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Remarks on a Paper of S. Lefschetz
A. F. Andreev
In the paper [1], S. Lefschetz considers, in the real domain, a differential system of the form
\[ \frac{dx}{dt}=X(x,y),\quad \frac{dy}{dt}=Y(x,y) \tag{1} \]
(\(X, Y\) are functions of \(x\) and \(y\) holomorphic at the point \(O(0,0)\), and \(O(0,0)\) is an isolated singular point of the system) and sets himself the task of giving a constructive process for reducing, in a finite number of steps, the study of the singular point \(O\) to the study of a finite set of ordinary points and isolated singular points of Bendixson type (for each of them at least one root of the characteristic equation is different from zero). Noting that for analytic \(X\) and \(Y\) difficulties arise here only in those cases when the curves \(X=0\), \(Y=0\) have a common tangent at the point \(O(0,0)\), S. Lefschetz studies the question of the existence in system (1) of trajectories approaching the point \(O\) with a definite tangent (he calls such trajectories \(TO\)-curves), precisely for the case when the direction of possible approach of the \(TO\)-curves to \(O\) is tangent to both curves \(X=0\), \(Y=0\).
First of all he proves that, by rotating the axes \(x\) and \(y\), changing the scale in \(x\) and \(t\), and making a special linear nonsingular transformation of the coordinates \(x,y\), system (1) can be brought to such a form that, after eliminating \(t\), we arrive at the equation
\[ \frac{dy}{dx}=\delta E(x,y)\frac{Y(x,y)}{X(x,y)}, \tag{2} \]
where \(\delta\ne 0\) is a constant, \(E(x,y)\) is holomorphic at \((0,0)\), \(E(0,0)=1\), and \(X(x,y)\) and \(Y(x,y)\) are polynomials in \(y\) of the same degree \(n\), of the form
\[ y^n+A_1(x)y^{n-1}+A_2(x)y^{n-2}+\ldots+A_n(x) \tag{3} \]
(\(A_i(x)\) is holomorphic at the point \(x=0\), \(A_i(0)=0\), \(i=1,2,\ldots,n\)), possessing the following properties: the curves \(X=0\), \(Y=0\) have the same number of imaginary and the same number of real branches passing through \((0,0)\), and have no multiple branches; each branch admits an analytic representation of the form
\[ y=\sum_{k=0}^{+\infty} a_k x^{1+\frac{k}{q}},\quad a_0\ne 0 \tag{4} \]
(\(q\ge 1\) is an integer, the same for all branches of both curves); if the direction \(y=mx\) (\(m\ne 0\) is a real constant) is tangent both
to \(X=0\), as well as to \(Y=0\), then the number of real branches of the curves \(X=0\) and \(Y=0\) tangent to this direction is the same, as is the number of imaginary branches; moreover, all these branches can be divided into pairs (one branch from the set \(X=0\), the other from the set \(Y=0\)) having in the representation (4) a fixed part (a full common part):
\[ \varphi(x)=\sum_{k=0}^{k_0} a_k x^{1+\frac{k}{q}},\qquad k_0 \geqslant 0,\quad a_0=m, \]
\[ y-\varphi(x)=\sum_{k=k_0+1}^{+\infty} a_k^{(j)} x^{1+\frac{k}{q}},\qquad a_{k_0+1}^{(j)}\ne 0,\quad j=1,2,\quad a_{k_0+1}^{(1)}\ne a_{k_0+1}^{(2)}. \]
The full common part of the representation of one pair of branches \(\varphi_1(x)\) may: 1) contain, as its first terms, the full common part of another pair \((\varphi_1 \supset \varphi_2)\); 2) coincide with it \((\varphi_1=\varphi_2)\); 3) be contained in it \((\varphi_1 \subset \varphi_2)\); or 4) intersect it \((\varphi_1\cap\varphi_2)\), i.e., contain only several of its first terms (in particular, the term \(mx\)).
Having carried out this extensive and delicate preliminary work (nos. 2–6), S. Lefschetz proceeds to investigate the question of the existence of \(TO\)-curves of equation (2), tangent at \((0,0)\) to the direction \(y=mx\) \((m\ne 0,\ x>0)\), under the assumption that at least one pair of real branches of the curves \(X=0,\ Y=0\) is tangent to this direction (nos. 7–10). This part of S. Lefschetz’s investigation contains errors.
