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UDC 517.917
INTEGRAL CURVES OF THE DARBOUX EQUATION
N. A. LUKASHEVICH
In [1] (see also [2], pp. 378–383) N. N. Bautin qualitatively investigated the system
\[ \frac{dx}{dt}=ax+by-x(x^2+y^2),\qquad \frac{dy}{dt}=cx+dy-y(x^2+y^2) \tag{1} \]
in connection with the problem of synchronization of a tube generator, and proved the existence of a limit cycle.
In [3] the equation\(^*\)
\[ \frac{dy}{dx}=\frac{y+P_n(x,y)}{x+Q_n(x,y)} \tag{2} \]
was considered in the complex domain under the assumption that \(P_n\) and \(Q_n\) are homogeneous polynomials of degree \(n\).
Below we shall consider the Darboux differential equation
\[ \frac{dy}{dx}=\frac{yP_m(x,y)-M_n(x,y)}{xP_m(x,y)+N_n(x,y)} \tag{3} \]
in the case when \(P_m, M_n\), and \(N_n\) are homogeneous polynomials, respectively of degrees \(m\) and \(n\), with real constant coefficients, \(m\ne n-1\), \(P_m(x,y)\not\equiv 0\).
- \(n=0,\quad M_0(x,y)\equiv -a,\quad N_0(x,y)\equiv b,\quad a^2+b^2\ne 0.\) The differential equation (2) has the form
\[ \frac{dy}{dx}=\frac{a+yP_m(x,y)}{b+xP_m(x,y)}. \tag{4} \]
Equation (4), in the finite part of the plane, can have only one or two singular points. Each of them lies on the integral straight line \(by-ax=0\) and is a node with roots of the characteristic equation
\[ \lambda_1=P_m(x_0,y_0),\qquad \lambda_2=(m+1)P_m(x_0,y_0). \]
The general integral of equation (4) has the form
\[ (xv)^{m+1}\left[ C+(m+1)\sum_{j=0}^{m} a_{m-j} b^{\,j-m-1} \sum_{s=0}^{m-j}(-1)^{m-j-s+1} C_{m-j}^{s} \frac{v^{-j-s-1}}{j+s-1} \right]=1, \]
\[ \text{} \]
\(^*\) For \(n=2\), equation (2) was investigated in [7].
if \(b\ne 0,\ v=b\dfrac{y}{x}+a,\ P_m(x,y)=\sum_{j=0}^{m}a_jx^jy^{m-j}\) and
\[ (m+1)\sum_{j=0}^{m}\frac{a_{m-j}}{m-j+1}x^jy^{m-j+1}+Cx^{m+1}=a,\quad b=0. \]
- \(n=1,\ m>1\). Equation (3) has the form\(^*\)
\[ \frac{dy}{dx}=\frac{ax+by+yP_m(x,y)}{cx+dy+xP_m(x,y)}. \tag{5} \]
We exclude from consideration the case \(a=d=b-c=0\) as trivial. We shall assume that all singular points of equation (5), other than \(O_0(0,0)\), do not lie on the coordinate axes. Such an arrangement of singular points can always be achieved by rotating the coordinate axes. The characteristic equation composed for the singular point \(O_0\) has the form
\[ \lambda^2-(b+c)\lambda+bc-ad=0. \tag{6} \]
Let us indicate the basic properties inherent in equation (5).
Property 1. If the roots \(\lambda_1,\lambda_2\) of equation (6) are real, then equation (5) has no singular points of focus or center type.
Proof. Let \(O_j(x_j,y_j)\) \((x_j,y_j\ne 0)\) be any singular point of equation (5), distinct from \(O_0\). Put \(y=kx\). From the system
\[ a+bk+kP_m(1,k)x^m=0,\quad c+dk+P_m(1,k)x^m=0 \tag{7} \]
we find
\[ dk^2+(c-b)k-a=0. \tag{8} \]
If \(\lambda_1,\lambda_2\) are real, then the roots \(k_1,k_2\) of equation (8) are also real. All singular points of equation (5) lie on the integral straight lines \(y=k_ix\) \((i=1,2)\). Consequently, none of the singular points of the equation can be a focus or a center.
