ON COMPLEX CYCLES OF A CERTAIN DIFFERENTIAL EQUATION

L. A. Cherkas

The purpose of this article is a more precise estimate of the number of cycles of the equation considered in [1]. Let us consider, in the complex domain, the equation

\[ \frac{dy}{dx}=\frac{P(x,y)}{Q(x,y)}, \tag{1} \]

where \(P\) and \(Q\) are polynomials of degree \(n\).

We shall denote by \(D\) the complex projective space of the coefficients of the polynomials.

By a solution \(y=\varphi(x)\) of equation (1) we shall mean a complete analytic function. We shall use the definitions of [1]. Consider the complex projective space \((x,y)\). The corresponding real space will be denoted by \(R_4\).

An integral curve \(\Phi\) will be called the graph of a solution \(y=\varphi(x)\) in the space \(R_4\). An integral curve is a two-dimensional surface in four-dimensional space.

A cycle \(L\) on an integral curve \(\Phi\) will be called homotopic to zero if it can be contracted to a point along the integral curve.

A cycle \(L\) will be called a \(k\)-fold limit cycle if the integral \(U(x,y)=C,\; U(x_0,y_0)=0,\; (x_0,y_0)\in L\), when continued around the cycle \(L\), gives the integral \(U^*(x,y)=\varphi(U),\; U^*(x_0,y_0)=0\), and in the function

\[ \varphi(C)-C=\sum_{k=1}^{\infty}\gamma_k C^k\gamma_i=0,\quad i=1,\ldots,k-1,\quad \gamma_k\ne 0. \]

If \(k=1\), then the cycle \(L\) will be called a simple limit cycle.

If the projection \(L_x\) of the cycle onto the \(x\)-plane has no points for which \(Q(x,y)=0\), then the multiplicity of \(L\) can be found as follows. Let \((x_0,y_0)\in L\). Then the solution \(y=\varphi(x)\), \(y_0=\varphi(x_0)\), will be single-valued when continued around \(L_x\), i.e. \(\varphi(x_0)=\varphi(x_0)e^{2\pi i}\). Make the change of variables \(y-y_1=\varphi(x)\). With respect to \(y_1\) we obtain the equation

\[ \frac{dy_1}{dx}=\sum_{k=1}^{\infty}\varphi_k(x)y_1^k, \]

where the series on the right-hand side converges in some neighborhood of the contour \(L_x\) for \(|y_1|<\rho\). In this case the solution of the equation can be found in the form

\[ y_1=\sum_{k=1}^{\infty}p_k(x)C^k, \]

where \(p_1(x_0)=1\), \(p_k(x_0)=0\) for \(k>1\), and the series converges for all values of \(x\) in some neighborhood \(L_x\) and for \(|C|<\rho_1\). If \(p_k(x)\) is the first multivalued function upon continuation along \(L_x\), then the cycle \(L\) will be a \(k\)-fold limit cycle.

For real limit cycles the definition of multiplicity coincides with the usual one. If there is a \(k\)-fold limit cycle, then by perturbing the numerator and denominator of the right-hand side of equation (1) by polynomials of degree \(n\) with sufficiently small coefficients in modulus, it can be split into \(k\) rough limit cycles.

A rough limit cycle may be homotopic to zero. For example, the equation

\[ \frac{dy}{dx}=-\frac{x}{y+x(x^2+y^2-1)} \]

has, on the integral curve \(x^2+y^2=1\), a rough limit cycle homotopic to zero. But under certain conditions, when an equation is perturbed, rough cycles not homotopic to zero may separate off from a rough limit cycle homotopic to zero. This can be shown, for example, for the equation just indicated.

If, however, we have a cycle not homotopic to zero, then it will be a rough limit cycle if we exclude from consideration those equations for which the corresponding points of the space \(D\) form some set \(M\) that does not divide \(D\). Thus, in order to estimate the number of cycles not homotopic to zero and to one another, simple, and properly situated, it suffices to estimate the number of rough limit cycles of the equation possessing the same property. We shall call the indicated property property \(P\). The estimate can be improved if one takes into account rough limit cycles that are certainly homotopic to zero.

We shall consider an equation integrable by quadratures and give an estimate for the number of rough limit cycles possessing property \(P\) that split off from cycles of this equation when its numerator and denominator are perturbed by polynomials of degree \(n\).

