x_1,y_1)\) is an integral of equation (40)). In view of equation (35) we have

\[ \delta(x_1,y_1)=\frac{dF(x_1,y_1)}{dt} =-\gamma_1 x_1 y_1^2(1+\varepsilon_1 x_1). \]

It is obvious that \(\delta(x_1,y_1)\) preserves its sign in each of the above-mentioned half-planes. The latter proves that limit cycles cannot surround \(M_{1\phi}\).

In the infinitely distant part of the plane there is only one singular point \(N_{1c}\). The disposition of the integral curves is given in Fig. 3, b.

Suppose that equation (35) has three singular points \(M_i\). The latter is possible only when the conditions \(\Delta=0,\ a_1\varepsilon_1\ne0\) are satisfied.

The singular points will be \(M_0(0,0)\), \(M_1\left(-\dfrac{1}{a_1},0\right)\), \(M_2\left(-\dfrac{1}{\varepsilon_1},\dfrac{-\gamma_1}{2a_1\varepsilon_1}\right)\).

The characteristic equation, formed for \(M_2\), has the form \(\lambda \times\)

\[ \times\left(\lambda+\frac{\gamma_1}{2a_1\varepsilon_1}\right)=0. \]

It is not difficult to verify that \(M_{2\mathrm{ocu}}\). Indeed, if we put \(x_1=u-\dfrac{1}{\varepsilon_1}\), \(y_1=v-\dfrac{\gamma_1}{2a_1\varepsilon_1}\), then in the variables \(u\), \(v\) equation (35) has the form

\[ \frac{dv}{du} = -\frac{u+a_1(u^2-v^2)+\gamma_1uv} {-\dfrac{\gamma_1}{2a_1}\,u+\varepsilon_1uv}. \]

Since \(a_1\ne0\), according to the results of [6], \(M_{1\mathrm{ocu}}\). The characteristic equation, formed for \(M_1\), has the form (39). Consequently, \(M_{1y}\) with one exceptional direction. In the infinitely distant part of the plane there are two singular points \(N_{2c}\) and \(N_{1\mathrm{ocu}}\).

It is easy to verify that the line \(y_1=\dfrac{\gamma_1(a_1x_1+1)}{2a_1(a_1-d_1)}\), passing through \(M_1\) and \(M_2\), is an integral line. The qualitative picture of the behavior of the integral curves is, as a whole, equivalent to the picture shown in Fig. 3, c.

Suppose that equation (35) has four singular points. The latter will be
\(M_0(0,0)\), \(M_1\left(-\dfrac{1}{a_1},0\right)\), \(M_2\left(-\dfrac{1}{\varepsilon_1},\dfrac{-\gamma_1+\sqrt{\Delta}}{2a_1\varepsilon_1}\right)\) and
\(M_3\left(-\dfrac{1}{\varepsilon_1},\dfrac{-\gamma_1-\sqrt{\Delta}}{2a_1\varepsilon_1}\right)\). The characteristic equation, formed to determine the character of the singular points \(M_2\) and \(M_3\), has the form

\[ \left[\lambda+\frac{1}{2a_1}\left(-\gamma_1\pm\sqrt{\Delta}\right)\right] \left[\pm\frac{\sqrt{\Delta}}{\varepsilon_1}-\lambda\right]=0. \]

It is not difficult to verify that if \(a_1(a_1-\varepsilon_1)>0\) and \(a_1\varepsilon_1>0\), then \(M_{1c}\), \(M_{2c}\) and \(M_{3c}\); whereas if \(a_1(a_1-\varepsilon_1)>0\) and \(a_1\varepsilon_1<0\), then \(M_{1c}\), \(M_{2y}\), \(M_{3y}\).

Let \(a_1(a_1-\varepsilon_1)<0\). The latter entails \(a_1\varepsilon_1>0\); moreover, if \(\gamma_1>0\), then \(M_{1y}\) and \(M_{2y}\), while \(M_{3c}\); if \(\gamma_1<0\), then \(M_{1y}\) and \(M_{3y}\), while \(M_{2c}\).

By direct calculations it is easy to verify that the straight lines passing through \(M_1\) and \(M_2\), \(M_2\) and \(M_3\), \(M_1\) and \(M_3\) are integral curves.

In the infinitely remote part of the plane there are three singular points \(N_i\) \((i=1,2,3)\), namely \(N_{1y}\) and \(N_{ic}\) \((i=2,3)\), if among \(M_i\) \((i=1,2,3)\) there are two nodes, and \(N_{iy}\) \((i=1,2,3)\), if \(M_{ic}\) \((i=1,2,3)\). The disposition of the integral curves as a whole is correspondingly equivalent to Fig. 3, г, if \(N_{1y}\), and \(N_{ic}\) \((i=2,3)\), and to Fig. 2, м, if \(N_{iy}\) \((i=1,2,3)\).