Corollary. Equation (5) has no limit cycles enclosing a singular point of node type.
Property 2. If the roots \(\lambda_1,\lambda_2\) of equation (6) are complex, then equation (5) has only one singular point \(O_0\)—a focus, if \(\operatorname{Re}\lambda_i\ne0\), and a center or a focus, if \(\operatorname{Re}\lambda_i=0\).
Property 2 follows from the fact that equation (6) has no real roots.
Suppose that \(\lambda_1=\alpha+\beta i,\ \lambda_2=\alpha-\beta i,\ \beta\ne0\). By a nondegenerate linear transformation we reduce equation (5) to canonical form, which in polar coordinates is rewritten as
\[ \frac{d\rho}{d\varphi}=\frac{1}{\beta}\left[\alpha\rho+\rho^{m+1}P_m^*(\cos\varphi,\sin\varphi)\right], \tag{9} \]
where \(P_m^*(\cos\varphi,\sin\varphi)\) is a homogeneous polynomial of degree \(m\) in \(\cos\varphi\) and \(\sin\varphi\).
Property 3. Equation (5) can have only one limit cycle. If the latter exists, it is an algebraic curve
\[ \sum_{j=0}^{m}A_jx^jy^{m-j}=1. \tag{10} \]
Property (3) follows from the form of equation (9).
\(\qquad\)
\(^*\) A special case of equation (5) was considered in [6].
Property (4). If the roots of equation (6) are purely imaginary and
\[ g=\int_0^{2\pi} P_m^*(\cos\varphi,\sin\varphi)\,d\varphi, \]
then \(O_0\) is a center if \(g=0\), and \(O_0\) is a focus if \(g\ne 0\).
Property (4) is obvious from (9) for \(\alpha=0\).
Property 5. If \(O_0\) is a center, then the center is isochronous. The period of the solution \(x=x(t)\), \(y=y(t)\), corresponding to any closed curve, is equal to
\[ \frac{2\pi}{\beta}. \]
Suppose that the roots \(\lambda_1,\lambda_2\) of equation (6) are real. From (7) we see that the greatest number of isolated singular points of equation (5) is equal to five, and the latter is possible only in the case when \(k_i\) \((i=1,2)\) are simple roots of equation (8); \(m\) is even and \(\lambda_i P_m(1,k_i)<0\), where
\[ \lambda_{1,2}=\frac{b+c\pm\sqrt{\delta}}{2},\qquad k_{1,2}=\frac{b-c\pm\sqrt{\delta}}{2};\qquad \delta\equiv (b-c)^2+4ad \]
and the plus sign is taken for \(\lambda_1,k_1\), while the minus sign is taken for \(\lambda_2,k_2\).
The characteristic equation, formed for the singular point \(O_j(x_j,y_j)\) \((x_j,y_j\ne 0)\), has the form
\[ \mu^2+(p+q)\mu-pq=0,\qquad p\equiv\frac{dy_j^2+ax_j^2}{x_jy_j},\qquad q\equiv mP_m(x_j,y_j). \tag{11} \]
From (11) we find \(\mu_1=-\sqrt{\delta}\), \(\mu_2=-m\lambda_1\), if \(O_j\) lies on the straight line \(y=k_1x\), and \(\mu_1=\sqrt{\delta}\), \(\mu_2=-m\lambda_2\), if \(O_j\) lies on the straight line \(y=k_2x\).
Suppose that \(\lambda_1\lambda_2>0\), \(\lambda_1\ne\lambda_2\), i.e. \(O_0\) is a node. Since \(\lambda_1\ne\lambda_2\), we have \(k_1\ne k_2\), and equation (5) can have either one, or three, or five singular points, all of them simple \((\mu_1\mu_2\ne 0)\). If (5) has only three singular points, and \(m\) is even, then the \(O_j\) lie on one of the straight lines \(y=k_ix\), and therefore are either nodes or saddles. One of them will be a node and the other a saddle if \(m\) is odd.
