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Mathematics
K. S. SIBIRSKII
ON THE NUMBER OF LIMIT CYCLES ARISING FROM A SINGULAR POINT OF FOCUS OR CENTER TYPE
(Presented by Academician I. G. Petrovskii on 13 X 1964)
1°. In the present note we consider the question of the number of limit cycles of the system of differential equations
\[ -\frac{dx}{dt}=b_{10}x+b_{01}y+\sum_{j+l=3} b_{jl}x^j y^l,\qquad \frac{dy}{dt}=c_{10}x+c_{01}y+\sum_{j+l=3} c_{jl}x^j y^l, \tag{1} \]
arising from the singular point \(O\) \((x=0,\ y=0)\) of the phase plane \(XOY\), in those cases when
\[ B\equiv (c_{01}-b_{10})^2-4(c_{10}b_{01}-b_{10}c_{01})<0. \tag{2} \]
Denote by \(E\) the space of coefficients of system (1) under condition (2), with the Euclidean metric, and by \(E_c\) \((E_f)\) the set of points of \(E\) to which correspond systems (1) having the point \(O\) as a center (focus). Then \(E=E_c\cup E_f\). By \(S(O,\delta)\) and \(S(\Sigma,\varepsilon)\) we shall denote, respectively, the \(\delta\)-neighborhood of the point \(O\) of the plane \(XOY\) and the \(\varepsilon\)-neighborhood of the point \(\Sigma\) of the space \(E\).
Following N. N. Bautin \((^{1,2})\), we shall say that, for a point \(\Sigma_0\in E\), the origin \(O\) of the phase plane \(XOY\) has, relative to \(E\), cyclicity of order \(k\), if the following two conditions are fulfilled: a) there exist numbers \(\varepsilon_0>0\) and \(\delta_0>0\) such that inside \(S(\Sigma_0,\varepsilon_0)\) there is no point to which there corresponds a system of the form (1) having inside \(S(O,\delta_0)\) more than \(k\) limit cycles; b) whatever positive numbers \(\varepsilon<\varepsilon_0\) and \(\delta<\delta_0\) may be, there exists a point \(\Sigma\in S(\Sigma_0,\varepsilon)\) to which there corresponds a system of the form (1) having \(k\) limit cycles inside \(S(O,\delta)\).
When conditions a) and b) are satisfied, one also says that, under variation of the coefficients of system (1) corresponding to the point \(\Sigma_0\), \(k\) limit cycles arise from the origin of the plane \(XOY\) \((^3)\).
Using the method of paper \((^2)\), the following is proved below.
Theorem 1. Whatever the number \(k=0,1,2,3,4,5\) may be, in \(E\) there exists a point for which the origin of the phase plane \(XOY\) has, relative to \(E\), cyclicity of order \(k\). In \(E\) there are no points to which there would correspond an order of cyclicity greater than 5.
The question of the birth of limit cycles from a singular point of the second group in the case of homogeneous polynomial nonlinear additions of the \(n\)-th degree was also considered by B. M. Peretyagin \((^{4,5})\). The main conclusion of these works is that, in some neighborhood of the origin, no more than \(n+1\) limit cycles can appear. As Theorem 1 shows, this result is erroneous.
2°. When condition (2) is fulfilled, by a nonsingular linear transformation and division of all coefficients of the right-hand sides by \(\sqrt{-B}\), one can arrange that the linear parts of system (1) take, respectively, the form \(y-\lambda x\) and \(x+\lambda y\). After this, by rotating the phase plane \(XOY\) one can pass to
to a system in which the coefficients of \(x^2y\) in the first equation and of \(xy^2\) in the second are the same. Such a system can be written in the form
\[ \begin{aligned} -\frac{dx}{dt}={}&y-\lambda x+(\omega+\theta-a)x^3+(\eta-3\mu)x^2y+(3\omega-3\theta\\ &+2a-\xi)xy^2+(\mu-\nu)y^3, \end{aligned} \tag{3} \]
\[ \begin{aligned} \frac{dy}{dt}={}&x+\lambda y+(\mu+\nu)x^3+(3\omega+3\theta+2a)x^2y+(\eta-3\mu)xy^2\\ &+(\omega-\theta-a)y^3. \end{aligned} \]
It is easy to see that the point \(O\) always has, for system (3), relative to the space \(\widetilde E\) of the coefficients \(\lambda, a, \mu, \nu, \xi, \eta, \theta, \omega\), the same cyclicity as for system (1) under condition (2) relative to \(E\).
