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UDC 517.916
ON PERTURBATIONS OF A DIFFERENTIAL EQUATION
L. A. CHERKAS
Consider, in the complex domain, the equation
\[ \frac{dy}{dx}=-\frac{x+P(x,y)}{y+Q(x,y)}, \tag{1} \]
where \(P(x,y)\) and \(Q(x,y)\) are polynomials of degree \(n>1\) with complex coefficients, possessing an algebraic integral
\[ x^2+y^2+F(x,y)=C^2, \tag{2} \]
where \(F(x,y)\) is a rational function of \(x,y\), holomorphic at the point \((0,0)\), and its expansion in powers of \(x\) and \(y\) begins with terms of no lower than the third degree. We shall assume that all singular points of equation (1), including the singular points at infinity, are simple, and that the number of singular points at infinity is greater than two.
Let us note that the integral curve (2) in the complex space \((x,y)\) intersects, for \(C\ne0\), the plane \(y=0\) at two points \((C_1,0)\), \((C_2,0)\), such that \(C_1\) and \(C_2\) tend to zero if \(C\) tends to zero. This fact will be needed below.
Together with equation (1) we shall consider the perturbed equation
\[ \frac{dy}{dx}=-\frac{x+\lambda y+P(x,y)+P'(x,y)} {y-\lambda x+Q(x,y)+Q'(x,y)}, \tag{3} \]
where \(P'\) and \(Q'\) are polynomials of degree not exceeding \(n\), and their expansions in powers of \(x,y\) begin with terms of the second degree.
We shall call the singular point \((0,0)\) of equation (1) cyclic of order \(l\) if, by perturbations of the numerator and denominator of equation (1) by polynomials of degree \(n\) with arbitrarily small coefficients, one can arrange that the point \((0,0)\) in equation (3) be a singular point for which the Lyapunov coefficients satisfy the conditions
\[ g_i=0 \quad (i=1,3,\ldots,2l-1), \qquad g_{2l+1}\ne0, \tag{4} \]
and it is impossible to arrange that the conditions
\[ g_i=0 \quad (i=1,3,\ldots,2m-1), \qquad g_{2m+1}\ne0, \quad m>l \]
be fulfilled.
A simple singular point of equation (3) will be called a saddle if the ratio of the roots of the characteristic equation for it has negative real part; if it has positive real part, then the singular point will be called a node.
Let \(a_0\) be a point in the complex space \(D\) of the coefficients of the polynomials in the numerator and denominator of equation (1). Then to the perturbed equation (3) there corresponds the perturbed point \(a\).
If equation (3) has a node with the ratio of the roots of the characteristic equation not equal to a positive integer \(n\) or \(1/n\) for all \(a\) from some closed domain \(D_1 \subset D\), then in some fixed neighborhood of the node there are no cycles not homotopic to zero. This follows from the fact that in a neighborhood of the node equation (3) has a general integral of the form
\[ u(x,y,a)=c[v(x,y,a)]^{\lambda(a)}, \]
where \(u(x,y,a)\) and \(v(x,y,a)\) are holomorphic in some fixed neighborhood of the node for arbitrary \(a\) from the domain \(D_1\).
For a saddle this is the case if one considers those \(a\) for which the ratio of the roots of the characteristic equation is not equal to a negative number. We shall study a neighborhood of the singular point \((0,0)\) of equation (3).
If we have a singular point \((0,0)\) of cyclicity \(l\) in the equation corresponding to \(a_0\), then the perturbed point \(a\), satisfying conditions (4), may be taken in the form \(a_0+\tau a(\tau)\), where \(a(\tau)\) is a point in the space \(D\), whose coordinates are holomorphic functions of \(\tau\) in some neighborhood of \(\tau=0\).
This follows directly from the fact that the Lyapunov quantities \(g_i\) \((i=1,\ldots)\) are analytic functions of the coefficients of equation (3), holomorphic in some neighborhood of the point \(a_0\).
