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Conditions for a Center in the Case of Homogeneous Nonlinearities of Third Degree
V. A. Lunkevich, K. S. Sibirskii
1. The center problem, posed in general form by Poincaré [1], consists in finding conditions under which there exists a neighborhood of the origin \(O\) containing no singular points other than the point \(O\), and such that all integral curves of the equation
\[ \frac{dy}{dx} = - \sum_{j+l\in A} c_{jl}x^j y^l \bigg/ \sum_{j+l\in A} b_{jl}x^j y^l, \]
which intersect this neighborhood at some point (different from the point \(O\)), are closed curves surrounding the origin. This problem has been solved completely only for the case when the set \(A\) consists of the numbers 1 and 2: \(A=\{1,2\}\). An explicit expression of the center conditions in this case directly in terms of the coefficients of the equation was published in [2].
In the present note we establish explicit expressions for the conditions for the existence at the origin of a singular point of center type for \(A=\{1,3\}\), i.e., for the equation
\[ \frac{dy}{dx} = - \frac{ c_{10}x+c_{01}y+c_{30}x^3+c_{21}x^2y+c_{12}xy^2+c_{03}y^3 }{ b_{10}x+b_{01}y+b_{30}x^3+b_{21}x^2y+b_{12}xy^2+b_{03}y^3 }. \tag{1} \]
We introduce the following notation:
\[ \varepsilon=c_{10}b_{01}-b_{10}c_{01};\quad \lambda=c_{01}-b_{10};\quad \alpha=c_{21}-3b_{30}; \]
\[ \beta=c_{12}-b_{21};\quad \gamma=3c_{03}-b_{12};\quad \sigma=b_{01}\beta-b_{10}\gamma; \]
\[ \tau=b_{01}\alpha-b_{10}\beta;\quad \omega=c_{01}\alpha-c_{10}\beta;\quad \theta=c_{01}\beta-c_{10}\gamma; \]
\[ M=b_{01}b_{21}-2b_{10}b_{12}+3c_{10}b_{03};\quad N=b_{01}c_{21}-2b_{10}c_{12}+3c_{10}c_{03}; \]
\[ P=c_{10}c_{12}-2c_{01}c_{21}+3b_{01}c_{30};\quad Q=c_{03}\alpha^2-b_{21}\alpha\beta+2b_{30}\beta^2+b_{30}\alpha\gamma-c_{30}\beta\gamma; \]
\[ R=9c_{21}^2-20c_{12}c_{30}+21b_{30}^2+66c_{21}b_{30}-80b_{21}c_{30}; \]
\[ S=2c_{03}\beta^2-c_{12}\beta\gamma+b_{30}\gamma^2. \]
2. The characteristic equation for the system of differential equations
\[ \frac{dx}{dt} = b_{10}x+b_{01}y+b_{30}x^3+b_{21}x^2y+b_{12}xy^2+b_{03}y^3, \]
\[ -\frac{dy}{dt} = c_{10}x+c_{01}y+c_{30}x^3+c_{21}x^2y+c_{12}xy^2+c_{03}y^3, \]
equivalent to equation (1), has the form
\[ \varkappa^2+\lambda\varkappa+\varepsilon=0. \tag{2} \]
Since a center is a non-asymptotically stable isolated point, it follows from the theorems of A. M. Lyapunov [3] (Theorem I, p. 127; Theorem III, p. 128; and Theorem V, p. 165) that, if not all the coefficients \(b_{10}, b_{01}, c_{10}\), and \(c_{01}\) are equal to zero, then for equation (1) to have a center at the origin it is necessary that equation (2) have either purely imaginary roots, i.e., that the conditions
\[ \lambda = 0,\quad \varepsilon > 0, \tag{3} \]
be satisfied, or a multiple zero root, which is equivalent to the equality
\[ \lambda = \varepsilon = 0. \tag{4} \]
When conditions (3) are satisfied,
\[ \varepsilon = c_{10}b_{01} - b_{10}^{2} > 0,\quad c_{10}\ne 0 \tag{5} \]
and by a nonsingular linear change of variables
\[ x=\varepsilon c_{10}x_{1}-\sqrt{\varepsilon}\,c_{10}b_{10}y_{1},\quad y=\sqrt{\varepsilon}\,c_{10}^{2}y_{1} \]
equation (1) is reduced to the case
\[ c_{10}=b_{01}=1,\quad b_{10}=c_{01}=0. \tag{6} \]
Equation (1) under conditions (6) was considered in the works of M. I. Al’mukhamedov [4] and N. A. Sakharnikov [5], but in them not all cases of a center were established. Correct center conditions for case (6) were obtained in the work of K. E. Malkin [6]. The completeness of K. E. Malkin’s center conditions was confirmed by another method in [7, 8].