If there are five singular points, then two of them are saddles, and the other three (including \(O_0\)) are nodes.
Suppose that \(\lambda_1=\lambda_2\ne 0\). Equation (5) has only one integral straight line \(y=k_1x\). Therefore, if \(m\) is odd, equation (5) has only two singular points. If \(m\) is even and \(\lambda_1P_m(1,k_1)<0\), then (5) has three singular points. \(O_0\) will be the unique singular point of equation (5) if \(m\) is even and \(\lambda_1P_m(1,k_1)>0\). It is not difficult to verify that the singular points different from \(O_0\) are open saddle-nodes.
Suppose that \(O_0\) is a saddle. Since \(\lambda_1\lambda_2<0\), all singular points different from \(O_0\) are nodes.
Suppose that \(\lambda_1=0\), \(\lambda_2\ne 0\). By a nondegenerate linear transformation equation (5) can always be reduced to the form
\[ \frac{du}{dv}=\frac{uP_m(v,u)}{-v+ku+vP_m(v,u)}, \tag{12} \]
for which the character of \(O_0\) is easy to establish (see, for example, [4]).
Finally, if \(\lambda_1=\lambda_2=0\), then, according to [5], \(O_0\) will be a node if \(m\) is even, and a closed saddle-node if \(m\) is odd. Moreover, in this case \(O_0\) is the only singular point of equation (5).
The behavior of the separatrices of saddles can be established each time by using the circumstance that equation (5) can be integrated in elementary functions. Thus, the qualitative picture of the behavior of the integral curves as a whole for equation (5) can always be constructed.
3. \(m=0,\quad n>2.\) The differential equation (1) can always be reduced to the form
\[ \frac{dy}{dx}=\frac{y+M_n(x,y)}{x+N_n(x,y)}. \tag{13} \]
Property 6. Equation (13) has no singular points of focus or center type.
Indeed, if on the axis \(ox\) or \(oy\) there are singular points different from \(O_0\), then the coordinate axes are integral straight lines; therefore the singular points lying on them cannot be centers or foci.
Let \(O_j(x_j,y_j)\) \((x_j,y_j\ne0)\) be any singular point different from \(O_0\). Put \(y=zx\). In the new variables equation (13) has the form
\[ \frac{dz}{dx}= \frac{x^{\,n-2}\left[M_n(1,z)-zN_n(1,z)\right]} {1+x^{\,n-1}N_n(1,z)}. \tag{14} \]
Let \(z=z_j\) be a root of the equation
\[ f(z)=M_n(1,z)-zN_n(1,z)=0. \tag{15} \]
Obviously, the straight line \(z=z_j\) is an integral line for equation (14), and, consequently, the straight line \(y=z_jx\) is an integral line for equation (13). Since \(O_j\) lies on the line, it cannot be a focus or a center.
Let \(O_j(x_j,z_j)\) be any singular point of equation (14). The roots of the characteristic equation composed for the singular point \(O_j\) have the form
\[ \lambda_1=(n-1)x_j^{\,n-2}N_n(1,z_j),\qquad \lambda_2=x_j^{\,n-2}f'_{z_j}(z_j). \tag{16} \]
Thus, \(O_j\) is a node if \(N_n(1,z_j)f'_{z_j}(z_j)>0\), and a saddle if
\[ N_n(1,z_j)f'_{z_j}(z_j)<0. \]
Since \(N_n(1,z_j)\ne0\), it follows that \(\lambda_1\ne0\). Suppose that
\[ f'_{z_j}(z_j)=0,\qquad f^{(k)}_{z_j}(z_j)\ne0,\quad k\ge2. \]
In the variables \(v=z-z_j,\ u=x-x_j\), equation (14) has the form
\[ \frac{dv}{du}= \frac{x_j^{\,n-2} f^{(k)}_{z_j}(z_j)v^k+\cdots} {\lambda_1u+x_j^{\,n-1}N'_n(1,z_j)v+\cdots}. \tag{17} \]
Thus, \(O_j\) is an open saddle-node if \(k\) is even; a node if \(k\) is odd and
\[ x_j^{\,n-2}f^{(k)}_{z_j}(z_j)<0, \]
and a saddle if \(k\) is odd but
\[ x_j^{\,n-2}f^{(k)}_{z_j}(z_j)>0. \]
Consider two cases: 1) \(n\) is even and 2) \(n\) is odd. Suppose that \(n\) is even. It is easy to see that to each root \(z=z_j\) of equation (15) such that \(N_n(1,z_j)\ne0\) there corresponds only one singular point \(O_j\), different from \(O_0\), of equation (13),
lying on the integral line \(y=z_jx\). If \(N_n(1,z_j)=0\), then on the integral line \(y=z_jx\) there are no singular points distinct from \(O_0(0,0)\).
