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ISOCHRONISM OF THE CENTER OF CERTAIN SYSTEMS OF DIFFERENTIAL EQUATIONS
N. A. LUKASHEVICH
GENERAL PROPOSITIONS
1. Consider the system of two differential equations:
\[ \frac{dx}{dt}=-y-P(x,y)\equiv P_1(x,y), \qquad \frac{dy}{dt}=x+Q(x,y)\equiv Q_1(x,y), \tag{1} \]
where \(P(x,y)\) and \(Q(x,y)\) are holomorphic functions of the variables \(x\) and \(y\), whose expansions in powers of \(x\) and \(y\) contain no constant or linear terms. In addition, we shall assume that \((0,0)\) is a singular point of center type. We note that any system
\[ \frac{dx}{dt}=X(x,y), \qquad \frac{dy}{dt}=Y(x,y) \tag{2} \]
with holomorphic right-hand sides, in the presence of a singular point of center type for which the characteristic equation has purely imaginary roots, is brought, by a nondegenerate affine transformation, to the form (1), for which \((0,0)\) is a center.
System (1) in polar coordinates has the form
\[ \frac{d\rho}{dt}=\rho^2 U(\rho,\varphi), \qquad \frac{d\varphi}{dt}=1+\rho V(\rho,\varphi). \tag{3} \]
Since \(\rho=0\), by assumption, is a center, it follows from [1] that system (3) has a solution of the form
\[ \rho=C+f_2(\varphi)C^2+f_3(\varphi)C^3+\cdots, \tag{4} \]
where the series (4) converges for all \(\varphi\) and for sufficiently small values of \(C\) (\(C\) is an arbitrary constant).
Denote
\[ \Phi(\varphi,C)\equiv 1+\bigl[C+f_2(\varphi)C^2+\cdots\bigr] V\bigl([C+f_2(\varphi)C^2+\cdots],\varphi\bigr). \tag{5} \]
In this case the period of the solutions is computed by the formula
\[ T(C)=\int_0^{2\pi}\frac{d\varphi}{\Phi(\varphi,C)} \equiv 2\pi\left(1+\sum_{j=1}^{\infty} h_{2j} C^{2j}\right). \tag{6} \]
Definition. We shall call the center isochronous if \(T(C)\equiv 2\pi\). Obviously, for the center to be isochronous it is necessary and sufficient that
\[ h_{2j}=0,\qquad j=1,2,\ldots . \]
Another criterion for isochronicity is given in [2]. Namely, in order that \((0,0)\) of system (1) be an isochronous center, it is necessary and sufficient that there exist a transformation holomorphic in a neighborhood of \((0,0)\),
\[ \eta=x+\sum_{i+j\geq 2}^{\infty}\alpha_{ij}x^i y^j,\quad \xi=y+\sum_{i+j\geq 2}^{\infty}\beta_{ij}x^i y^j, \tag{7} \]
which carries system (1) into the system
\[ \frac{d\eta}{dt}=-\xi,\quad \frac{d\xi}{dt}=\eta. \tag{8} \]
From the foregoing it is clear that, in complexity, the solution of the problem of isochronicity of a center is equivalent to solving the center and focus problem and, in the general case, requires checking the fulfillment of an infinite number of conditions \(h_{2j}=0,\ j=1,2,\ldots\). In this connection it seems expedient to note some special cases of systems (1) with an isochronous center.
As is known [3], the center and focus problem for systems of the form (1), in the case where the right-hand sides of the latter are polynomials, is solved by a finite number of operations, the number of which depends in some way on the degrees of the polynomials in the right-hand sides of the system. An analogous result for systems with polynomial right-hand sides also holds in the solution of the problem of isochronicity.
Indeed, since the right-hand sides of the system are polynomials, all \(h_{2j}\) will be polynomials in a finite number of coefficients of the polynomials of the system. In this case, according to Hilbert’s basis theorem for polynomial ideals [4], there exists such a natural number \(m\) that all \(h_{2j},\ j=1,2,\ldots,\) can be written in the form
\[ h_{2j}=\sum_{k=1}^{m}\varepsilon_{jk}h_{2k},\quad j=1,2,\ldots, \tag{9} \]
where \(\varepsilon_{jk}\) are polynomials with respect to the coefficients of the right-hand sides of the system.
