UDC 517.916

STABILITY OF A SINGULAR CYCLE IN THE CRITICAL CASE

L. A. CHERKAS

Let there be given the system

\[ \frac{dx}{dt}=X(x,y), \qquad \frac{dy}{dt}=Y(x,y), \tag{1} \]

where \(X, Y\) are holomorphic in some domain \(G\) of the real variables \(x\) and \(y\). By a singular cycle of system (1) we shall mean a closed curvilinear polygon whose sides are separatrices of singular points of saddle type or of hyperbolic sectors of compound singular points, with each interior angle of the polygon forming one hyperbolic sector.

In the case when \(X, Y\) are polynomials, one of the sides of a singular cycle may be a part of the line at infinity.

A singular cycle \(L\) bounding a domain \(S\) will be called stable as \(t \to +\infty\) if for any \(\varepsilon>0\) there exists a neighborhood \(O(\delta,L)\) of it such that the trajectories of the system passing at \(t=t_0\) through a point \((x_0,y_0)\in O(\delta,L)\cap S\) do not leave the \(\varepsilon\)-neighborhood of \(L\) for \(t>t_0\).

Analogously one can introduce the notion of asymptotic stability, as well as instability, of a singular cycle.

If \(n\) simple singular points of saddle type lie on the singular cycle, then in this case Doliak [1] gave a sufficient condition for stability of the cycle. Let us formulate this condition. Number the vertices of the singular cycle in order of increasing time. Let these be the points \(P_i(x_0^i,y_0^i)\) \((i=1,\ldots,n)\). In a neighborhood of the point \(P_i(x_0^i,y_0^i)\), by means of the substitution

\[ x-x_0^i=x_i(u,v)=\alpha_{10}u+\alpha_{01}v+\cdots, \]

\[ y-y_0^i=y_i(u,v)=\beta_{10}u+\beta_{01}v+\cdots, \]

where \(\alpha_{10}\beta_{01}-\alpha_{01}\beta_{10}\ne0\) and \(x_i(u,v), y_i(u,v)\) are holomorphic in some neighborhood of the point \((0,0)\), \(|u|<\rho\), \(|v|<\rho\), the differential equation corresponding to system (1) can be represented in the form

\[ u\,dv+v\,[\lambda_i+f_i(u,v)]\,du=0, \]

where \(f_i(u,v)\) is holomorphic in some neighborhood of \((0,0)\), and its expansion in powers of \(u\) and \(v\) begins with terms of not lower than first order; moreover, the curve \(u=0,\ |v|<\rho\) will coincide with a part of the arc \(P_{i-1}P_i\), while the curve \(v=0,\ |u|<\rho\) will coincide with a part of the arc \(P_iP_{i+1}\).

Let \(\lambda=\lambda_n\lambda_{n-1}\cdots\lambda_1\). Then if \(\lambda-1>0\) the singular cycle will be stable as \(t\to+\infty\). The result remains valid if one of the vertices

polygon will be a compound singular point of saddle-node type, in which the ratio of the roots of the characteristic equation, taken in the above-established order, is equal to zero. In this case the singular cycle will be unstable as \(t \to +\infty\).

If on the singular cycle there lies only one singular point of saddle type and it is a loop formed by a separatrix of the saddle going into the same saddle \(P_1(x_0, y_0)\), and if \(\delta = \dfrac{\partial X}{\partial x} + \dfrac{\partial Y}{\partial y}\) at the point \((x_0, y_0)\) is nonzero, then for \(\delta > 0\) the loop is stable, and for \(\delta < 0\) it is unstable as \(t \to +\infty\) [2].

If, however, \(\delta = 0\) or \(\lambda - 1 = 0\), then the question of the stability of the singular cycle remains open. We shall regard this case as critical.