Let us take as the initial equation

\[ \frac{dy}{dx}=\frac{p_{n-1}(x)y+\gamma y^n}{p_n(x)}, \tag{2} \]

where \(p_{n-1}(x)\), \(p_n(x)\) are polynomials of degrees \(n-1\) and \(n\), respectively. We shall impose some restrictions on their coefficients later. The general integral of equation (2) has the form

\[ y^{n-1}= \frac{ \displaystyle \prod_{i=1}^{n}(x-x_i)^{-\lambda_i} }{ \displaystyle C+\rho\int_{x_0}^{x}\prod_{i=1}^{n}(x-x_i)^{-\lambda_i-1}\,dx }, \tag{3} \]

where the numbers \(x_i,\lambda_i\), \(i=1,\ldots,n\), and \(\rho\) are determined by the coefficients of \(p_{n-1}(x)\), \(p_n(x)\), and \(\gamma\). It is easy to see that, by choosing the coefficients of the polynomials \(p_{n-1}(x)\), \(p_n(x)\), the numbers \(x_1,\ldots,x_n\) and \(\lambda_1,\ldots,\lambda_n\) can be made arbitrary. Choose the numbers \(x_1,\ldots,x_n\) real and distinct, and choose the numbers \(\lambda_1,\ldots,\lambda_n\) real and irrational in such a way that the function

\[ w(p)=\int_{p_0}^{p}\prod_{i=1}^{n}(1-px_i)^{-\lambda_i-1}\, p^{\sum_{i=1}^{n}(\lambda_i+1)-2}\,dp \]

mapped the upper half-plane \(p\) onto a convex \((n+1)\)-gon in the \(w\)-plane. Then the function

\[ w_1(x)=C+p\int_{x_0}^{x}\prod_{i=1}^{n}(x-x_i)^{-\lambda_i-1}\,dx \]

will map the upper half-plane \(x\) onto a curvilinear polygon whose sides are arcs of circles. Then, if one considers one branch of the function \(w_1(x)\), it vanishes at no more than one point of the \(x\)-plane, and hence the specified branch of the function standing on the right-hand side of (3) will have a single pole in the \(x\)-plane.

Equation (2) has algebraic solutions \(x=x_i,\ i=1,\ldots,n,\ y=0\). There are no other algebraic solutions. This can be proved in the same way as was done in [2].

Let us now prove that on the solutions of equation (2) there are no more than \((n-1)^2\) cycles possessing property \(P\). To this end consider the general integral of equation (3). Let \(L_1,\ldots,L_r\) be cycles in the \(x\)-plane with the points \(x_1,\ldots,x_n\) removed. Then to each cycle one may assign a certain value of the constant \(C\) in such a way that, upon continuation around it, the right-hand side of the integral (3) is a single-valued function. To this contour there will correspond \(n-1\) cycles on the solution of equation (2), since, extracting the root of degree \(n-1\) from (3), we obtain \(n-1\) values. Thus the number of cycles not homotopic to zero and to one another, simple, correctly situated on the solutions of equation (2), does not exceed \((n-1)r\), where \(r\) is the number of cycles homeomorphic to a circle, not intersecting one another and not homologous to zero on the \(x\)-plane with the points \(x_1,\ldots,x_n\) removed. If a cycle in the \(x\)-plane contains inside it only one singular point \(x_i\), then it can be shifted into a sufficiently small neighborhood of the singular point \((x_i,y_j)\), \(y_j\ne0\), \(i=1,\ldots,n,\ j=1,\ldots,n-1\), of equation (2), and therefore the cycles corresponding to this contour are homotopic to zero. The number of such cycles is \(n^2-n\). Hence we obtain that the number of cycles on the solutions of equation (2) possessing property \(P\) does not exceed
\(N_1=(n-1)r-n^2+n\). From [1] we have \(r\le 2n-1\); consequently, for \(N_1\) one may take \((n-1)^2\). All the cycles on the solutions of equation (2) that we have considered are rough, and therefore they remain under perturbation of the equation by polynomials of degree \(n\) with coefficients sufficiently small in modulus.

We shall now estimate the number of cycles possessing property \(P\) which can split off from the algebraic solutions of equation (2). It suffices to estimate the number of such cycles splitting off from the solution \(y=0\) under perturbation of equation (2). Denote the point of the space \(D\) corresponding to equation (2) by \(\alpha_0\), and the point corresponding to the perturbed equation by \(\alpha\). Let us have an arbitrary cycle \(L\), a topological image of a circle in the \(x\)-plane.