Unfortunately, the number \(m\), as a function of the degrees of the polynomials in the right-hand sides of the system, is not known a priori.
For systems of the form (1), under the condition that their right-hand sides satisfy certain additional conditions, there are other criteria for the presence of isochronicity of a center. For example, if system (1) is canonical, then for isochronicity of the center [2] it is necessary and sufficient that the condition
\[ \int_{0}^{2\pi}\rho^2(\varphi,C)\,d\varphi\equiv 2\pi C^2, \tag{10} \]
be fulfilled, where in place of \(\rho\) one should substitute the series (4).
Below we shall solve the question of isochronicity of the center \((0,0)\) for some special cases of system (1). In the presence of isochronicity we shall give a qualitative picture of the behavior of the integral curves as a whole.
2. The right-hand sides of system (1) satisfy the conditions:
\[ \frac{\partial P_1}{\partial x}\equiv\frac{\partial Q_1}{\partial y},\quad \frac{\partial P_1}{\partial y}\equiv-\frac{\partial Q_1}{\partial x}. \tag{11} \]
In [5] the system (2) was considered under the assumption that \(X(x,y)\) and \(Y(x,y)\) are polynomials of the second degree with respect to \(x\) and \(y\), satisfying conditions (11), and the possible behavior of the integral curves was established.
Consider system (1) when conditions (11) are fulfilled. Let \(M_0(x_0,y_0)\) be any singular point of it. We shall call \(M_0\) simple if the roots of the characteristic equation drawn up to determine its character are distinct from zero. Otherwise we shall call \(M_0\) multiple.
The characteristic equation drawn up to determine the character of \(M_0\) has the form
\[ \lambda^2-2P'_{1x}(M_0)\lambda+P_{1x}^{\prime 2}(M_0)+P_{1y}^{\prime 2}(M_0)=0. \tag{12} \]
Thus, if \(M_0\) is a simple singular point, then:
a) \(M_0\) is a focus if \(P'_{1x}(M_0)P'_{1y}(M_0)\ne0\);
b) \(M_0\) is a node if \(P'_{1x}(M_0)\ne0,\quad P'_{1y}(M_0)=0\);
c) \(M_0\) is a center if \(P'_{1x}(M_0)=0,\quad P'_{1y}(M_0)\ne0\).
Let us prove the last assertion. Let
\[ P(x,y)\equiv \sum_{k+j\ge2}^{\infty} a_{kj}x^k y^j,\qquad Q(x,y)\equiv \sum_{k+j\ge2}^{\infty} b_{kj}x^k y^j. \]
If condition c) is fulfilled, then the right-hand sides of system (3) have the form
\[ \rho^2 U(\rho,\varphi) = -\sum_{k=2}^{\infty}\rho^k \bigl[a_{k0}\cos(k-1)\varphi+ \]
\[ +\,b_{k0}\sin(k-1)\varphi\bigr] \equiv -\sum_{k=2}^{\infty}\rho^k u_k(\varphi), \tag{13} \]
\[ \rho V(\rho,\varphi) \equiv \sum_{k=2}^{\infty}\rho^{k-1} \bigl[-a_{k0}\sin(k-1)\varphi+ \]
\[ +\,b_{k0}\cos(k-1)\varphi\bigr] \equiv \sum_{k=2}^{\infty}\rho^{k-1}v_k(\varphi). \]
Since \(v'_k(\varphi)\equiv(k-1)u_k(\varphi)\), it follows, according to [3], that \(M_0\) is a center.
Let \(M_0\) be a multiple singular point. Suppose that \(z=x_0+iy_0\) is a root of multiplicity \(s\) of the equation
\[ f(z)=-[y+P(x,y)]+i[x+Q(x,y)]=0. \tag{14} \]
According to [6], \(M_0\) will be a point of the type of a multiple point of order \(2(s-1)\). The latter means that only elliptic regions adjoin \(M_0\), and the number of these is \(2(s-1)\). The Poincaré index of such a singular point is \(+s\).
Theorem 1. System (1), when conditions (11) are fulfilled, has no singular points of saddle type in the finite part of the plane.
Suppose that system (1) has a finite number of singular points. In this case, from Theorem 1 and the work [7] (pp. 32–47) it follows that all infinitely distant singular points are saddles.