Consider the system

\[ \frac{dy}{dt}=xy\,\frac{dx}{dt}=a_{00}+a_{01}x+ax^2+a_{11}xy+a_{10}y+a_{20}y^2. \tag{2} \]

If the following conditions are imposed on the coefficients of the system:

\[ \begin{aligned} 1)&\quad a_{01}^2-4a_{00}a<0,\\ 2)&\quad a_{11}^2-4a_{20}(a-1)<0,\\ 3)&\quad \alpha=\frac{-1+a}{a}<0,\\ 4)&\quad a_{00}a_{20}<0, \end{aligned} \tag{3} \]

then the system has two singular cycles, one of which encloses the upper half-plane \(y>0\), the other the lower \(y<0\). Both cycles give us the critical case of stability. We shall give sufficient conditions for the stability of the singular cycle enclosing the upper half-plane. Along with system (2) we shall consider the equation

\[ \frac{dy}{dx}= \frac{xy}{a_{00}+a_{01}x+ax^2+a_{11}xy+a_{10}y+a_{20}y^2}. \tag{4} \]

Condition 1) of (3) guarantees the absence of singular points on the axis \(OY\); conditions 2), 3) give us one singular point at infinity of saddle type in the direction of the axis \(OY\), and, finally, condition 4) means that equation (4) has two singular points of focus type located on the axis \(x=0\) and on different sides of the line \(y=0\). The integral curves of equation (4), in the absence of a center, are: the line \(y=0\), spirals located in the upper or lower half-planes, and limit cycles. Moreover, it has been proved [5] that, in the case \(a_{11}a_{01}<0\), limit cycles can occur only in the half-plane \(y>0\), while in the case \(a_{11}a_0>0\) they, respectively, can occur only in the lower half-plane.

In order to study a neighborhood of the infinitely distant point, we make the Poincaré transformation

\[ x=\frac{1}{\xi},\qquad y=\frac{\eta}{\xi}. \]

We obtain the following equation:

\[ \frac{d\eta}{d\xi}= \frac{\eta\left[(-1+a)+a_{00}\xi^2+a_{11}\eta+a_{10}\xi\eta+a_{01}\xi+a_{20}\eta^2\right]} {\xi\left(a+a_{00}\xi^2+a_{01}\xi+a_{11}\eta+a_{10}\xi\eta+a_{10}\eta^2\right)}. \tag{5} \]

If the number \(\alpha=\dfrac{-1+a}{a}\) is sufficiently poorly approximated by rational fractions, for example
\[ |\alpha m+n|>\frac{c}{(m+n)} \]
for positive integers \(m\) and \(n\) [3, 4], then there exists a general integral of equation (5) of the form
\[ \eta \xi^{\frac{1-a}{a}}\left(1+b_{10}\xi+b_{01}\eta+\ldots\right)=C, \tag{6} \]
where the series \(1+b_{10}\xi+b_{01}\eta+\ldots\) converges for \(|\xi|<\rho,\ |\eta|<\rho\). Passing again to the variables \(x\) and \(y\), we obtain an integral of equation (4) of the form
\[ \frac{y}{x^{\frac{1}{a}}}\left(1+\frac{b_{10}}{x}+\frac{b_{01}y}{x}+\ldots\right)=C, \tag{7} \]
where the series converges for \(|x|>\dfrac{1}{\rho},\ \left|\dfrac{y}{x}\right|<\rho\). We shall first consider those values of \(\alpha\) for which the integral (7) exists.

Let us now consider, in the phase plane, the solution of system (2) passing at \(t=0\) through the point \((M_1,1)\), \(M_1>0\). If \(M_1\) is sufficiently large, then the solution will first intersect the straight line \(y=1\), as time increases and as time decreases, respectively at the points \((-M_3,1)\), \((-M_2,1)\). If for all sufficiently large values of \(M_1\) the inequality
\[ \frac{1}{M_2}-\frac{1}{M_3}>0 \]
holds, then the singular cycle will be stable; if
\[ \frac{1}{M_2}-\frac{1}{M_3}<0, \]
it will be unstable as \(t\to+\infty\). In the case
\[ \frac{1}{M_2}-\frac{1}{M_3}\equiv 0 \]
we obtain a center, and hence nonasymptotic stability. The center conditions for system (2) are known, and we shall not consider this case.