Suppose that \(n\) is odd. On the integral line \(y=z_jx\) there are three singular points \(O_0(0,0)\), \(O_1(x_1,y_1)\), and \(O_2(-x_1,-y_1)\), if \(N_n(1,z_j)<0\). If, however, \(N_n(1,z_j)>0\), then on the integral line \(y=z_jx\) there are no singular points distinct from \(O_0(0,0)\). Moreover, if on the line \(y=z_jx\) there are three singular points, then the points \(O_1\) and \(O_2\) are simultaneously either nodes, or saddles, or open saddle-nodes.
To determine the possible number and character of infinitely distant singular points, put
\[ x=\frac{1}{\xi},\qquad y=\frac{\tau}{\xi}. \tag{18} \]
In the variables \(\xi,\tau\), equation (13) has the form
\[ \frac{d\tau}{d\xi}=-\frac{f(\tau)}{\xi^n+\xi N_n(1,\tau)}. \tag{19} \]
Let \(\tau=\tau_j\) be a root of the equation \(f(\tau)=0\). To this root there corresponds an infinitely distant singular point with roots of the characteristic equation
\[ \mu_1=N_n(1,\tau_j),\qquad \mu_2=-f'_{\tau_j}(\tau_j). \tag{20} \]
If on the integral line \(y=\tau_jx\) there are singular points \(O_j(x_j,y_j)\) distinct from \(O_0(0,0)\), then the infinitely distant singular point \((\xi=0,\tau=\tau_j)\) is a node if \(O_j\) is a saddle; \((\xi=0,\tau=\tau_j)\) is a saddle if \(O_j\) is a node; and, finally, \((\xi=0,\tau=\tau_j)\) is an open saddle-node if \(O_j\) is an open saddle-node.
Thus, the qualitative picture of the behavior of the integral curves as a whole for equation (13) will be completely determined if we settle the question of limit cycles. From what was set forth above it is clear that limit cycles of equation (15) can be expected only in the case when \(O_0\) is its only singular point and there are no integral lines \(y=z_jx\).
Equation (13) in polar coordinates has the form
\[ \frac{d\rho}{d\varphi} = -\frac{u(\varphi)}{v(\varphi)}\rho - \frac{1}{v(\varphi)}\rho^{2-n}, \tag{21} \]
where
\[ u(\varphi)\equiv \cos\varphi\, N_n(\cos\varphi,\sin\varphi) +\sin\varphi\, M_n(\cos\varphi,\sin\varphi), \]
\[ v(\varphi)\equiv \sin\varphi\, N_n(\cos\varphi,\sin\varphi) -\cos\varphi\, M_n(\cos\varphi,\sin\varphi). \tag{22} \]
Property 7. Equation (13) can have no more than one limit cycle. The latter exists only under the condition \(v(\varphi)<0\) for all \(\varphi\in[0,2\pi]\).