Definition. A simply connected domain \(G\) of the \(xOy\) plane will be called a domain of a center if it contains only one singular point of center type and is entirely filled with closed integral curves.
According to [8], for systems of type (1), \(G\) is always open. In that case, by virtue of Theorem 1, we must conclude that \(G\) for system (1), when conditions (11) are satisfied, is always unbounded.
Theorem 2. System (1), when conditions (11) are satisfied, has no limit cycles.
Proof. Since the Poincaré index of a closed integral curve is equal to \(+1\), on the basis of Theorem 1 we must conclude that the system under consideration has no limit cycles enclosing several singular points. Thus, if cycles exist, they can enclose only one singular point of focus or node type. We shall show that the latter is impossible.
Let \(M_0\) be a singular point. After shifting the origin of coordinates to \(M_0\), system (1) has the form
\[ \frac{dx_1}{dt} = -P'_{1x}(M_0)x_1 - P'_{1y}(M_0)y_1 - P^*(x_1,y_1), \]
\[ \frac{dy_1}{dt} = P'_{1y}(M_0)x_1 - P'_{1x}(M_0)y_1 + Q^*(x_1,y_1), \tag{15} \]
where \(P^*(x_1,y_1)\) and \(Q^*(x_1,y_1)\) satisfy conditions (11).
As a topographic Poincaré system we take the family of closed integral curves of system (15), if in the latter we put \(P'_{1x}(M_0)=0\). This family, according to [1], has the form
\[ \Phi(x_1,y_1) \equiv x_1^2+y_1^2+F(x_1,y_1)=C^2. \tag{16} \]
By virtue of the preceding, the family (16) fills an unbounded domain.
Let us compute \(d\Phi(x_1,y_1)\). By virtue of system (15),
\[ d\Phi(x_1,y_1) \equiv (2x_1+F'_{x_1}) \left[-P'_{1x}(M_0)x_1-P'_{1y}(M_0)y_1-P^*(x_1,y_1)\right]+ \]
\[ +(2y_1+F'_{y_1}) \left[P'_{1y}(M_0)x_1-P'_{1x}(M_0)y_1+ \right. \]
\[ \left. +Q^*(x_1,y_1)\right] \equiv -2P'_{x_1}(M_0)(x_1^2+y_1^2). \tag{17} \]
From (17) it is clear that there are no limit cycles enclosing a singular point of focus or node type.
Theorem 3. For system (1), when conditions (11) are satisfied, all centers are isochronous.
Proof. Consider the system of partial differential equations
\[ \frac{\partial u}{\partial x}P_1+ \frac{\partial u}{\partial y}Q_1 = v, \qquad \frac{\partial v}{\partial x}P_1+ \frac{\partial v}{\partial y}Q_1 = -u. \tag{18} \]
It is not difficult to show that (18) has a solution \(u(x,y)\) and \(v(x,y)\) such that
\[ \frac{\partial u}{\partial x} \equiv \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} \equiv - \frac{\partial v}{\partial x}. \tag{19} \]
Indeed, differentiating the first equation partially with respect to \(y\), and the second with respect to \(x\), and adding the expressions obtained, we obtain an identically zero expression by virtue of conditions (11) and (19). In an analogous way we obtain an identically zero expression also in the case when the first equation is differentiated partially with respect to \(x\), and the second with respect to \(y\), and the obtained expressions are subtracted.
Thus the transformation (7) \(\eta(x,y)\equiv u(x,y)\), \(v(x,y)\equiv \xi(x,y)\), where \(u(x,y)\) and \(v(x,y)\) are a solution of system (18) satisfying conditions (19), exists and transforms system (1) into system (8).
- System (1) is canonical, i.e.
\[ \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}=0. \tag{20} \]
It is known [9] that a canonical system in the finite part of the plane can have among its singular points only points of center and saddle type. Below we shall show that in some cases the character of the isochronism of a center is predetermined by the character of the distribution of the other singular points of the system, situated both in the finite and in the infinitely distant parts of the plane.
Theorem 4. For the isochronism of \((0,0)\) of system (1), under fulfillment of condition (20), it is necessary that in the infinitely distant part of the plane
a) either there be no singular points at all; then in the finite part of the plane \((0,0)\) must be the only singular point—an isochronous center;
b) or, among the infinitely distant singular points, there must be complex ones distinct from a node.