In [9], the authors of the present note obtained explicit expressions for the center conditions for equation (1) in case (3) directly in terms of the coefficients of equation (1). It turned out that, for equation (1) to have a center at the origin when the characteristic equation has purely imaginary roots, it is necessary and sufficient that at least one of the following two series of conditions be fulfilled:
\[ \begin{aligned} \text{I.}\quad &\varepsilon>0,\quad \lambda=\tau-\theta=B=C=0;\\ \text{II.}\quad &\varepsilon>0,\quad \lambda=\tau-\theta=E=G=H=R=0, \end{aligned} \]
where
\[ \begin{aligned} B={}&(2b_{10}c_{03}-b_{01}b_{21}-c_{10}b_{03})\alpha +2(c_{10}c_{03}+b_{01}b_{30})\beta\\ &+(2b_{10}b_{30}-c_{10}c_{12}-b_{01}c_{30})\gamma;\\ C={}&(b_{10}b_{03}-2b_{01}c_{03})\alpha^{2} -4b_{01}b_{30}\beta^{2}\\ &-2b_{01}b_{30}\alpha\gamma +2(b_{01}b_{21}-b_{10}c_{03})\alpha\beta\\ &+(2b_{01}c_{30}+b_{10}c_{21}-b_{10}b_{30})\beta\gamma;\\ E={}&\varepsilon(5c_{21}+\alpha) +15(c_{10}^{2}c_{03}-c_{10}b_{10}c_{12}+b_{10}^{2}c_{21}-b_{10}b_{01}c_{30});\\ G={}&\varepsilon(5b_{21}+\beta) +15(c_{10}^{2}b_{03}-c_{10}b_{10}b_{12}+b_{10}^{2}b_{21}-b_{10}b_{01}b_{30});\\ H={}&\varepsilon(5c_{12}-\beta) +15(b_{01}^{2}c_{30}-b_{10}b_{01}c_{21}+b_{10}^{2}c_{12}-c_{10}b_{10}c_{03}). \end{aligned} \]
Since in the case under consideration \(\lambda=0\) and conditions (5) are satisfied, taking into account the identities
\[ c_{10}C+b_{10}\alpha B+b_{10}(b_{21}\alpha-2b_{30}\beta+c_{30}\gamma)(\tau-\theta)=-2\varepsilon Q; \]
\[ b_{10}E+c_{10}H=\varepsilon(\omega+5P),\quad b_{01}E+b_{10}H=\varepsilon(\tau+5N); \]
\[ G=c_{10}(\sigma+5M)-b_{10}(\tau+5N)+6b_{10}(\tau-\theta); \]
\[ -3B=\alpha M-2\beta N+\gamma P+2c_{12}(\tau-\theta), \]
conditions I and II can be replaced respectively by the following:
\[ \begin{aligned} &1)\quad \varepsilon>0,\quad \lambda=\tau-\theta=\alpha M-2\beta N+\gamma P=Q=0;\\ &2)\quad \varepsilon>0;\quad \lambda=\tau-\theta=\sigma+5M=\tau+5N=\omega+5P=R=0. \end{aligned} \]
- If not all the coefficients \(b_{10}, b_{01}, c_{10}\), and \(c_{01}\) are equal to zero and conditions (4) are satisfied, then at least one of the coefficients \(c_{10}\) and \(b_{01}\) must be nonzero. If \(c_{10}\ne0\), then by the nonsingular linear change of variables
\[ x_1=y,\qquad y_1=c_{10}x+b_{10}y \tag{7} \]
equation (1) is reduced to the case
\[ c_{10}=c_{01}=b_{10}=0,\qquad b_{01}=1. \tag{8} \]