Take a number \(M>0\) and consider the value \(y_1=y(M)\). Assuming \(M_1\) and \(M\) such that, as \(x\) varies on the interval \([M,M_1]\), the solution falls into the domain of convergence of the series (7), for \(y_1\) we obtain the equation
\[ \frac{y_1}{M^{\frac{1}{a}}}\left(1+\frac{b_{10}}{M}+\frac{b_{01}y_1}{M}+\ldots\right)=\gamma_0, \]
where
\[ \gamma_0=\frac{1}{M_1^{\frac{1}{a}}} \left(1+\frac{b_{10}+b_{01}}{M_1}+\ldots\right) = \frac{1}{M_1^{\frac{1}{a}}} \left(1+\frac{\beta_1}{M_1}+\ldots\right). \]

Hence we find \(y_1\) in the form of a series in powers of \(\gamma_0\):
\[ y_1=\alpha_1(M)M^{\frac{1}{a}}\gamma_0+\alpha_2(M)M^{\frac{2}{a}}\gamma_0^2+\ldots \]

The resulting series converges for all sufficiently large values of \(M_1\). We find the value \(y_0=y(0)\) of the solution under consideration. Since the right-hand side of equation (4) is holomorphic in some neighborhood of the solution \(y=0\) for \(x\in[0,M]\), the solution \(y(x)\) satisfying the initial condition \(y(M)=y_1\) can be represented in the form of a series

\[ y=\varphi_1(x)y_1+\varphi_2(x)y_1^2+\cdots, \]

where \(\varphi_1(M)=1,\ \varphi_i(M)=0\ (i=2,\ldots)\), and the series converges for \(x\in[0,M]\) and \(|y_1|<\rho_1\). Hence

\[ y_0=\varphi_1(0)y_1+\varphi_2(0)y_1^2+\cdots . \tag{8} \]

The computations give

\[ \varphi_1(x)=\exp\left(\int_M^x \frac{x}{P}\,dx\right),\qquad \varphi_2(x)=-\varphi_1(x)\int_M^x \frac{a_{11}x^2+a_{10}x}{P^2}\,\varphi_1(x)\,dx, \]

\[ P=a_{00}+a_{01}x+ax^2. \]

Substituting the expression found for \(y_1\) into (8), we obtain

\[ y_0=\varphi_1(0)\alpha_1(M)M^{\frac{1}{a}}\gamma_0+ \left[\alpha_2(M)\varphi_1(0)+\varphi_2(0)\alpha_1^2(M)\right]M^{\frac{2}{a}}\gamma_0^2+\cdots . \tag{9} \]

The series converges for all sufficiently large values of \(M_1\). Since the coefficients of the powers of \(\gamma_0\) in the series (9) must not depend on the choice of \(M\), we have

\[ \varphi_1(0)\alpha_1(M)M^{\frac{1}{a}}=\mu_1, \]

\[ \left[\varphi_1(0)\alpha_2(M)+\varphi_2(0)\alpha_1^2(M)\right]M^{\frac{2}{a}}=\mu_2, \]

\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]

where \(\mu_i\ (i=1,\ldots)\) are constants independent of \(M\). It is easy to compute the value of \(\mu_1\). Indeed, we have

\[ \alpha_1(M)=\frac{1}{1+\dfrac{b_{10}}{M}+\dfrac{b_{20}}{M^2}+\cdots}. \tag{10} \]

Hence

\[ \mu_1=\lim_{M\to+\infty}\varphi_1(0)M^{\frac{1}{a}}. \]

Thus, we have \(y_0=\sum_{k=1}^{\infty}\mu_k\gamma_0^k\), where the series converges for all sufficiently large values of \(M_1\).