Consider equation (3) corresponding to \(a_0+\tau a(\tau)\). After introducing polar coordinates \(x=\rho\cos\varphi,\ y=\rho\sin\varphi\), the solution \(\rho(\varphi)\) of the equation satisfying the initial condition \(\rho(0)=c\) can be represented in the form
\[ \rho(\varphi)=\sum_{k=1}^{\infty} p_k[a_0+\tau a(\tau),\varphi]c^k, \tag{5} \]
where the series converges for \(0\le \varphi\le 2\pi,\ |c|<c_0\), the \(p_k\) are \(2\pi\)-periodic functions of \(\varphi\) for \(k=1,\ldots,2l-1\), and \(p_{2l+1}\) is the first nonperiodic function. Then the displacement function \(\Delta\rho=\rho(2\pi)-\rho(0)\) has the form
\[ \Delta\rho=\sum_{k=2l+1}^{\infty} p_k[a_0+\tau a(\tau),2\pi]c^k, \]
where the series converges for \(|\tau|<\tau_0,\ |c|<c_0\), and \(p_{2l+1}=g_{2l+1}[a_0+\tau a(\tau)]\) [3, 4].
Let the expansions of the coefficients \(p_k[a_0+\tau a(\tau)]\), \(k=2l+1,\ldots\), in powers of \(\tau\) begin with terms of order \(\tau^{\rho_k}\), i.e.,
\[ p_k=\tau^{\rho_k}(\gamma_{0k}+\gamma_{1k}\tau+\ldots),\qquad \gamma_{0k}\ne 0, \]
and let \(\rho_0\) be the least of the numbers \(\rho_k\), and \(N+1\) the least index of a coefficient \(p_k\) whose expansion contains a first term of order \(\rho_0\).
Now consider equation (3) corresponding to the point \(a_0+\tau a(\tau)+a'\tau^{\rho_0}\), where \(a'\) does not depend on \(\tau\). For it the displacement function has the form
\[ \Delta\rho=\sum_{k=1}^{\infty} p_k[a_0+\tau a(\tau)+a'\tau^{\rho_0},2\pi]c^k-c \]
and is a holomorphic function in some neighborhood of \(\tau=0,\ c=0,\ a'=0\). Note that \(p_k[\alpha_0+\tau a(\tau)+a'\tau^{\rho_0}],\ k\ne 1\), have the common factor \(\tau^{\rho_0}\), and \(\Delta\rho\) can be represented in the form
\[ \Delta\rho=\tau^{\rho_0}c\sum_{k=1}^{\infty}p_k'(\tau,a')c^{k-1}. \]
Consider the series \(\sum_{k=1}^{\infty}p_k'(\tau,a')c^{k-1}\). Since for \(\tau=0,\ a'=0\) it begins with the \(N\)-th power of \(c\), by the Weierstrass theorem [1] it can be represented in the form
\[ \sum_{k=1}^{\infty}p_k'(\tau,a')c^{k-1} = \Phi(\tau,a',c)\sum_{k=1}^{N+1}\gamma_k'(\tau,a')c^{k-1}, \]
where \(\Phi(0,0,0)\ne 0\). Hence it follows that \(\Delta\rho=0\) for \(\tau\ne 0\) if
\[ \sum_{k=1}^{N+1}\gamma'(\tau,a')c^{k-1}=0, \tag{6} \]
and the coefficients of equation (6) \(\gamma_k(\tau,a')\) satisfy the equalities \(\gamma_k'(0,0)=0\) \((k=1,\ldots,N)\), \(\gamma_{N+1}(0,0)\ne 0\), and are holomorphic functions in some neighborhood of \(\tau=0,\ a'=0\). Suppose that the vector \(a'\) can be chosen in such a way that all roots of equation (6) for \(\tau=0\) are distinct and lie in the domain of convergence of the series (5). There are examples of equations [6] for which the vector \(a'\) can be chosen in the indicated manner.