The conditions for the presence of a center at the origin of equation (1), under conditions (8), were established in the work of A. F. Andreev [10] and consist in the fulfillment of the relations
\[ c_{30}>0,\qquad \alpha=2b_{30}\beta-c_{30}\gamma=\beta W=0, \]
where
\[ W=2b_{30}^{3}-b_{30}c_{30}c_{12}+c_{31}^{2}c_{03}. \]
Since the identity
\[ 2\beta^{2}W=c_{30}^{2}S+(2b_{30}\beta-c_{30}\gamma)(c_{30}b_{30}\gamma+2b_{30}^{2}\beta-c_{30}c_{12}\beta) \]
holds, the conditions of A. F. Andreev can be written in the form
\[ c_{30}>0,\qquad \alpha=2b_{30}\beta-c_{30}\gamma=S=0. \tag{9} \]
Substituting here, instead of \(c_{30}, c_{21}, c_{12}, c_{03}, b_{30}, b_{21}, b_{12}\), and \(b_{03}\), the coefficients of the equation obtained after transformation (7), equal (after multiplication by \(c_{10}^{3}\)), respectively, to
\[ c_{10}^{4}b_{03}-c_{10}^{3}b_{10}(b_{12}+c_{03})+c_{10}^{2}b_{10}^{2}(b_{21}+c_{12})-c_{10}b_{10}^{3}(b_{30}+c_{21})+b_{10}^{4}c_{30}; \]
\[ c_{10}^{3}b_{12}-c_{10}^{2}b_{10}(2b_{21}+c_{12})+c_{10}b_{10}^{2}(3b_{30}+2c_{21})-3b_{10}^{3}c_{30}; \]
\[ c_{10}^{2}b_{21}-c_{10}b_{10}(3b_{30}+c_{21})+3b_{10}^{2}c_{30},\qquad c_{10}b_{30}-b_{10}c_{30}; \]
\[ c_{10}^{3}c_{03}-c_{10}^{2}b_{10}c_{12}+c_{10}b_{10}^{2}c_{21}-b_{10}^{3}c_{30}; \]
\[ c_{10}^{2}c_{12}-2c_{10}b_{10}c_{21}+3b_{10}^{2}c_{30},\qquad c_{10}c_{21}-3b_{10}c_{30},\qquad c_{30}, \]
and taking into account conditions (4), we obtain, instead of \(c_{30}>0\), the relation
\[ 5c_{10}^{-1}\,[c_{10}M-2b_{10}N+b_{01}P+b_{10}(\tau-\theta)]>0, \]
instead of \(\alpha\) the expression \(-c_{10}^{2}(\tau-\theta)\), instead of \(3(2b_{30}\beta-c_{30}\gamma)\) the polynomial
\[ c_{10}^{3}(b_{10}\alpha-P)(\tau-\theta)+c_{10}^{4}(\alpha M-2\beta N+\gamma P), \]
and instead of \(S\) the expression \(c_{10}^{4}(c_{30}\beta-b_{30}\alpha)(\tau-\theta)+c_{10}^{5}Q\). Hence it follows that, for \(c_{10}\ne0\), in the case of a multiple zero root of the characteristic equation, the origin will be a center of equation (1) if and only if
\[ 3)\quad c_{10}(c_{10}M-2b_{10}N+b_{01}P)>0,\qquad \varepsilon=\lambda=\tau-\theta=\alpha M-2\beta N+\gamma P=Q=0. \]
If, however, under the fulfillment of conditions (4) \(c_{10}=0\), while \(b_{01}\ne 0\), then \(b_{10}=0\), \(c_{01}=0\), and the center conditions (9) take the form
\[ 4)\quad b_{01}c_{30}>0,\quad c_{10}=b_{10}=c_{01}=\alpha=2b_{30}\beta-c_{30}\gamma=S=0. \]
- Let us now assume that \(c_{10}=c_{01}=b_{10}=b_{01}=0\). Passing in this case to polar coordinates
\[ x=r\cos\varphi,\qquad y=r\sin\varphi, \]