We shall now express \(y_0\) in terms of \(M_2\). For this it suffices to consider the equation

\[ \frac{dy}{dx}=\frac{xy}{a_{00}-a_{01}x+ax^2-a_{11}xy+a_{10}y+a_{20}y^2} \tag{11} \]

and to apply to it the formula for \(y_0\) obtained above, replacing \(a_{01}\) by \(-a_{01}\), \(a_{11}\) by \(-a_{11}\), and leaving the remaining coefficients unchanged. All coefficients occurring in the preceding formulas for equation (11) will be denoted by an overbar. Then we obtain \(y_0=\sum_{k=1}^{\infty}\bar{\mu}_k\bar{\gamma}_0^{\,k}\). Comparing the two expansions for \(y_0\), we obtain a relation between \(\gamma_0\) and \(\bar{\gamma}_0\). It is easy to compute that \(\mu_1=\bar{\mu}_1\). Taking this into account, we obtain

\[ \overline{\gamma}_0=\gamma_0+p_2\gamma_0^2+p_3\gamma_0^3+\cdots, \tag{12} \]

where

\[ p_2=\frac{\mu_2-\overline{\mu}_2}{\mu_1},\qquad p_3=\frac{\mu_3-\overline{\mu}_3}{\mu_1} -\frac{2\overline{\mu}_2(\mu_2-\overline{\mu}_2)}{\mu_1^2}. \]

Using (12), we express \(\dfrac{1}{M_2}\) as a series in powers of \(\dfrac{1}{M_1}\), \(\dfrac{1}{M_1^{1/a}}\).

We have

\[ \frac{1}{M_2}\left(1+\frac{\overline{\beta}_1}{M_2} +\frac{\overline{\beta}_2}{M_2^2}+\cdots\right)^a = \gamma_0^a(1+p_2\gamma_0+\cdots)^a . \]

Hence we obtain

\[ \begin{aligned} \frac{1}{M_2} &=\frac{1}{M_1}\left[1+\frac{a\beta_1}{M_1}+\cdots\right] \left[1+\frac{a(\mu_2-\overline{\mu}_2)}{M_1^{1/a}\mu_1} \left(1+\frac{\beta_1}{M_1}+\cdots\right)^2+\cdots\right] \\ &\quad -\frac{a\beta_1}{M_1^2} \left[1+\frac{a\beta_1}{M_1}+\cdots\right]^2 \left[1+\frac{a(\mu_2-\overline{\mu}_2)}{M_1^{1/a}\mu_1} \left(1+\frac{\beta_1}{M_1}+\cdots\right)^2+\cdots\right]^2 +\cdots \end{aligned} \]

or

\[ \frac{1}{M_2}=\sum_{i+j=1}^{\infty}\delta_{ij}M_1^{-\left(i+\frac{j}{a}\right)}. \]

Making the change of variables \(y=\dfrac{1}{y_1}\), \(x=\dfrac{x_1}{y_1}\), we arrive at the equation

\[ \frac{dy_1}{dx_1} = \frac{x_1y_1} {-a_{20}-a_{11}x_1+(1-a)x_1^2-a_{01}x_1y_1-a_{10}y_1-a_{00}y_1^2}. \tag{13} \]

Expressing \(\dfrac{1}{M_3}\) as a series in powers of \(\dfrac{1}{M_1}\), \(\dfrac{1}{M_1^{1/(1-a)}}\), we obtain

\[ \frac{1}{M_3} = \sum_{i+j=1}^{\infty}\delta'_{ij}M_1^{-\left(i+\frac{j}{1-a}\right)}, \]

where the coefficients \(\delta'_{ij}\) are obtained from the coefficients \(\delta_{ij}\) by replacing \(a_{01}\) by \(-a_{11}\), \(a_{11}\) by \(-a_{01}\), \(a_{00}\) by \(-a_{20}\), and \(a\) by \(1-a\). Thus, the sign of the difference \(\dfrac{1}{M_2}-\dfrac{1}{M_3}\) will be determined by the sign of the series

\[ \sum_{i+j=1}^{\infty} \delta_{ij}M_1^{-\left(i+\frac{j}{a}\right)} - \delta'_{ij}M_1^{-\left(i+\frac{j}{1-a}\right)}. \tag{14} \]

Without loss of generality, one may assume \(\dfrac{1}{2}<a<1\), since if this condition is not satisfied we can perform the Poincaré transformation and first consider equation (13).

