Full Text
On Conditions for a Center
V. A. Lunkevich, K. S. Sibirskii
1. Consider the differential equation
\[ \frac{dy}{dx} = -\frac{c_{10}x+c_{01}y+c_{30}x^3+c_{21}x^2y+c_{12}xy^2+c_{03}y^3} {b_{10}x+b_{01}y+b_{30}x^3+b_{21}x^2y+b_{12}xy^2+b_{03}y^3} \tag{1} \]
under the condition
\[ c_{10}b_{01}-b_{10}c_{01}\ne 0. \tag{2} \]
The purpose of the present note is to express the conditions for the existence of a center at the origin for equation (1), under condition (2), directly in terms of the coefficients of this equation.
The characteristic equation of the system of differential equations
\[ \frac{dx}{dt} = b_{10}x+b_{01}y+b_{30}x^3+b_{21}x^2y+b_{12}xy^2+b_{03}y^3, \]
\[ \frac{dy}{dt} = -c_{10}x-c_{01}y-c_{30}x^3-c_{21}x^2y-c_{12}xy^2-c_{03}y^3, \]
which is equivalent to equation (1), will be
\[ k^2+\lambda k+\varepsilon=0, \tag{3} \]
where
\[ \lambda=c_{01}-b_{10}, \qquad \varepsilon=c_{10}b_{01}-b_{10}c_{01}. \]
As Poincaré showed [1], in order for equation (1) to have a center at the origin, under condition (2), it is necessary that the roots of equation (3) be purely imaginary, that is, that the conditions
\[ \lambda=0, \qquad \varepsilon>0 \]
be satisfied.
In what follows we shall assume these conditions to be fulfilled. Then
\[ \varepsilon=c_{10}b_{01}-b_{10}^{2}, \tag{4} \]
and, with the nonsingular linear change of variables
\[ x=\varepsilon c_{10}\bar{x}-\sqrt{\varepsilon}\,c_{10}b_{10}\bar{y}, \qquad y=\sqrt{\varepsilon}\,c_{10}^{2}\bar{y}, \]
equation (1) is brought to the form
\[ \frac{d\bar{y}}{d\bar{x}} = - \frac{\bar{x}+\bar{c}_{30}\bar{x}^{3}+\bar{c}_{21}\bar{x}^{2}\bar{y}+\bar{c}_{12}\bar{x}\bar{y}^{2}+\bar{c}_{03}\bar{y}^{3}} {\bar{y}+\bar{b}_{30}\bar{x}^{3}+\bar{b}_{21}\bar{x}^{2}\bar{y}+\bar{b}_{12}\bar{x}\bar{y}^{2}+\bar{b}_{03}\bar{y}^{3}}. \tag{5} \]
where
\[ \left. \begin{aligned} \bar c_{30}&=\varepsilon^{2}c_{10}c_{30};\\ \bar c_{21}&=\varepsilon\sqrt{\varepsilon}\,c_{10}(c_{10}c_{21}-3b_{10}c_{30});\\ \bar c_{12}&=\varepsilon c_{10}(c_{10}^{2}c_{12}-2c_{10}b_{10}c_{21}+3b_{10}^{2}c_{30});\\ \bar c_{03}&=\sqrt{\varepsilon}\,c_{10}(c_{10}^{3}c_{03}-c_{10}^{2}b_{10}c_{12} +c_{10}b_{10}^{2}c_{21}-b_{10}^{3}c_{30});\\ \bar b_{30}&=\varepsilon\sqrt{\varepsilon}\,c_{10}(c_{10}b_{30}-b_{10}c_{30});\\ \bar b_{21}&=\varepsilon c_{10}\bigl[c_{10}^{2}b_{21}-c_{10}b_{10}(c_{21}+3b_{30}) +3b_{10}^{2}c_{30}\bigr];\\ \bar b_{12}&=\sqrt{\varepsilon}\,c_{10}\bigl[c_{10}^{3}b_{12} -c_{10}^{2}b_{10}(2b_{21}+c_{12})\\ &\qquad\qquad\qquad\quad +c_{10}b_{10}^{2}(2c_{21}+3b_{30})-3b_{10}^{3}c_{30}\bigr];\\ \bar b_{03}&=c_{10}\bigl[c_{10}^{4}b_{03}-c_{10}^{3}b_{10}(b_{12}+c_{03}) +c_{10}^{2}b_{10}^{2}(b_{21}+c_{12})\\ &\qquad\qquad\qquad\quad -c_{10}b_{10}^{3}(c_{21}+b_{30})+b_{10}^{4}c_{30}\bigr]. \end{aligned} \right\} \tag{6} \]
By rotating the coordinate axes through an angle \(\varphi\) satisfying the relation