Proof. Indeed, if there exists \(\varphi=\varphi_j\) such that \(v(\varphi_j)=0\), then there are no limit cycles, since \(\varphi=\varphi_j\), \(\varphi=\varphi_j+\pi\) are integral. Let \(v(\varphi)\ne 0\), \(\varphi\in[0,2\pi]\). From (21) we find
\[ \rho^{\,n-1} = \exp\left[(1-n)\int \frac{u(\varphi)}{v(\varphi)}\,d\varphi\right] \times \]
\[ \times \left\{ C-(n-1)\int \frac{1}{v(\varphi)} \exp\left[(n-1)\int \frac{u(\varphi)}{v(\varphi)}\,d\varphi\right]d\varphi \right\}. \tag{23} \]
To a real periodic solution of (23) with period \(2\pi\) (the latter is possible only under the condition that \(v(\varphi)<0\) for all \(\varphi\in[0,2\pi]\)) there corresponds a limit cycle of equation (13).
Remark. In Sec. 2 it was proved that the limit cycle of equation (5) is always an algebraic curve of degree \(m\). In contrast to this, the limit cycle of equation (13) may also fail to be an algebraic curve. For example, the differential equation
\[ \frac{dy}{dx}= \frac{-y+x^3+x^2y+xy^2+\frac12 y^3} {-x+\frac12 x^3-x^2y-y^3} \]
has the algebraic limit cycle \(x^2+y^2=4\). The limit cycle of the equation
\[ \frac{dy}{dx}= \frac{-y-x^3+y^3} {-x+x^3+2x^2y+2xy^2+y^3} \]
has the form
\[ \rho^2=-e^{2\varphi}\left[ \frac{4e^{4\pi}}{1-e^{4\pi}} \int_0^{2\pi}\frac{e^{2\varphi}\,d\varphi}{2+\sin 2\varphi} + \int_0^\varphi\frac{e^{2\vartheta}\,d\vartheta}{2+\sin 2\vartheta} \right]. \]
The limit cycle of system (1) has the form
\[ (d^2+c^2+\delta)x^2-2(ac+bd)xy+(a^2+b^2+\delta)y^2=(a+d)\delta,\qquad \delta=ad-bc. \]
- \(m\geqslant 1,\quad n\geqslant 2.\) Let \(O_j(x_j,y_j)\) be any singular point such that \(x_jy_j\ne0\). Putting \(y=kx\) to determine the values of \(k\), we obtain the equation
\[ kN_n(1,k)+M(1,k)=0. \tag{24} \]
Obviously, the straight lines \(y=k_jx\), where \(k_j\) is any root of equation (24), are integral curves. Since all singular points \(O_j\) lie on these straight lines, none of them can be a focus or a center.
It is easy to see that singular points different from \(O_0\) and lying on the coordinate axes likewise cannot be foci or centers. Moreover, on any of the integral straight lines \(y=k_jx\) there cannot be more than three (including \(O_0(0,0)\)) singular points.
In the same way as was done in Secs. 1, 2, 3, for equation (3) one can establish necessary and sufficient conditions for the presence of such facts as: 1) when \(O_0(0,0)\) is a center; 2) when equation (3) has a limit cycle; 3) when equation (3) has an algebraic integral, etc.
References
- N. N. Bautin, On a differential equation having a limit cycle. ZhTF, 9, 610, 1939.
- A. A. Andronov, A. A. Vitt, S. E. Khaikin, Theory of Oscillations. Fizmatgiz, 1959.
- L. A. Cherkas, DAN BSSR, 7, No. 8, 1963.
- N. A. Lukashevich, Differential Equations, vol. I, No. 2, 196—198, 1965.
- A. F. Andreev, Investigation of the behavior of integral curves of a system of two differential equations in a neighborhood of a singular point. Vestnik LSU, ser. math., phys. and chem., No. 8, 1955.
- Kh. R. Latipov, Sh. Sharipov, Proceedings of Samarkand University, 144, 1964.
- A. N. Berlinskii, Differential Equations, 2, No. 3, 353—360, 1966.
Received by the editors
April 12, 1965
Belorussian State University
named after V. I. Lenin