It is easy to establish that \(\delta_{0j}=\delta'_{0j}=0\). We also have

\[ \delta_{10}-\delta'_{10}=0,\qquad \delta_{20}=a(\beta-\bar{\beta}_1)=\frac{a_{11}}{1-a}-\frac{a_{01}}{a},\qquad \delta'_{20}-\delta_{20}=0. \]

It has been established by calculation that also \(\delta_{i0}-\delta'_{i0}=0\) \((i=3,\ldots,8)\). We shall not use the latter fact.

Thus, for sufficiently large positive values of \(M_1\), the sign of the series (14) will be determined by the sign of the principal term, which will be

\[ \frac{\delta_{11}}{M_1^{1+\frac{1}{a}}} \]

in the case where

\[ \delta_{11}=\frac{a(\mu_2-\bar{\mu}_2)}{\mu_1}\ne 0. \]

Assuming that all the conditions formulated above on the coefficients of system (2) are fulfilled, we have the following: if \(\mu_2-\bar{\mu}_2>0\), then the singular cycle enclosing the half-plane \(y>0\) will be stable as \(t\to +\infty\); if \(\mu_2-\bar{\mu}_2<0\), then it will be stable as \(t\to -\infty\). If, however, \(\mu_2-\bar{\mu}_2=0\), then the question of the stability of the cycle remains open.

We proceed to the computation of \(\mu_2\). Fix all the coefficients of equation (4) except \(a\). The coefficient \(a\) will be regarded as a complex variable. Then \(\mu_2(a)\) will be a function of the variable \(a\), defined for all complex values for which the real part \(\operatorname{Re} a>\frac12\), \(\operatorname{Im}a\ne 0\), for all real values \(a>1\), for which \(a>0\) and is not equal to an integer, and also for those \(\frac12<a<1\) for which the integral (7) exists. For real values of \(a\) the function \(\mu_2\) assumes real values. The function \(\mu_2(a)\) is continuous in the domain of variation of \(a\) under consideration and is holomorphic for all \(a\) in this domain, except those values which satisfy the inequality \(\frac12<a<1\). Therefore we may compute it for the real values under consideration \(a>1\), and then continue it analytically.

Let us compute \(\mu_2\) for real values \(a>1\), for which \(a\) is not equal to a positive integer. We have

\[ \mu_2= \lim_{M\to+\infty} \left[ \varphi_1(0)\alpha_2(M)-\varphi_1^2(0)\alpha_1^2(M) \int_M^0 \frac{a_{11}x^2+a_{10}x}{P^2} \exp\left(\int_0^x \frac{x}{P}\,dx\right)\,dx \right]M^{\frac{2}{a}} . \]

Since \(\alpha_2(M)\) is an infinitesimal of order \(\frac{1}{M}\) as \(M\to+\infty\), we obtain

\[ \mu_2= -\mu_1^2 \int_{+\infty}^{0} \frac{a_{11}x^2+a_{10}x}{P^2} \exp\left(\int_0^x \frac{x}{P}\,dx\right)\,dx, \]

\[ \bar{\mu}_2= -\mu_1^2 \int_{-\infty}^{0} \frac{a_{11}x^2+a_{10}x}{P^2} \exp\left(\int_0^x \frac{x}{P}\,dx\right)\,dx. \]

Then

\[ \mu_2-\bar{\mu}_2=\mu_1^2\int_{-\infty}^{\infty}\frac{a_{11}x^2+a_{10}x}{P^2} \exp\left(\int_0^x \frac{x}{P}\,dx\right)\,dx . \]