\[ 2(\bar b_{21}-\bar c_{12})\cos 2\varphi+ (\bar b_{12}-3\bar c_{03}+\bar c_{21}-3\bar b_{30})\sin 2\varphi=0, \]
one can arrange that, in the new equation, the equality
\[ \bar b_{21}=\bar c_{12} \tag{7} \]
holds. In this case equation (5) can always be written in the form
\[ \frac{dy}{dx}= -\frac{x+(\mu+\nu)x^{3}+(3\omega+3\theta+2a)x^{2}y+(\eta-3\mu)xy^{2}+(\omega-\theta-a)y^{3}} {y+(\omega+\theta-a)x^{3}+(\eta-3\mu)x^{2}y+(3\omega-3\theta+2a-\xi)xy^{2}+(\mu-\nu)y^{3}}. \tag{8} \]
As shown in [2], (Corollary 1) implies
Theorem 1. For the existence of a center at the origin for equation (8), it is necessary and sufficient that at least one of the following three series of conditions be satisfied:
\[ \left. \begin{aligned} &1)\quad \xi=a=0; \qquad 2)\quad \xi=\nu=\theta=0;\\ &3)\quad \xi=\nu=\omega=\eta=4(\mu^{2}+\theta^{2})-a^{2}=0. \end{aligned} \right\} \tag{9} \]
- Using the invariants of the rotation group of the plane [3], one can readily obtain the necessary and sufficient conditions for the existence of a center at the origin for equation (5).
Indeed, introducing the variable \(w=x+iy\), equation (5) can be replaced by the equation
\[ -i\,\frac{dw}{dt}=w+\frac{1}{8}\bigl(z_{30}\bar w^{3}+z_{21}\bar w^{2}w+z_{12}\bar w w^{2}+z_{03}w^{3}\bigr), \tag{10} \]
where
\[ \left. \begin{aligned} z_{30}&=\bar c_{30}-\bar b_{21}-\bar c_{12}+\bar b_{03} +i(\bar b_{30}+\bar c_{21}-\bar b_{12}-\bar c_{03}),\\ z_{21}&=3\bar c_{30}-\bar b_{21}+\bar c_{12}-3\bar b_{03} +i(3\bar b_{30}+\bar c_{21}+\bar b_{12}+3\bar c_{03}),\\ z_{12}&=3\bar c_{30}+\bar b_{21}+\bar c_{12}+3\bar b_{03} +i(3\bar b_{30}-\bar c_{21}+\bar b_{12}-3\bar c_{03}),\\ z_{03}&=\bar c_{30}+\bar b_{21}-\bar c_{12}-\bar b_{03} +i(\bar b_{30}-\bar c_{21}-\bar b_{12}+\bar c_{03}). \end{aligned} \right\} \tag{11} \]
The equation (8) obtained after rotation of the coordinate axes can also be written in the form (10), but for it
\[ \left. \begin{gathered} z_{30}=8\mu-2\eta+i(8\theta+\xi),\quad z_{12}=2\eta-i\xi,\\ z_{21}=6\nu+i(12\omega-2a-\xi),\quad z_{03}=2\nu-i(4\omega+6a-\xi). \end{gathered} \right\} \tag{12} \]
Solving these equalities with respect to \(\nu,\eta,\xi,\mu,a,\omega\), and \(\theta\), we obtain
\[ \left. \begin{gathered} 6\nu=\operatorname{Re} z_{21},\quad 2\eta=\operatorname{Re} z_{12},\quad \xi=-\operatorname{Im} z_{12},\\ 8\mu=\operatorname{Re} z_{30}+\operatorname{Re} z_{12},\quad 20a=-3\operatorname{Im} z_{03}-\operatorname{Im} z_{21}-2\operatorname{Im} z_{12},\\ 40\omega=3\operatorname{Im} z_{21}-\operatorname{Im} z_{03}-4\operatorname{Im} z_{12},\quad 8\theta=\operatorname{Im} z_{30}+\operatorname{Im} z_{12}, \end{gathered} \right\} \tag{13} \]
where the condition is satisfied
\[ \operatorname{Re} z_{21}=3\operatorname{Re} z_{03}, \tag{14} \]
which can also be obtained from (7) taking (11) into account.