To each root of equation (6) there corresponds a cycle on the solution of equation (1) which does not disappear under perturbations of the equation. Since the cycle passes through the points \(x=C_1,\ C_2\) for \(y=0\), together with the root \(C_1\) equation (6) has the root \(C_2\). Hence it follows that \(N\) is an even number. All the cycles that have arisen are rough in the sense of the definition of [7]. Indeed, suppose equation (5) has a simple root \(c_0\).
Consider the general integral \(U(x,y)=c\) of equation (3) in a neighborhood of the point \(x=c_0,\ y=0\); \(U(x,y)\) is holomorphic at the point \((c_0,0)\) and \(U(0,c)\equiv c\). Such an integral exists and is uniquely determined [5]. Under analytic continuation along the cycle determined by the formula
\[ \rho(\varphi)=\sum_{k=1}^{\infty}p_k[\alpha_0+\tau a(\tau)+a'\tau^{\rho_0},\varphi]c_0^k,\qquad 0\leq \varphi\leq 2\pi, \]
the integral passes into an integral \(U^*(x,y)\).
Consider the general integrals \(\Phi=U(x,y)-c_0,\quad \Phi^*=U^*(x,y)-c_0\).
Then, if \(\rho(2\pi)=c_0+\sum_{k=1}^{\infty}m_k a^k\) under the condition that \(\rho(0)=c_0+a\), the integral \(\Phi\) is expressed in terms of \(\Phi^*\) by the formula
\[ \Phi=\sum_{k=1}^{\infty}m_k(\Phi^*)^k. \]
Since, by assumption, \(m_1\) is different from unity, this proves the assertion about the roughness of the cycle corresponding to a simple root of the succession function.
For what follows we shall need two lemmas.
Lemma 1. Let the cycles \(L_1\) and \(L_2\) be projected onto the \(x\)-plane as nonintersecting topological circles and bound a compact set \(K\) lying on a solution of equation (3). Then its projection onto the \(x\)-plane has as its boundary either the union of the projections of the cycles \(L_1\) and \(L_2\), or one of the projections of these cycles onto the \(x\)-plane.
Proof. Let \(K_x\) be the projection of the compact set \(K\), and let \(L_{1x}, L_{2x}\) be the projections of the cycles \(L_1, L_2\) onto the \(x\)-plane. Then if \(x_0 \in K_x\) and is a boundary point of \(K_x\) not belonging to the curves \(L_{1x}, L_{2x}\), then in this case the point \((x_0,y_0)\) lying on the compact set will be an algebraic branch point of the solution of equation (3) and, consequently, will make the denominator of equation (3) vanish. But the compact set \(K\) has only a finite number of branch points; therefore \((x_0,y_0)\) cannot be projected to a boundary point of \(K_x\), since \(K_x\) in the case under consideration cannot have isolated boundary points not lying on the curves \(L_{1x}\) and \(L_{2x}\). This proves the lemma.
Remark. The assertion of the lemma is valid in the case where the compact set is bounded by a single cycle \(L\).
Let the cycle be projected onto the \(x\)-plane as a topological circle \(L_x\). Then the projection of the compact set onto the \(x\)-plane will be a closed connected domain of the extended \(x\)-plane with boundary \(L_x\), and the solution on which the compact set lies has no singular branch points in this domain.
Lemma 2. For a cycle \(L\) homotopic to zero (and on the compact set which it bounds there are no singular points of the equation), the succession function is identically zero.
Proof. Let the cycle \(L\) be the boundary of a compact set \(K\) containing no singular points of equation (3). Take a point \(p \in K\), not lying on the cycle, and choose in some manner a family of cycles \(L(\alpha)\), \(0 \leq \alpha \leq 1\), \(L(\alpha) \to p\) as \(\alpha \to 1\), \(L(0)=L\), with \(L(\alpha)\) depending continuously on \(\alpha\). Since \(p\) is an ordinary point of equation (3), there exists an integral \(\Phi(x,y)\), which is a holomorphic function in some neighborhood of the point \(p\). This means that, for all \(\alpha\) sufficiently close to unity, the succession function along the cycle \(L(\alpha)\) will be identically zero. But then the succession function along the cycle \(L(0)\) will also be identically zero, since under analytic continuation of \(\Phi(x,y)\) we shall not encounter singular points in a sufficiently small neighborhood of the compact set \(K\). This proves the lemma.