We have

\[ \frac{\mu_2-\bar{\mu}_2}{\mu_1^2} = A\int_{-\infty}^{\infty}\frac{a_{11}x^2+a_{10}x}{P^2}\,dx + a^{-\frac{1}{2a}}\int_{-\infty}^{\infty} \frac{a_{11}x^2+a_{10}x}{P^{2-\frac{1}{2a}}} \times \]

\[ \times \exp\left( -\frac{a_{01}}{2a^2\sqrt{\Delta}}\, \operatorname{arctg}\frac{x+\dfrac{a_{01}}{2a}}{\sqrt{\Delta}} \right)\,dx, \]

where

\[ \Delta=a_{00}-\frac{a_{01}^2}{4a^2},\qquad A=-\exp\left( -\frac{a_{01}}{2a^2\sqrt{\Delta}}\, \operatorname{arctg}\frac{a_{01}}{2a\sqrt{\Delta}} \right) \left(\frac{a_{00}}{a}\right)^{\frac{1}{2a}} . \tag{15} \]

The first of the integrals is readily calculated; we shall give the second a form convenient for calculation. By making the substitution \(x+\dfrac{a_{01}}{2a}=p\sqrt{\Delta}\) and integrating the second integral in formula (15) by parts, it takes the form

\[ B\int_{-\infty}^{\infty} \frac{e^{\beta\operatorname{arctg}p}}{(p^2+1)^{1-\frac{1}{2a}}}\,dp + C\int_{-\infty}^{\infty} \frac{e^{\beta\operatorname{arctg}p}}{(p^2+1)^{2-\frac{1}{a}}}, \tag{16} \]

where \(\beta=-\dfrac{a_{01}}{2a^2\sqrt{\Delta}}\); the coefficients \(B\) and \(C\) are elementary functions of the coefficients of equation (4), and we do not write them out because of their bulkiness. Both integrals in (16) are reduced, by the substitution \(\operatorname{arctg}p=x\), to the integral

\[ \Phi(\alpha,\beta)=\int_{-\pi/2}^{\pi/2}(\cos x)^\alpha e^{\beta x}\,dx \]

with \(\alpha=1-\dfrac{1}{a}\) for the first and \(\alpha=2-\dfrac{1}{a}\) for the second integral. Expressing \(\cos x\) through the exponential function by Euler’s formulas and integrating by parts, we obtain the recurrence formulas

\[ \Phi(\alpha,\beta)= \Phi(\alpha+1,\beta-i)\frac{\alpha+i\beta+2}{\alpha+1}, \]

\[ \Phi(\alpha,\beta)= \Phi(\alpha+1,\beta+i)\frac{\alpha-i\beta+2}{\alpha+1}. \]

Hence

\[ \Phi(\alpha,\beta)= \Phi(\alpha+2,\beta)\, \frac{(\alpha+2)^2+\beta^2}{(\alpha+1)(\alpha+2)} . \tag{17} \]

Formula (17) is convenient in that, for real \(\alpha\) and \(\beta\), it contains real functions.

Take the expansion of \(e^{\beta x}\) in a Fourier series on \([-\pi,\pi]\)

\[ e^{\beta x}=\frac{a_0}{2}+\sum_{k=1}^{\infty}a_k\cos kx+b_k\sin kx. \]

The series converges uniformly on the interval \(\left[-\dfrac{\pi}{2},\,\dfrac{\pi}{2}\right]\). Hence

\[ \Phi(\alpha,\beta)=\left(\frac{a_0}{2}J_{\alpha+20}+\sum_{k=1}^{\infty}a_kJ_{\alpha+2k}\right) \frac{(\alpha+2)^2+\beta^2}{(\alpha+1)(\alpha+2)}, \]

where

\[ J_{\alpha k}=\int_{-\pi/2}^{\pi/2}(\cos x)^\alpha\cos kx\,dx. \]