It follows from (13) that the relations (9) are equivalent, respectively, to the following three series of equalities:
\[ \left. \begin{aligned} 1)\quad & \operatorname{Im} z_{12}=\operatorname{Im}(3z_{03}-\overline{z}_{21})=0;\\ 2)\quad & \operatorname{Im} z_{12}=\operatorname{Re} z_{21}=\operatorname{Im} z_{30}=0;\\ 3)\quad & z_{12}=\operatorname{Re} z_{21}=\operatorname{Im}(3z_{21}-\overline{z}_{03})=4z_{21}\overline{z}_{21}-z_{30}\overline{z}_{30}=0. \end{aligned} \right\} \tag{15} \]
Introducing the notation \(z'_{03}\equiv 3z_{03}-\overline{z}_{21}\), one can rewrite condition (14) in the form
\[ \operatorname{Re} z'_{03}=0. \tag{16} \]
As is easy to see, when equality (16) is satisfied, the conditions (15) are equivalent to the following:
\[ \left. \begin{aligned} 1)\quad & \operatorname{Im} z_{12}=\operatorname{Im}(z_{21}z'_{03})=\operatorname{Im}(z_{30}z_{03}^{\prime 2})=0;\\ 2)\quad & z_{12}=10\overline{z}_{21}+z'_{03}=2|z_{21}|-|z_{30}|=0. \end{aligned} \right\} \tag{17} \]
Here condition 1) combines cases 1) and 2) from (15).
The notation (17) for the center conditions for equation (8) is advantageously distinguished from the notation (15) by the fact that the expressions
\[ z_{12},\quad z_{21}z'_{03},\quad z_{30}z_{03}^{\prime 2},\quad 2|z_{21}|-|z_{30}|=2(z_{21}\overline{z}_{21})^{\frac12}-(z_{30}\overline{z}_{30})^{\frac12} \]
and
\[ |10\overline{z}_{21}+z'_{03}|^2 = 100z_{21}\overline{z}_{21} +10z_{21}z'_{03} +10\overline{z}_{21}\overline{z}'_{03} +z'_{03}\overline{z}'_{03} \]
are invariants of rotation of the coordinate axes [3], and therefore, for the equation (5) also to have a center at the origin, it is necessary and sufficient that at least one of the two series of conditions (17) be satisfied (as for equation (8)).