Proof. If one assumes that in the infinitely distant part of the plane all singular points are nodes, then the domain \(G\) is finite and bounded by the separatrices of saddles. By virtue of the continuous dependence of the period on the initial data [10], the center is not isochronous.
Suppose that in the infinitely distant part of the plane there are no singular points.
Two cases are possible:
a) In the finite part of the plane there are singular points of saddle type. In that case the domain \(G\) is finite, bounded by the separatrices of the saddles, and consequently the center is not isochronous.
b) In the finite part of the plane there are no singular points of saddle type. Since the domain \(G\) is open, \((0,0)\) is the only singular point and the whole plane is filled with closed integral curves. In that case the domain of holomorphy of the integral coincides with the whole plane and, consequently, there always exists a transformation (7) transforming system (1) into system (8), i.e. the center is isochronous.
Corollary. If in the infinitely distant part of the plane all singular points are simple, then in the finite part of the plane there are no isochronous centers. The latter follows from the fact that every simple infinitely distant singular point for system (1), under fulfillment of (20), is a node.
Example. It is easy to verify that the system
\[
\frac{dx}{dt}=-y-ax^2,\qquad \frac{dy}{dt}=x+2axy+2a^2x^3
\tag{21}
\]
has an isochronous center at \((0,0)\). Indeed, the transformation (7)
\[
\eta=x,\qquad \xi=y+ax^2
\tag{22}
\]
transforms system (21) into system (8). In the infinitely distant part of the plane, system (21) has only one singular point—a two-separatrix saddle; in the finite part of the plane there is only one singular point, \((0,0)\).
In what follows we shall use the following notation:
\[ \omega_j(x,y)\equiv \sum_{k=0}^{j} a_{j-k,k}x^{j-k}y^k,\qquad H(x,y)\equiv \sum_{j\geqslant 3}^{\infty}\omega_j(x,y), \]
\[ P(x,y)\equiv \frac{\partial H}{\partial y},\qquad Q(x,y)\equiv \frac{\partial H}{\partial x}. \tag{23} \]
The integral of the equation of the trajectories of system (1) has the form
\[ x^2+y^2+2H(x,y)=C^2. \tag{24} \]
In polar coordinates (24) assumes the form
\[ \rho^2+2\sum_{j\geqslant 3}^{\infty}\rho^j\omega_j(\cos\varphi,\sin\varphi)=C^2. \tag{25} \]
Theorem 5. If
\[ \int_{0}^{2\pi}\omega_j(\cos\varphi,\sin\varphi)\,d\varphi\leqslant 0,\qquad j=3,4,\ldots, \]
then \((0,0)\) is a non-isochronous center.
Proof. From (25) we find
\[ \rho^2=C^2-2\omega_3(\varphi)C^3+\{6\omega_3^2(\varphi)-2\omega_4(\varphi)\}C^4+\cdots . \tag{26} \]
Substituting (26) into (10), we obtain
\[ h_2=2\int_{0}^{2\pi}[3\omega_3^2(\varphi)-\omega_4(\varphi)]\,d\varphi. \]
Since
\[ \int_{0}^{2\pi}\omega_4(\varphi)\,d\varphi\leqslant 0, \]
then \(h_2=0\) entails \(\omega_3(\varphi)\equiv 0\).
In an analogous way it is easy to show that the conditions
\[ \int_{0}^{2\pi}\omega_5(\varphi)\,d\varphi\leqslant 0 \]
and \(h_4=0\) entail \(\omega_4(\varphi)\equiv 0\), and so on. Consequently, the center cannot be isochronous if the conditions of Theorem 5 are satisfied and \(H(x,y)\not\equiv 0\).
Corollary. System (1), under condition (20), in the case when \(P(x,y)\) is a homogeneous polynomial, is not isochronous.
4. Consider the system
\[ \frac{dx}{dt}=-y-bx^3-(c-\beta)x^2y-(3d-\gamma)xy^2-fy^3, \]
\[ \frac{dy}{dt}=x+ax^3+(3b+\alpha)x^2y+(c+\beta)xy^2+dy^3. \tag{27} \]
Necessary and sufficient conditions for the existence of a center at \((0,0)\) for system (27) are given in [11–13].