By means of the substitution \(\operatorname{tg}x=\sqrt{\dfrac{t}{1-t}}\), this is reduced to the gamma function

\[ J_{\alpha+20}=2\sqrt{\pi}\, \frac{\alpha+1}{\pi(\alpha+3)} \frac{\Gamma\left(\dfrac{\alpha}{2}+\dfrac{1}{2}\right)} {\Gamma\left(\dfrac{\alpha}{2}\right)}. \]

We split the integral \(J_{\alpha+2k}\) into two, applying the formula

\[ \cos kx=\cos(k-1)x\cos x-\sin(k-1)x\sin x, \]

and integration by parts in the second integral gives

\[ J_{\alpha+2k}=\frac{\alpha-k+4}{\alpha+3}\,J_{\alpha+3\,k-1}. \]

We use this formula to compute \(J_{\alpha+2k}\):

\[ J_{\alpha+2k}= \frac{ 2^k\left(\dfrac{\alpha+2-k}{2}+1\right)\cdots \left(\dfrac{\alpha+2-k}{2}+k\right) }{ (\alpha+3)(\alpha+4)\cdots(\alpha+k+2) } J_{\alpha+k+20}. \]

Applying the properties of the gamma function, we obtain the final formula

\[ J_{\alpha+2k}= \frac{\Gamma(\alpha+3)}{2^{\alpha-1}}\, \frac{\sin\pi\gamma_k}{(\alpha+k-2)(\alpha+k)(\alpha+k+3)}\, \frac{\Gamma(\gamma_k)}{\Gamma(\gamma_k+\alpha)}, \]

\[ \gamma_k=\frac{k-\alpha-2}{2}. \]

We shall also prove that

\[ \frac{\Gamma(\gamma_k)}{\Gamma(\gamma_k+\alpha)}\to 0 \quad \text{as } k\to+\infty \]

under the condition that \(\operatorname{Re}\alpha>0\). We prove the more general assertion

\[ \frac{\Gamma(z)}{\Gamma(z+\alpha)}\to 0 \quad \text{as } \operatorname{Re}z\to+\infty,\quad |\operatorname{Im}z|<L,\quad \operatorname{Re}\alpha>0,\quad L=\mathrm{const}. \]

We have

\[ \frac{\Gamma(z)}{\Gamma(z+\alpha)} = \left(1+\frac{\alpha}{z}\right)\exp(C\alpha) \prod_{k=1}^{\infty} \left(1+\frac{\alpha}{k+z}\right)\exp\left(-\frac{\alpha}{k}\right), \]

$C$ is Euler’s constant.

Consider the function

\[ \exp \left[\sum_{k=1}^{\infty} \ln \left(1+\frac{\alpha}{k+z}\right)-\frac{\alpha}{k}\right]=\psi(z). \]

To prove the assertion, it suffices to show that $\psi(z)\to 0$ under the indicated tendency of $z$ to infinity. If $\alpha=\alpha^{*}+i\alpha^{**}$, $z=z^{*}+iz^{**}$, then

\[ \operatorname{Re}\sum_{k=1}^{\infty}\frac{-\alpha z}{k(k+z)} = \sum_{k=1}^{\infty} \frac{-\alpha[(k+z^{*})z^{*}+z^{**2}]+\alpha^{**}z^{**}} {k[(k+z^{*})^{2}+z^{**2}]} . \]

It follows from this that

\[ \operatorname{Re}\sum_{k=1}^{\infty}\frac{-\alpha z}{k(k+z)}\to -\infty \]

as $z^{*}\to +\infty$, $|z^{**}|<L$, $\alpha^{*}>0$. We also have

\[ \ln \left(1+\frac{\alpha}{k+z}\right)-\frac{\alpha}{k} = -\frac{\alpha z}{k(k+z)} + \ln \left(1+\frac{\alpha}{k+z}\right)-\frac{\alpha}{k+z}. \]

But for all $z$ for which $\operatorname{Re} z>0$ is sufficiently large, uniformly in $k$ we have

\[ \left|\ln \left(1+\frac{\alpha}{k+z}\right)-\frac{\alpha}{k+z}\right| < \frac{M}{|k+z|^{2}} . \]

Here we take the principal branch of the logarithmic function. Thus,

\[ \left| \sum_{k=1}^{\infty}\ln \left(1+\frac{\alpha}{k+z}\right)-\frac{\alpha}{k+z} \right| < M\sum_{k=1}^{\infty}\frac{1}{|k+z|^{2}} \to 0 \quad \text{as } \operatorname{Re} z\to +\infty . \]

The assertion follows from this.