Taking into account (11) and the fact that
\[ z'_{03}\equiv 3z_{03}-\overline{z}_{21} = 4(\overline{b}_{21}-\overline{c}_{12}) +2i(3\overline{b}_{30}-\overline{c}_{21}-\overline{b}_{12}+3\overline{c}_{03}), \]
one can express conditions (17) through the coefficients of equation (5). In doing so it is convenient to introduce the notation
\[ \left. \begin{aligned} 2\bar\mu &\equiv \bar c_{30}+\bar b_{03};& 2\bar\nu &\equiv \bar c_{30}-\bar b_{03};& 2\bar\psi &\equiv \bar b_{30}+\bar c_{03};& 2\bar\theta &\equiv \bar b_{30}-\bar c_{03};\\ 2\bar\chi &\equiv \bar c_{12}+\bar b_{21};& 2\bar\beta &\equiv \bar c_{12}-\bar b_{21};& \bar\alpha &\equiv \bar c_{21}-3\bar b_{30};& \bar\gamma &\equiv 3\bar c_{03}-\bar b_{12}. \end{aligned} \right\} \tag{18} \]
Then we obtain that
\[ z_{30}=2(\bar\mu-\bar\chi)+i(8\bar\theta+\bar\alpha+\bar\gamma);\quad z_{21}=6\bar\nu+2\bar\beta+i(12\bar\psi+\bar\alpha-\bar\gamma); \]
\[ z_{12}=6\bar\mu+2\bar\chi-i(\bar\alpha+\bar\gamma);\quad z_{03}=-8\bar\beta-2i(\bar\alpha-\bar\gamma), \]
and there holds
Theorem 2. For the equation
\[ \frac{dy}{dx} = -\frac{x+(\bar\mu+\bar\nu)x^3+(3\bar\psi+3\bar\theta+\bar\alpha)x^2y+(\bar\chi+\bar\beta)xy^2+(\bar\psi-\bar\theta)y^3} {y+(\bar\psi+\bar\theta)x^3+(\bar\chi-\bar\beta)x^2y+(3\bar\psi-3\bar\theta-\bar\gamma)xy^2+(\bar\mu-\bar\nu)y^3} \]
it is necessary and sufficient, for the origin to be a center, that at least one of the following two series of conditions be satisfied:
\[ \left. \begin{aligned} 1)\quad& \bar\alpha+\bar\gamma =\bar\alpha\bar\nu+\bar\alpha\bar\beta+4\bar\beta\bar\psi =6\bar\alpha^2+(\bar\chi-\bar\mu)\bar\alpha\bar\beta-4\bar\theta\bar\beta^2=0;\\ 2)\quad& \bar\alpha+\bar\gamma =\bar\chi+3\bar\mu =\bar\alpha+5\bar\psi =\bar\beta+5\bar\nu =\bar\psi^2+4(\bar\nu^2-\bar\mu^2-\bar\theta^2)=0. \end{aligned} \right\} \tag{19} \]
Thus we arrive at center conditions equivalent to the conditions of K. E. Malkin [4]. The earlier works of M. I. Al’mukhamedov [5] and N. A. Sakharnikov [6] do not contain all cases of a center [4].
Comparison of conditions (17) with those given in [3] in the form
\[ 1)\quad \operatorname{Im}z_{12}=\operatorname{Im}(z_{21}z'_{03})=\operatorname{Im}(z_{30}z_{03}^{\prime\,2})=0; \]
\[ 2)\quad z_{12}=10\bar z_{21}+z'_{03}=z_{30}^2\bar z_{21}+4z_{21}^3=0 \]
with the conditions of N. A. Sakharnikov shows that in [6] the center cases
\[ z_{12}=10\bar z_{21}+z'_{03}=2|z_{21}|-|z_{30}|=0, \]
\[ \arg z_{30}\ne 2\arg z_{21}+(2k+1)\pi/2 \]
were omitted.
- To express the conditions for the origin to be a center of equation (1) under condition (2) directly through the coefficients of this equation, it is now necessary to substitute expressions (18), taking (6) into account, into conditions (19).
After collecting like terms and reducing by \(V\varepsilon^3c_{10}^3\), we obtain the condition \(\bar\alpha+\bar\gamma=0\) in the form
\[ A=0, \tag{20} \]
where
\[ A=b_{01}\alpha-2b_{10}\beta+c_{10}\gamma;\quad \alpha\equiv c_{21}-3b_{30};\quad \beta\equiv c_{12}-b_{21};\quad \gamma\equiv 3c_{03}-b_{12}. \]
The condition \(\bar\alpha\bar\nu+\bar\alpha\bar\beta+4\bar\beta\bar\psi=0\), after reduction by \(\varepsilon\sqrt{\varepsilon}\,c_{10}^5\), gives