In no. 7 a proposition is proved which can be formulated as follows: if the real function
\[ \psi_1(x)=\sum_{k=0}^{p-q-1} a_k x^{1+\frac{k}{q}},\qquad p>q,\quad a_0=m, \]
is a common part of the representation of one or several pairs of branches \(X=0,\ Y=0\), not coinciding with any full common part of paired branches \(X=0,\ Y=0\), and moreover the first common part \(\psi_2(x)\) containing \(\psi_1(x)\) is
\[ \psi_2(x)=\psi_1(x)+a_{p-q}x^{\frac{p}{q}} \]
(\(\psi_2(x)\) may coincide with the full common part \(\varphi(x)\) of one or several pairs of branches), then equation (2) has no \(TO\)-curves of the form
\[ y=\psi_1(x)+x^\mu y_1(x),\qquad \frac{p-1}{q}<\mu\leqslant \frac{p}{q}; \tag{5} \]
\[ y_1(x)=y_1^0+O(1),\qquad y_1^0\ne 0\text{ — constant},\quad \lim_{x\to 0}O(1)=0. \tag{6} \]
From this proposition the conclusion is drawn (end of no. 7) that if the functions
\[ \varphi_i(x)=\sum_{k=0}^{p_i-q-1} a_k^{(i)} x^{1+\frac{k}{q}},\qquad p_i>q,\quad a_0^{(i)}=m \]
\((i=1,2,\ldots,n_1;\ n_1\leq n)\) represent the set of full common parts of all pairs of real branches \(X=0,\ Y=0\), tangent to \(y=mx\), then any \(TO\)-curve of equation (2) adjoining the point
\(O\) in the direction \(y=mx\), admits a representation of the form (for some fixed \(i\in\{1,2,\ldots,n_1\}\))
\[ y=\varphi_i(x)+x^{\frac{p_i}{q}}y_i(x), \tag{7} \]
where one may assume that the functions \(y_i(x)\) have finite \((\ne 0)\) limits \(c_i\) as \(x\to 0\) (p. 22, lines 5–8).
However, the assertion formulated above is, generally speaking, false. Indeed, transforming equation (2) by means of the substitution (5), we arrive at the equation
\[ \frac{dy_1}{dx}= \frac{ \delta E_1Y_1- \left( m+\sum_{k=1}^{p-q-1}\left(1+\frac{k}{q}\right)a_k x^{\frac{k}{q}}+\mu x^{\mu-1}y_1 \right)X_1 }{ x^\mu X_1 }, \tag{8} \]
where
\[ E_1=E_1(x,y_1,\mu)\equiv E(x,\psi_1+x^\mu y_1),\qquad E_1(0,y_1,\mu)\equiv 1, \]
and \(X_1=X_1(x,y_1,\mu)\) and \(Y_1=Y_1(x,y_1,\mu)\) are polynomials in \(y_1\) of degree \(n\), whose linear factors, corresponding to paired branches \(X=0,\ Y=0\), have one of the two forms:
\[ y_1-a x^{r-\mu}-O(x^{r-\mu}),\quad r>\mu,\quad a\text{ constant}, \tag{9_1} \]
or
\[ x^{\mu-r}y_1-a-O(1),\quad r<\mu,\quad a\ne 0\text{ constant}. \tag{9_2} \]
Thus,
\[ X_1(0,y_1,\mu)\equiv Y_1(0,y_1,\mu)\equiv y_1^l,\quad l\le n. \]
If \(\delta\ne m\), then equation (8) can be represented in the form
\[ \frac{dy_1}{dx}= \frac{(\delta-m)y_1^l+x^{\lambda_2(\mu)}f_2(x,y_1,\mu)} {x^\mu\left[y_1^l+x^{\lambda_1(\mu)}f_1(x,y_1,\mu)\right]}; \tag{10} \]
\[ \lambda_j(\mu)>0,\quad 1\le \frac{p-1}{q}<\mu<\frac{p}{q}, \]
\(f_j(x,y_1,\mu)\) are series in integral positive powers of \(y_1\) and in nonnegative powers of \(x\), convergent for any of the \(\mu\) under consideration in some domain of the plane \(xy_1\) containing the axis \(y_1\), \(j=1,2\). Equation (10) has no solutions of the form (6). Consequently, equation (2), for \(\delta\ne m\), has no \(TO\)-curves of the form (5), (6). S. Lefschetz’s assertion in this case is true.
If, however, \(\delta=m\), then in the numerator of the right-hand side of equation (8) there occurs mutual cancellation of the terms containing the lowest powers of \(x\), and this equation takes the form
\[ \frac{dy_1}{dx}= \frac{P(y_1,\mu)+x^{\lambda(\mu)}f(x,y_1,\mu)} {x^{\sigma(\mu)}\left[y_1^l+x^{\lambda_1(\mu)}f_1(x,y_1,\mu)\right]}, \tag{11} \]
where \(\sigma(\mu)\le \mu,\ \lambda(\mu)>0\), \(P(y_1,\mu)\) is a polynomial in \(y_1\) with coefficients in the form of linear functions of \(\mu\), and \(f(x,y_1,\mu)\) is of the same kind as \(f_1(x,y_1,\mu)\).