Remark. The integral (2) defines a certain function \(y=f(x,C)\), and for sufficiently small values of \(C\) in modulus it has two branch points of the second order, the order of whose modulus as \(C\) tends to zero is equal to \(|C|\). From the second lemma it follows directly that the cycles, from which the cycles considered above split off under a perturbation of the equation, can be shifted along the solution so that their projections onto the \(x\)-plane are topological circles encircling exactly the two indicated branch points of the solution.
We shall now prove that the cycles arising under a perturbation of the equation are not homotopic to one another. Indeed, suppose they are homotopic to one another and bound a compact set \(K\) which is projected onto the \(x\)-plane as a closed connected domain whose boundary is the union of the projections of the cycles onto the \(x\)-plane or one of the projections of these cycles. We ...
we regard as topological neighborhoods that do not intersect each other and encircle exactly the two branch points of the solution indicated in the remark to the lemma.
If the set of points \((x, y)\) forms a cycle \(L_1(\alpha)\), then the set of points \((x, y^*)\) also forms a cycle \(L'_1(\alpha)\), homotopic to \(L_1(\alpha)\). Indeed, if the projection of the cycles \(L_1(\alpha)\) and \(L'_1(\alpha)\) onto the \(x\)-plane is contracted to the curvilinear segment connecting the indicated two branch points, then the cycles \(L_1(\alpha)\) and \(L'_1(\alpha)\) coincide. This means that the cycles can be contracted to one another along the solution. If now the compact set \(K\) contains branch points, then they are branch points of second order (the boundary of the compact set consists of two components!). The cycles \(L_1(\alpha)\) and \(L_2(\alpha)\) lie on different branches of the solution, differing only in sign. Consequently, in this case \(L_2(\alpha)\) is homotopic to \(L'_1(\alpha)\), and they form a compact set without branch points of the solution \(y=\varphi(x,\alpha)\). If the compact set contains no branch points, then in this case the cycles \(L_1(\alpha)\) and \(L_2(\alpha)\) lie on the solution \(y=\varphi(x,\alpha)\), holomorphic in the domain \(G\) with boundary \(L_{1x}\cup L_{2x}\). As \(\alpha\to\alpha_0\), the function \(y=\varphi(x,\alpha)\) converges to the function \(y=\varphi(x,\alpha_0)\) uniformly in \(x\) on the boundary of the domain \(G\). By the Weierstrass theorem, we have that \(\varphi(x,\alpha_0)\) is holomorphic in the domain \(G\), and the cycles \(L_1(\alpha_0)\) and \(L_2(\alpha_0)\) lie on the solution \(y=\varphi(x,\alpha_0)\).
Thus it has been proved that, in the limit as \(\alpha\to\alpha_0\), we must obtain a compact set bounded by the cycles \(L_1(\alpha_0)\) and \(L_2(\alpha_0)\). This means that the cycles \(L_1(\alpha_0)\) and \(L_2(\alpha_0)\) must lie on one solution of equation (1). The contradiction obtained proves the assertion.