we represent equation (1) in the form
\[ \frac{dr}{d\varphi}=r\,\frac{Z(\varphi)}{N(\varphi)}, \tag{10} \]
where
\[ Z(\varphi)=c_{03}\sin^4\varphi+(c_{12}-b_{03})\sin^3\varphi\cos\varphi+ \]
\[ +(c_{21}-b_{12})\sin^2\varphi\cos^2\varphi+(c_{30}-b_{21})\sin\varphi\cos^3\varphi-b_{30}\cos^4\varphi, \]
\[ N(\varphi)=b_{03}\sin^4\varphi+(b_{12}+c_{03})\sin^3\varphi\cos\varphi+ \]
\[ +(b_{21}+c_{12})\sin^2\varphi\cos^2\varphi+(b_{30}+c_{21})\sin\varphi\cos^3\varphi+c_{30}\cos^4\varphi. \]
As Forster showed [11], the origin is a center for equation (10) if and only if the equation \(N(\varphi)=0\) has no real roots, and
\[ \int_{0}^{2\pi}\frac{Z(\varphi)}{N(\varphi)}\,d\varphi=0. \tag{11} \]
Here it is necessary that \(b_{03}\ne 0\), since otherwise \(N(\varphi)\) vanishes for \(\cos\varphi=0\).
As is easily seen,
\[ 4Z(\varphi)\equiv(\alpha\cos^2\varphi+2\beta\cos\varphi\sin\varphi+\gamma\sin^2\varphi)-N'(\varphi), \]
and therefore condition (11) takes the form
\[ \int_{0}^{2\pi} \frac{\alpha\cos^2\varphi+2\beta\cos\varphi\sin\varphi+\gamma\sin^2\varphi}{N(\varphi)}\,d\varphi=0. \]
By the substitution \(z=\operatorname{tg}\varphi\), this condition reduces to
\[ \int_{-\infty}^{+\infty}\frac{\alpha+2\beta z+\gamma z^2}{N_1(z)}\,dz=0, \tag{12} \]
where the polynomial
\[ N_1(z)=b_{03}z^4+(b_{12}+c_{03})z^3+(b_{21}+c_{12})z^2+(b_{30}+c_{21})z+c_{30} \]
must have no real roots. Moreover, obviously \(b_{03}c_{30}>0\).
Condition (12) is also easily obtained from the results of Frommer [12].
Denote by \(z_1\) and \(z_2\) the roots of \(N_1(z)\) lying in the upper half-plane, and suppose that \(z_1\ne z_2\). Then condition (12) takes the form
\[ \operatorname*{res}_{z=z_1}\frac{\alpha+2\beta z+\gamma z^2}{N_1(z)} + \operatorname*{res}_{z=z_2}\frac{\alpha+2\beta z+\gamma z^2}{N_1(z)} =0, \]
or, equivalently,
\[ (\alpha+2\beta z_1+\gamma z_1^2)(z_2-\bar z_2)(z_2-z_1) - (\alpha+2\beta z_2+\gamma z_2^2)(\bar z-z_1)(z_1-\bar z_2)=0. \]
After division by \(z_2-z_1\), from this we obtain
\[ \alpha (z_1+z_2-\bar z_1-\bar z_2)+2\beta (z_1z_2-\bar z_1\bar z_2)+ \]
\[ +\gamma [z_1z_2(\bar z_1+\bar z_2)-\bar z_1\bar z_2(z_1+z_2)]=0. \tag{13} \]
If \(z_1=z_2\), then condition (12) is equivalent to the equality
\[ \operatorname*{res}_{z=z_1}\frac{\alpha+2\beta z+\gamma z^2}{N_1(z)}=0, \]
or, what is the same thing,
\[ (\beta+\gamma z_1)(z_1-\bar z_1)^2-(\alpha+2\beta z_1+\gamma z_1^2)(z_1-\bar z_1)=0. \]
The last equality coincides with (13), if in it one sets \(z_2=z_1\).