Remark. In [7] Frommer showed that equation (35) is integrable in elementary functions. For this purpose it is sufficient to put

\[ x_1=\frac{u}{1+a_1u},\qquad y_1=\frac{v}{1+a_1u}. \]

1. Suppose that the conditions \(A_5\) are satisfied. In what follows we shall assume that \(ab\ne0\). Indeed, if \(b=0\), then from \(A_5\) we have \(d=0\), \(\alpha=0\), and we obtain a special form of equation (26). If, however, \(a=0\), then from \(A_5\) we have \(\alpha=d=b=0\). Thus, let \(ab\ne0\). From \(A_5\) we find

\[ b=-d,\qquad c=-\frac{2a^2+d^2}{a},\qquad \beta=\frac{5}{a}(a^2+b^2). \tag{41} \]

Taking (41) into account, equation (2) has the form

\[ \frac{dy}{dx} = - \frac{ x+ax^2+2bxy-\dfrac{2a^2+b^2}{a}y^2 }{ y+bx^2+\dfrac{a^2+3b^2}{a}xy-by^2 }. \tag{42} \]

Put \(x_1=ax+by,\ y_1=bx-ay\). In the variables \(x_1,y_1\), equation (42) has the form

\[ \frac{dy_1}{dx_1} = \frac{x_1+x_1^2-2y_1^2+3\mu x_1y_1}{-y_1-x_1y_1+\mu x_1^2}, \qquad \frac{b}{a}=\mu\ne0. \tag{43} \]

To determine the abscissas of the singular points of equation (43), we have

\[ (1+\mu^2)x_1^3+3(1+\mu^2)x_1^2+3x_1+1=0. \tag{44} \]

It is easy to see that equation (44), for any real value \(\mu\ne0\), has only one real root \(x_{10}\), which always lies in the interval \((-3,-1)\) and is determined by the formula

\[ x_{10} = -1+\sqrt[3]{\alpha-1} \left( \sqrt[3]{1+\sqrt{\alpha}}+\sqrt[3]{1-\sqrt{\alpha}} \right), \qquad \alpha=\frac{1}{1+\mu^2}. \tag{45} \]

Thus, equation (43), for any \(\mu\ne0\), has only two singular points \(M_0(0,0)\) and \(M_1(x_{10},y_{10})\). The characteristic equation, composed for determining the character of \(M_1\), has the form

\[ \lambda^2-\frac{5\mu}{1+x_{10}}\lambda +6\mu^2\frac{x_{10}^2}{(1+x_{10})^2}=0. \tag{46} \]

From (46) we find

\[ \lambda_1=3\mu\frac{x_{10}}{1+x_{10}},\qquad \lambda_2=2\mu\frac{x_{10}}{1+x_{10}}. \]

Thus, for any \(\mu \ne 0\), \(M_{1y}\).

The possible number and character of the infinitely distant singular points are easily established by considering the equations:

\[ \frac{d\tau}{dz} = \frac{z(1+\tau^{2})-\tau^{2}+2\mu\tau+1}{z(\tau z+\tau-\mu)} \left( x_{1}=\frac{1}{z_{1}},\ y_{1}=\frac{\tau}{z} \right) \tag{48} \]

and

\[ \frac{d\xi}{dz} = \frac{z(1+\xi^{2})+\xi^{3}+2\mu\xi^{2}-\xi}{z(\xi z+z^{2}+3\mu\xi-2)} . \tag{49} \]

From (49) it is obvious that equation (43), for any value \(\mu \ne 0\), has three infinitely distant singular points. The characteristic equation constructed to determine the character of \(N_{1}(0,0)\) has the form
\[ (\lambda+2)(\lambda+1)=0, \]
i.e. \(N_{1y}\). For the points \(N_{2}(\tau_{2}=\mu+\sqrt{1+\mu^{2}},\ z=0)\) and \(N_{3}(\tau_{3}=\mu-\sqrt{1+\mu^{2}},\ z=0)\), the characteristic equation has the form
\[ (\lambda+\mu-\tau_i)(\lambda+2\tau_i-2\mu)=0. \]

From the latter it is obvious that, for any \(\mu \ne 0\), \(N_{2c}\) and \(N_{3c}\).

In [8] it is proved that, for equation (1), a singular point of node type cannot be surrounded by limit cycles. Thus, the qualitative picture of the behavior of the integral curves as a whole is equivalent to the picture shown in Fig. 3, d.

The fact that, in equation (43), a limit cycle cannot surround a singular point of node type also follows from the results of [9], where a general integral of equation (43) is found. The latter is an algebraic curve of the third degree.

From the above there follows the following theorem.

Theorem. If equation (1) has at least one singular point of center type, then: 1) equation (1) cannot have limit cycles in the entire plane of the variables \(x,y\); 2) the equation is integrable in elementary functions.

Remark. Most of the results of the present article are contained in works [10, 11]. Separate results were published in [12].

References

1. Lyapunov A. M. Investigation of one of the special cases of the problem of stability of motion. Collected Works, vol. II. Moscow—Leningrad, 1956.

2. Sakharnikov N. A. PMM, vol. XII, issue 5, 1948.

3. Andreev A. F. Vestnik LGU, No. 8, 1955.

4. Poincaré A. On curves defined by differential equations. Moscow—Leningrad, 1947.

5. Sakharnikov N. A. PMM, vol. XV, issue 3, 1951.

6. Vorob’ev A. P. DAN BSSR, 3, No. 8, 1959.

7. Frommer M. Math. Ann., 1934, Bd. 109.

8. Vorob’ev A. P. DAN BSSR, vol. IV, No. 9, 1960.

9. Korkine A. Math. Ann., B. 48, H. 3, 1896.

10. Lukashevich N. A. Behavior of integral curves in the large for certain systems of ordinary differential equations in the presence of a singular point of center type. Candidate dissertation. Minsk, Academy of Sciences of the BSSR, 1962.

11. Lukashevich N. A. Behavior of integral curves in the large for certain systems of ordinary differential equations in the presence of a singular point of center type. Author’s abstract of candidate dissertation. Minsk, 1962.

12. Lukashevich N. A. DAN BSSR, vol. IV, No. 12, 1960.

1. Correspondence is understood in the sense of Poincaré [4]. ↩