We shall also prove that the cycles which arise are not homotopic to zero on the solutions of equation (3). The proof will be by contradiction. Let the cycle \(L(\alpha)\) be homotopic to zero. Its projection onto the \(x\)-plane may be assumed to be one and the same for all \(\alpha\) sufficiently close to \(\alpha_0\), and to be a topological neighborhood encircling two branch points of the solution. Then the projection of the compact set bounded by the cycle \(L(\alpha)\) is a closed domain \(G\) with boundary \(L_x\), whose interior points fill the part of the extended plane exterior with respect to \(L_x\). The compact set \(K\) cannot contain branch points, since its boundary consists of a single component. Thus it is the graph of some function, holomorphic everywhere in the domain \(G\), except for the point \(x=\infty\). We assume here that the solutions of equation (3) have no finite poles. This can always be achieved by a linear change of the variables \(x\) and \(y\). Indeed, equation (1), and hence also equation (3), under small perturbations have a finite number of simple singular points at infinity. Then, replacing the variables \(x\) and \(y\) in equation (3) by the formulas
\[ x=x_1\cos\alpha-y_1\sin\alpha,\qquad y=x_1\sin\alpha+y_1\cos\alpha, \]
we arrive at the equation
\[ \frac{dy_1}{dx_1} = -\frac{ x_1+\lambda y_1+\displaystyle\sum_{i+j=2}^{n} a_{ij}x_1^{i}y_1^{j} }{ y_1-\lambda x_1+\displaystyle\sum_{i+j=2}^{n} b_{ij}x_1^{i}y_1^{j} }. \tag{7} \]
The number \(\alpha\) can be chosen in such a way that \(b_{0n}\) is nonzero. For an equation with real coefficients this means that we can rotate the coordinate system in such a way that equation (3) will have no singular points at infinity in the direction of the \(OY\) axis.
Making in equation (7) the substitution \(y=\dfrac{1}{u}\), we obtain the equation
\[ \frac{d u}{d x_1} = \frac{ u^2\left(x_1 u^n+\lambda u^{n-1}+\displaystyle\sum_{i+j=2}^{n} a_{ij}x_1^i u^{\,n-j}\right) }{ u^{n-1}-\lambda x_1 u^n+\displaystyle\sum_{i+j=2}^{n} b_{ij}x_1^i u^{\,n-j} }. \tag{8} \]
If the solution \(y_1(x_1)\) of equation (7) has a pole at the point \(x_0\), then \(u(x_1)\) has a zero at this point. But for \(b_{0n}\ne 0\), equation (8) has the unique solution \(u\equiv 0\) satisfying the initial condition \(u(x_0)=0\). This proves the assertion.
Taking into account the absence of finite poles, we can represent the solution on which the cycle lies in the form
\[ y=\psi(x,\alpha)+\eta(\alpha)x, \]
where \(\psi(x,\alpha)\) is holomorphic in the domain \(G\), and \(\eta(\alpha)\) is the coordinate \(\eta\) of the singular point at infinity, obtained with the aid of the Poincaré substitution
\[ x=\frac{1}{\xi}, \qquad y=\frac{\eta}{\xi}. \]
Applying the Weierstrass theorem once again, in the limit as \(\alpha\to\alpha_0\) we obtain a compact set \(K(\alpha_0)\), which is the graph of a function holomorphic in the domain \(G\), excluding \(x=\infty\), and obtained from the integral (2). But any function that is defined by the integral (2) for \(C\ne 0\) and sufficiently close to zero has more than two singular points. This proves the assertion.
It can be shown that the arising \(\dfrac{N}{2}>l\) cycles are simple and properly situated, i.e., they can be shifted along the solution in such a way that their projections onto the \(x\)-plane are topological circles that do not intersect one another; likewise, they can be shifted along the solutions so that their projections onto the \(y\)-plane satisfy the same requirement stated above.
References
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- Lyapunov A. M. The General Problem of the Stability of Motion. Academy of Sciences of the USSR, 1956.
- Poincaré A. On Curves Defined by Differential Equations. Gostekhizdat, 1947.
- Otrokov N. F. Matematicheskii Sbornik, 41, no. 4, 1957.
- Bautin N. N. Matematicheskii Sbornik, 30:1, 1952, pp. 181–196.
- Cherkas L. A. Differential Equations, 1, No. 2, 182–186, 1965.
Received by the editors
September 30, 1965
Minsk Radio Engineering
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