\[ (c_{10}c_{12}+b_{01}c_{30}+b_{10}b_{30}-b_{10}c_{21})A+c_{10}B=0, \]
where
\[ B=(2b_{10}c_{03}-b_{01}b_{21}-c_{10}b_{03})\alpha +2(c_{10}c_{03}+b_{01}b_{30})\beta \]
\[ +(2b_{10}b_{30}-c_{10}c_{12}-b_{01}c_{30})\gamma. \]
Hence, taking (20) into account, we obtain
\[ B=0. \tag{21} \]
Dividing the condition \(\bar{\theta}\bar{\alpha}^{2}+(\bar{\chi}-\bar{\mu})\bar{\alpha}\bar{\beta}-4\bar{\psi}\bar{\beta}^{2}=0\) by \(\varepsilon^{2}\sqrt{\varepsilon}\,c_{10}^{7}\), we obtain
\[ \begin{aligned} &(2b_{10}^{2}c_{30}\beta+b_{10}^{2}\alpha^{2}-b_{10}b_{01}c_{30}\alpha+2b_{01}c_{10}b_{30}\alpha+2c_{10}b_{10}b_{30}\beta-\\ &\quad -c_{10}b_{01}c_{30}\beta+c_{10}b_{10}c_{30}\gamma-2c_{10}b_{10}c_{21}\beta+c_{10}^{2}c_{12}\beta)A+c_{10}^{2}\beta B+c_{10}^{2}C=0, \end{aligned} \tag{22} \]
where
\[ \begin{aligned} C &\equiv (b_{10}b_{03}-2b_{01}c_{03})\alpha^{2}-4b_{01}b_{30}\beta^{2}-b_{10}c_{30}\gamma^{2}-2b_{01}b_{30}\alpha\gamma+\\ &\quad +2(b_{01}b_{21}-b_{10}c_{03})\alpha\beta+(2b_{01}c_{30}+b_{10}c_{21}-b_{10}b_{30})\beta\gamma. \end{aligned} \]
Taking (20) and (21) into account, condition (22) is equivalent to the equality \(C=0\). The condition \(\bar{\chi}+3\bar{\mu}=0\), after division by \(c_{10}^{3}\), gives
\[ D=0, \tag{23} \]
where
\[ \begin{aligned} D &\equiv 3b_{01}^{2}c_{30}-3b_{10}b_{01}c_{21}+(c_{10}b_{01}+2b_{10}^{2})c_{12}-3c_{10}b_{10}c_{03}-\\ &\quad -3b_{10}b_{01}b_{30}+(c_{10}b_{01}+2b_{10}^{2})b_{21}-3c_{10}b_{10}b_{12}+3c_{10}^{2}b_{03}. \end{aligned} \]
Dividing the condition \(\bar{\alpha}+5\bar{\psi}=0\) by \(\sqrt{\varepsilon}\,c_{10}^{2}\), we obtain
\[ 5b_{10}b_{01}c_{30}-(2c_{10}b_{01}+3b_{10}^{2})c_{21}+5c_{10}b_{10}c_{12}-5c_{10}^{2}c_{03}+(c_{10}b_{01}-b_{10}^{2})b_{30}=0. \]
Taking (4) into account, this condition can be written in the form
\[ E=0, \tag{24} \]
where
\[ E\equiv \varepsilon(5c_{21}+\alpha)+15(c_{10}^{2}c_{03}-c_{10}b_{10}c_{12}+b_{10}^{2}c_{21}-b_{10}b_{01}c_{30}). \]
The condition \(\bar{\beta}+5\bar{\nu}=0\), after division by \(c_{10}^{2}\), can be written in the form \(2b_{10}E+3c_{10}F=0\), where
\[ \begin{aligned} F &=5b_{01}^{2}c_{30}-5b_{01}b_{10}c_{21}+(c_{10}b_{01}+4b_{10}^{2})c_{12}-5c_{10}b_{10}c_{03}+\\ &\quad +5b_{01}b_{10}b_{30}-(c_{10}b_{01}+4b_{10}^{2})b_{21}+5c_{10}b_{10}b_{12}-5c_{10}^{2}b_{03}. \end{aligned} \]
Taking (24) into account, the condition \(\bar{\beta}+5\bar{\nu}=0\) is then equivalent to the equality
\[ F=0. \tag{25} \]
The system of conditions (23) and (25) can be replaced by the conditions
\[ G\equiv \frac{1}{2}(5D-3F)=0,\qquad H\equiv \frac{1}{2}(5D+3F)=0, \tag{26} \]
where
\[ G=\varepsilon(5b_{21}+\beta)+15(c_{10}^{2}b_{03}-c_{10}b_{10}b_{12}+b_{10}^{2}b_{21}-b_{10}b_{01}b_{30}), \]
\[ H=\varepsilon(5c_{12}-\beta)+15(b_{01}^{2}c_{30}-b_{10}b_{01}c_{21}+b_{10}^{2}c_{12}-c_{10}b_{10}c_{03}). \]
The condition \(\bar\psi^{\,2}+4(\bar\nu^{\,2}-\bar\mu^{\,2}-\bar\theta^{\,2})=0\), after division by \(\varepsilon c_{10}^{4}\), can be written in the form
\[ 4K\varepsilon^{2}+(80c_{30}G-44b_{30}E-12c_{21}E)\varepsilon+E^{2}=0, \]
where
\[ K=9c_{21}^{2}-20c_{12}c_{30}+21b_{30}^{2}+66c_{21}b_{30}-80b_{21}c_{30}. \]
Taking (24) and (26) into account, from this we obtain \(K=0\).