Introduce the succession function \(\Delta_L y(\Delta y,\alpha)\), equal to the increment of the solution \(y(x,\alpha)\) upon continuation around the cycle \(L\), where the solution satisfies the initial condition \(y(x_0,\alpha)=\Delta y,\ y(x_0,\alpha_0)=0\). If, for example,

vary along the ray \(a_0+A\tau\), then

\[ y(x,a_0+A\tau)=\sum_{j+k=0}^{\infty}\varphi_{jk}(x)\tau^j(\Delta y)^k, \tag{4} \]

where the series converges in some neighborhood of the contour \(L\) and for \(|\tau|<\rho\), \(|\Delta y|<\rho\). It is easy to see that
\[ \varphi_{01}(x)=k\prod_{i=1}^{n}(x-x_i)^{\lambda_i},\quad k=\mathrm{const}. \]
One can always choose \(\varphi_{jk}(x_0)=0\), \(j+k\ge 2\), \(\varphi_{10}(x_0)=0\), \(\varphi_{01}(x_0)=1\). If the contour \(L\) bounds a domain containing none of the coordinates \(x_i\), \(i=1,\ldots,n\), of the singular points lying on the solution \(y=0\), then the cycle arising for \(a\ne a_0\) is homotopic to zero, since the cycle present for \(a=a_0\) is homotopic to zero, and, for sufficiently small \(|a-a_0|\), no essentially singular or algebraic points of the solution \(y(x,x_0,\Delta y,a)\) of the perturbed equation arise inside the contour \(L\). That is, in this case all the functions \(\varphi_{jk}(x)\) in the series (4) will be single-valued in the domain bounded by the contour \(L\). If, however, the contour \(L\) bounds a domain containing some of the points \(x_i\), \(i=1,\ldots,n\), then in this case we have

\[ \Delta_L y(\Delta y,a_0+A\tau)=\sum_{j+k=1}^{\infty}\gamma_{jk}\tau^j(\Delta y)^k, \]

where \(\gamma_{01}\ne0\). Hence we obtain that from each cycle \(L\) enclosing some of the points \(x_i\), \(i=1,\ldots,n\), under perturbation of equation (2) there arises a rough limit cycle. Since the number of cycles on the solution \(y=0\) with the punctured points \(x_1,\ldots,x_n\) and possessing property \(P\) does not exceed, according to [1], \(2n-1\), the number of cycles separating from the solution \(y=0\) and possessing property \(P\) also does not exceed \(2n-1\). Of these, \(n\) cycles can be shifted into sufficiently small neighborhoods of the singular points of the perturbed equation, and one cycle can be shifted into a neighborhood of the point at infinity of the equation. Then we obtain that the number of cycles under consideration does not exceed \(n-2\). And since there are \(n+1\) straight lines that are solutions of equation (2), no more than \((n-2)(n+1)\) cycles possessing property \(P\) separate from algebraic solutions.

Let us estimate the number of cycles possessing property \(P\) that separate, under perturbation of the equation, from nonalgebraic solutions of equation (2). Let the contour \(L\) enclose one and only one of the points \(x_i\), \(i=1,\ldots,n\), and suppose that the function on the right-hand side of (3) is single-valued upon traversal of the contour \(L\). Considering one branch of the function, we obtain for the solution \(y=\varphi(x)\) only one critical pole, say \(x_i^0\). Upon traversal of a contour in the \(x\)-plane that goes around the point \(x_i\) \(k\) times and the point \(x_i^0\) \(n-1\) times, the solution will be single-valued. To each such contour there corresponds a rough limit cycle on a solution of equation (2), and from it a cycle not homotopic to zero may separate. Let the value \(x_i\) correspond to the singular points of equation (2)
\[ (x_i,y_i^j),\quad y_i^j\ne0,\quad j=1,\ldots,n-1. \]
Then, upon traversal of the indicated rough cycles, they will correspond to certain contours in the \(y\)-plane. One can count the number of contours that are homeomorphic to a circle, do not intersect, and are not homologous to zero or to one another in the \(y\)-plane with the punctured points \(y_i',\ldots,y_i^{\,n-1},\infty\). Their number does not exceed \(2n-1\). If we discard \(n\) contours enclosing only one point each, we obtain \(n-1\) contours, to which there correspond \(n-1\) cycles on the solutions

equation. The number of such cycles for all points \(x_i\) does not exceed \(n(n-1)\). Suppose now that a contour in the \(x\)-plane encloses more than one point \(x_i\), and that it corresponds to a single-valued function in the right-hand side of integral (3). If one traverses a contour enclosing, additionally, a critical pole of the solution \(n-1\) times, then on the corresponding solution we obtain a multiple cycle. Under a perturbation of equation (2), a cycle not homotopic to zero may separate from it. The number of such contours does not exceed \(n-1\). Thus, the number of cycles which separate from nonalgebraic solutions of equation (2) and have property \(P\) does not exceed \(n^2-1\).

Summing all cases, we obtain that the number of all cycles having property \(P\) which separate under a perturbation of equation (2) does not exceed \(3n^2-3n-2\).

Considering separately the case \(n=2\), one can show that the number of the indicated cycles does not exceed three.

References

1. E. M. Landis and I. G. Petrovsky, Matematicheskii sbornik, 43, no. 2, 1957.
2. L. A. Cherkas, Doklady AN BSSR, 7, no. 8, 1963.

Received by the editors
September 10, 1964

Minsk Radio Engineering Institute