The first two necessary conditions for isochronicity have the form
\[ h_2=\varepsilon_2(3f+3a+2c)=0, \tag{28} \]
\[ \begin{aligned} h_4={}&\varepsilon_4(51a^2+12c^2+51f^2+240b^2-12ac-12fc-90af\\ &+288bd+32a\beta-32f\beta+160b\alpha+160d\alpha+32\alpha^2)=0, \end{aligned} \tag{29} \]
where \(\varepsilon_2\cdot\varepsilon_4\ne0\) are constants independent of the parameters of the problem.
Thus, if \(h_2^2+h_4^2\ne0\), the solutions are not isochronous. Suppose that \(h_2=h_4=0\). Taking into account that \(h_2=0\), the condition \(h_4=0\) has the form
\[ 3a^2+3f^2+7.5b^2+7.5d^2+\alpha^2+90bd+(a-f)\beta+5(b+d)\alpha=0. \tag{30} \]
The first necessary and sufficient set of conditions for the existence of a center at \((0,0)\) is
\[ \alpha=\beta=\gamma=0. \tag{31} \]
According to the remark to Theorem 5, system (27), when conditions (31) are satisfied, is not isochronous.
Let us take the second set of necessary and sufficient conditions for the existence of a center at \((0,0)\):
\[ \alpha+\gamma=c+\frac{3}{2}(a+f)=\alpha+\frac{5}{2}(b+d)= \]
\[ =\beta+\frac{5}{2}(a-f)=bd-4af=0. \tag{32} \]
By means of a rotation transformation one can always arrange that in the transformed system the condition \(a=f\) is satisfied. Then conditions (30) and (32) are compatible only in the case when all coefficients of the system are equal to zero, i.e., the system degenerates into a linear one.
The third and fourth sets of necessary and sufficient conditions for the existence of a center at \((0,0)\), respectively, have the form
\[ \alpha=\gamma=b=d=0 \tag{33} \]
and
\[ \alpha+\gamma=0,\qquad k=\frac{2\beta}{\alpha}=\frac{f-a}{\alpha+2b+2d}, \]
\[ 2(b-d)(k^2-1)+(a+f-2c)k=0. \tag{34} \]
It is easy to verify that, by means of a rotation transformation, case (34) is reduced to case (33).
Therefore, in what follows it is enough to consider case (33), and to write the isochronicity conditions obtained here in the corresponding form for case (34).
Thus, under conditions (33), the necessary condition for isochronicity takes the form
\[ 3a^2+3f^2+\beta(a-f)=0. \tag{35} \]
To simplify the subsequent computations, put
\[ a=\frac{1}{8}(A+B),\qquad f=\frac{1}{8}(A-B),\qquad \beta=\frac{1}{8}(2C-B). \tag{36} \]
In the notation (36), the third necessary condition for isochronicity has the form
\[ 90A^4+18B^4+199A^2B^2+81B^2C^2+172A^2C^2+236A^2BC+ \]
\[ +72B^3C+18BC^3=0. \tag{37} \]
Consider two cases: 1) \(B=0\); 2) \(B\ne0\).
If \(B=0\), then from (28), (35), and (37) it follows that \(a=f=c=0\). Accordingly, system (27) in polar coordinates has the form
\[ \frac{d\rho}{dt}=\beta\rho^3\cos\varphi\sin\varphi,\qquad \frac{d\varphi}{dt}=1. \tag{38} \]
The motions are isochronous.
If \(B\ne0\), then, taking (35) into account, condition (37) is rewritten in the form
\[ 32B^6+864A^2B^4+1332A^4B^2+130A^6=0. \]
Since the latter is impossible for \(B\ne0\), the solutions are not isochronous.
Theorem 6. If system (27) has a singular point of center type at the origin, then the periodic solutions are isochronous only in the following cases:
\[ \text{1) } a=b=c=d=f=\alpha=\gamma=0, \]
\[ \text{2) } \alpha+\gamma=\beta=a=c=f=0,\qquad b=d,\qquad \alpha+4b=0. \]
Remark. If the periodic solutions of system (1) are isochronous, then the system has only one singular point, a center, and an algebraic integral (an algebraic curve of the second degree).
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Received by the editors
December 20, 1964
Belorussian State University
named after V. I. Lenin