Thus Theorem 3 is proved.
Theorem 3. For the existence of a center at the origin for equation (1), under condition (2), it is necessary and sufficient that at least one of the following two series of conditions be fulfilled:
\[ \begin{aligned} 1)\quad &\varepsilon>0,\quad \lambda=b_{01}\alpha-2b_{10}\beta+c_{10}\gamma\\ &=(2b_{10}c_{03}-b_{01}b_{21}-c_{10}b_{03})\alpha+{}\\ &\quad +2(c_{10}c_{03}+b_{01}b_{30})\beta+(2b_{10}b_{30}-c_{10}c_{12}-b_{01}c_{30})\gamma\\ &=(b_{10}b_{03}-2b_{01}c_{03})\alpha^{2}-4b_{01}b_{30}\beta^{2}-b_{10}c_{30}\gamma^{2}-2b_{01}b_{30}\alpha\gamma+{}\\ &\quad +2(b_{01}b_{21}-b_{10}c_{03})\alpha\beta+(2b_{01}c_{30}+b_{10}c_{21}-b_{10}b_{30})\beta\gamma=0; \end{aligned} \]
\[ \begin{aligned} 2)\quad &\varepsilon>0,\quad \lambda=b_{01}\alpha-2b_{10}\beta+c_{10}\gamma\\ &=\varepsilon(5b_{21}+\beta)+15(c_{10}^{2}b_{03}-c_{10}b_{10}b_{12}+b_{10}^{2}b_{21}-b_{10}b_{01}b_{30})\\ &=\varepsilon(5c_{21}+\alpha)+15(c_{10}^{2}c_{03}-c_{10}b_{10}c_{12}+b_{10}^{2}c_{21}-b_{10}b_{01}c_{30})\\ &=\varepsilon(5c_{12}-\beta)+15(b_{01}^{2}c_{30}-b_{10}b_{01}c_{21}+b_{10}^{2}c_{12}-c_{10}b_{10}c_{03})\\ &=9c_{21}^{2}-20c_{12}c_{30}+21b_{30}^{2}+66c_{21}b_{30}-80b_{21}c_{30}=0, \end{aligned} \]
where
\[ \varepsilon\equiv c_{10}b_{01}-b_{10}c_{01};\quad \lambda\equiv c_{01}-b_{10}; \]
\[ \alpha=c_{21}-3b_{30};\quad \beta=c_{12}-b_{21};\quad \gamma=3c_{03}-b_{12}. \]
References
- Poincaré A. On curves defined by differential equations, GITTL, Moscow—Leningrad, 1947.
- Sibirskii K. S. Differential Equations, No. 1, 1965, pp. 53—66.
- Sibirskii K. S. Doklady AN SSSR, 151, No. 3, 1963, pp. 497—500; with a correction, 156, No. 2, 1964, p. 238.
- Malkin K. E. Volga Mathematical Collection, issue 2, 1964, pp. 87—91.
- Almukhamedov M. I. Kazan, Izvestiya of the Physico-Mathematical Society, 9, ser. 3, 1937, pp. 105—121.
- Sakharnikov N. A. PMM, 14, issue 6, 1950, pp. 651—658.
Received by the editors
October 5, 1964
Institute of Mathematics
with the Computing Center of the Academy of Sciences of the Moldavian SSR