If for some \(\mu\in\left(\dfrac{p-1}{q},\,\dfrac{p}{q}\right)\) in equation (11) \(\sigma(\mu)\ge 1\), and \(P(y_1,\mu)\) has a real root \(y_1=y_1^0\ne 0\) of odd multiplicity, then equation (11) has, for this \(\mu\), a solution of the form (6), and equation (2)—
solution of the form (5), (6). If for some \(\mu \in \left(\dfrac{p-1}{q},\, \dfrac{p}{q}\right)\) in equation (11) \(\sigma(\mu) < 1\), then it has a solution of the form (6) for each \(y_1^0 \ne 0\) for which \(P(y_1^0,\mu) \ne 0\). In this case equation (2) has an infinite set of \(TO\)-curves of the form (5), (6). As the following examples show, both of these possibilities occur.
Example 1. An equation of the form (2)
\[ y'=\frac{y-x-x^3+5x^4}{y-x-x^3-5x^4} \tag{2°} \]
(here \(\delta=m=1,\ \psi_1(x)=x,\ \psi_2(x)=x+x^3=\varphi(x)\)—the greatest common part of the representation of the unique pair of branches of the curves \(X=0,\ Y=0\)) as a result of a substitution of the form (5)
\[ y=x+x^\mu y_1,\quad 1<\mu<3; \]
for \(\mu=\dfrac{5}{2}\) is transformed into an equation of the form (11)
\[ y_1'=\frac{5\left[4-y_1^2+\sqrt{x}(1+5x\sqrt{x})y_1\right]}{2x\left[y_1-\sqrt{x}(1+5x)\right]}, \]
which has solutions of the form (6)
\[ y_1(x)=\pm 2+O(1). \]
Consequently, equation (2°) has \(TO\)-curves of the form (5), (6)
\[ y=x\pm 2x^{\frac{5}{2}}+O\!\left(x^{\frac{5}{2}}\right). \]
Example 2. An equation of the form (1)
\[ y'=\sqrt{2}\,\frac{y^2-2x^2y+x^4}{xy-2x^3} \tag{1°} \]
has a special order of curvature \(\nu^0=\sqrt{2}\) (see\(^1\) [2], p. 19) and therefore has a family of solutions
\[ y=\gamma x^{\sqrt{2}}+o\!\left(x^{\sqrt{2}}\right),\quad \gamma\ne 0\text{—parameter.} \tag{5°} \]
It is obvious that, as a result of transforming equation (1°) by S. Lefschetz’s method to the form (2), the \(TO\)-curves (5°) will receive an analytic representation of the form (5), (6), and not of the form (7).
In no. 8 S. Lefschetz studies the question of the existence, for equation (2), of \(TO\)-curves of the form (7). Here there are analogous errors.
\(^1\) In [2], the second paragraph on p. 19 is better formulated as follows: “The existence for the equation
\[ \frac{dy}{dx}=\frac{P(x,y)}{Q(x,y)} \]
of a special order of curvature \(\nu^0\) is equivalent to the representability of this equation in the form
\[ \frac{dy}{dx}=\nu^0\,\frac{y\Pi(x,y)+Y(x,y)}{x\Pi(x,y)+X(x,y)}, \]
where \(\Pi(x,y),\ X(x,y),\ Y(x,y)\) are holomorphic at the point \((0,0)\) and are such that \(X(x,ux^{\nu^0})\) has a higher order of smallness as \(x\to 0\) than \(x\Pi(x,ux^{\nu^0})\); \(u\) is a variable parameter, while \(Y(x,ux^{\nu^0})\) has the same property with respect to \(ux^{\nu^0}\Pi(x,ux^{\nu^0})\).”
Let us note, moreover, that in paper [1] the question of the existence, for equation (2), of \(TO\)-curves of the form (5) and (7), where \(y_1(x)\) has the property
\[ \lim_{x \to 0} y_1(x)=0,\qquad \lim_{x \to 0}\frac{y_1(x)}{x^\alpha}=\infty \]
or
\[ \lim_{x \to 0} y_1(x)=\infty,\qquad \lim_{x \to 0} y_1(x)x^\alpha=0 \]
for any fixed \(\alpha\), is not studied at all; such curves, as is known (see [2]), may exist.
References
- Lefschetz S. On a theorem of Bendixson. Boletin de la Sociedad Matematica Mexicana, 1, ser. 2, No. 1, 1956, 13—27.
- Andreev A. F. Vestnik LGU, No. 1, 1962, pp. 5—21.
Received by the editors
November 3, 1964
Leningrad Institute
of Precision Mechanics and Optics