We note that, since \(N_1(z)=0\) is an equation of the fourth degree, for the roots \(z_1\) and \(z_2\) there exists a known expression in radicals directly in terms of the coefficients of the equation.
- Thus, the following has been proved.
Theorem. For the equation (1) to have a center at the origin, it is necessary and sufficient that at least one of the following five series of conditions be fulfilled:
\[ \begin{aligned} &1)\quad \varepsilon>0,\quad \lambda=\tau-\theta=\alpha M-2\beta N+\gamma P=Q=0;\\[4pt] &2)\quad \varepsilon>0,\quad \lambda=\tau-\theta=\sigma+5M=\tau+5N=\omega+5P=R=0;\\[4pt] &3)\quad c_{10}(c_{10}M-2b_{10}N+b_{01}P)>0,\quad \varepsilon=\lambda=\tau-\theta=\alpha M-2\beta N+\gamma P=\\ &\qquad =Q=0;\\[4pt] &4)\quad b_{01}c_{30}>0,\quad c_{10}=b_{10}=c_{01}=\alpha=2b_{30}\beta-c_{30}\gamma=S=0;\\[4pt] &5)\quad b_{03}c_{30}>0,\quad b_{10}=b_{01}=c_{10}=c_{01}=0,\quad \text{the equation} \end{aligned} \]
\[ b_{03}z^4+(b_{12}+c_{03})z^3+(b_{21}+c_{12})z^2+(b_{30}+c_{21})z+c_{30}=0 \]
has no real roots, and its roots \(z_1\) and \(z_2\), lying in the upper half-plane, satisfy the condition
\[ \operatorname{Im}\,[\alpha(z_1+z_2)+2\beta z_1z_2+\gamma z_1z_2(\bar z_1+\bar z_2)]=0. \]
As an example, consider the differential equation
\[ \frac{dy}{dx}= \frac{d x^3+(a-e+ed)x^2y+(b+d^2+1)xy^2+cy^3} {a x^3+(b+e^2)x^2y+(c-e+ed)xy^2-dy^3}. \]
The equation \(N_1(z)=0\) has roots
\[ z=\frac{1}{2}\left(e\pm\sqrt{e^2-4d}\right) \quad \text{and} \quad z=\frac{1}{2d}\left(-e\pm\sqrt{e^2-4d}\right). \]
For the origin to have a singular point of center type, it is necessary and sufficient that the conditions
\[ e^2<4d,\qquad e(d-1)+2(a+c)=0 \]
be fulfilled.
A particular case of this equation for \(d=1,\ e^2=2\) was considered by Frommer in [12].
References
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- Sibirskii, K. S. Izvestiya AN MSSR, No. 11, 86–91, 1963.
- Lyapunov, A. M. The general problem of the stability of motion. Gostekhizdat, 1950.
- Almukhamedov, M. I. Izvestiya Kazansk. fiziko-matem. ob-va, 9, ser. 3, 105–121, 1937.
- Sakharnikov, N. A. PMM, 14, issue 6, 651–658, 1950.
- Malkin, K. E. Volzhsk. matem. sb., issue 2, 1964, pp. 87–91.
- Sibirskii, K. S. DAN SSSR, 161, No. 2, 304–307, 1965.
- Sibirskii, K. S. Differential Equations, No. 1, 53–66, 1965.
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Received by the editors
April 22, 1965.
Institute of Mathematics with the Computing Center of the Academy of Sciences of the Moldavian SSR,
Kishinev Polytechnic Institute