The expression for $\Phi(\alpha,\beta)$ takes the form

\[ \Phi(\alpha,\beta) = \frac{(\alpha+2)^{2}+\beta^{2}}{(\alpha+1)(\alpha+2)} \left[ \frac{ a_{0}\sqrt{\pi}(\alpha+1)\Gamma\left(\frac{\alpha}{2}+\frac{1}{2}\right) }{ \alpha(\alpha+3)\Gamma\left(\frac{\alpha}{2}\right) } + \sum_{k=1}^{\infty} \frac{ a_{k}\Gamma(\alpha+3)\sin \pi\gamma_{k} }{ 2^{\alpha-1}(\alpha+k-2)(\alpha+k)(\alpha+k+3) } \, \frac{\Gamma(\gamma_{k})}{\Gamma(\gamma_{k}+\alpha)} \right]. \tag{18} \]

The series converges for all $\alpha$ for which $\operatorname{Re}\alpha>0$ and is not equal to a positive integer or zero. The function $\Phi(\alpha,\beta)$ can be analytically continued, with the aid of formula (18), also to those values of $\alpha$ for which $-1<\operatorname{Re}\alpha<0$. For this it is necessary to use formula (17). We can now write

\[ \mu_2-\bar{\mu}_2=\mu_1^2\left[ A\int_{-\infty}^{\infty}\frac{a_{11}x^2+a_{10}x}{P^2}\,dx +B\Phi\left(1-\frac{1}{a},-\frac{a_{01}}{2a^2\sqrt{\Delta}}\right) +\right. \]

\[ \left. +C\Phi\left(2-\frac{1}{a},-\frac{a_{01}}{2a^2\sqrt{\Delta}}\right) \right], \tag{19} \]

where \(B, C\) are elementary functions of the coefficients of equation (4) and can easily be computed. This representation is valid for all values of \(a\) for which \(\frac12<a<1\). The convergence of the series appearing in the expression \(\Phi(\alpha,\beta)\) can be made rapid by using formula (17) several times.

Up to now we have considered sufficient stability conditions for those \(a\) for which the integral (6) exists. But the stability conditions can also be extended to those \(a\) for which the integral (6) does not exist. (The set of such \(a\) has Lebesgue measure zero.) Indeed, let the coefficients of equation (3) be such that there is no center, and let \(\mu_2-\bar{\mu}_2>0\), as determined by formula (19), for \(a=a_0\), for which the integral (6) does not exist. Then there is a sequence \(a_n\to a_0\) as \(n\to\infty\), for which the integral (6) exists and the singular cycle is stable. Since the function of the sequence \(\frac{1}{M_2}-\frac{1}{M_3}\) is a continuous function of \(a\) at the point \(a_0\), it follows that, also for \(a=a_0\) and for all sufficiently large values of \(M_1\), we have \(\frac{1}{M_2}-\frac{1}{M_3}>0\), i.e. stability of the singular cycle.

References

1. Dulac H. Bull. Soc. math. t. 51, 1923.
2. Andronov A. A., Leontovich E. A. Matem. sb., 48 (90): 3, 1959, pp. 335—376.
3. Siegel K. L. Collection of Translations. Mathematics, 5, issue 2, 1961.
4. Pliss V. A. Differential Equations, 1, No. 2, 153—161, 1965.
5. Cherkas L. A. DAN BSSR, 6, No. 6, 1962.

Received by the editors
January 8, 1966

Minsk Radio Engineering Institute