We denote an arbitrary point of the space \(\widetilde E\) by \(\sigma\), the images of the sets \(E_c\) and \(E_f\) in \(\widetilde E\) respectively by \(\widetilde E_c\) and \(\widetilde E_f\), and the expression \(4(\mu^2+\theta^2)-a^2\) by \(\varkappa\).
Lemma 1. In each of the following three cases:
\[ \begin{gathered} 1)\ \lambda=\xi=a=0;\qquad 2)\ \lambda=\xi=\nu=\theta=0;\\ 3)\ \lambda=\xi=\eta=\omega=\nu=4(\mu^2+\theta^2)-a^2=0, \end{gathered} \tag{4} \]
system (3) has a center at the origin.
The validity of this lemma is not difficult to verify. Indeed, when conditions 1) are fulfilled, the derivatives of the right-hand sides of the two equations of system (3) with respect to \(x\) and to \(y\), respectively, coincide. In case 2), the straight line \(y=-x\) is an axis of symmetry of the direction field determined by system (3) in the phase plane. In case 3), for \(\mu\theta\ne0\), as was shown by K. E. Malkin \((^6)\), system (3) has an A. M. Lyapunov integral of the form \(F^2 f^{-3}=C\), where \(F\) and \(f\) are polynomials of degrees 6 and 4. By direct differentiation it is not hard to verify that one may take
\[ F=1-\frac{3a}{32\mu^3}\left(bu^2+cv^2-\frac{bu^3v}{2\mu}+\frac{b^3u^6}{96\mu^4}\right), \qquad f=1-\frac{uv}{2\mu}+\frac{b^2u^4}{64\mu^4}, \]
where \(b\equiv a-2\theta,\ c\equiv a+2\theta,\ u\equiv cx-2\mu y,\ v\equiv bx-2\mu y\). For \(\mu\theta=0\) the existence of a center follows from the closedness of the set \(\widetilde E_c\).
Eliminating \(t\) from system (3) and passing to polar coordinates \(\rho,\varphi\), we can write the solution \(\rho=\rho(\varphi)\) of the resulting equation, satisfying the initial condition \(\rho(0)=\rho_0\), as a series in powers of \(\rho_0\)
\[ \rho(\varphi,\sigma)=\rho_0v_0(\varphi,\sigma)+\rho_0^3v_1(\varphi,\sigma)+\rho_0^5v_2(\varphi,\sigma)+\cdots, \tag{5} \]
where, whatever \(\varepsilon>0\) and \(\sigma_0\in\widetilde E\) may be, there exists a positive number \(r=r(\varepsilon,\sigma_0)\) such that the series (5) converges for all \(\varphi\in[0,2\pi]\) and \(\sigma\in S(\sigma_0,\varepsilon)\) for values of \(\rho_0\) satisfying the relation \(|\rho_0|<r\).
From (5), for \(\varphi=2\pi\), we obtain the so-called succession function
\[ \rho(2\pi,\sigma)=\rho_0v_0(2\pi,\sigma)+\rho_0^3v_1(2\pi,\sigma)+\rho_0^5v_2(2\pi,\sigma)+\cdots \tag{6} \]
The positive roots of the difference \(\rho(2\pi,\sigma)-\rho_0\) correspond, obviously, to closed integral curves of system (3).
\(3^\circ\). The following two propositions hold:
Lemma 2. The coefficients \(v_j(2\pi,\sigma)\) \((j=0,1,2,\ldots)\) of the succession function are entire functions of the coordinates of the point \(\sigma\in\widetilde E\), which, for \(\lambda=0\), become homogeneous polynomials of degree \(j\) with respect to the remaining coordinates of the point \(\sigma\), and \(v_0(2\pi,\sigma)=\exp(2\pi\lambda)\).
Lemma 3. The coefficients \(v_j(2\pi,\sigma)\) for \(j>0\) have the form
\[ v_j(2\pi,\sigma)=a^2\theta\varkappa\Theta_j^{(6)}+a^2\theta\eta\Theta_j^{(5)}+a\theta\omega\Theta_j^{(4)}+a\nu\Theta_j^{(3)}+\xi\Theta_j^{(2)}+\lambda\Theta_j^{(1)}, \tag{7} \]
where \(\Theta_j^{(l)}\) are entire functions of the coordinates of the point \(\sigma\in\widetilde E\), with \(\Theta_j^{(j+l)}=0\) for \(l>1\), and \(\Theta_j^{(j+1)}\) are nonzero real numbers.
Lemma 2 is established directly from the recurrence relations obtained for determining the functions \(v_j(\varphi,\sigma)\).
To prove Lemma 3, note that, on the basis of Lemma 1, when each of the three series of conditions (4) is satisfied, all coefficients \(v_j(2\pi,\sigma)\) \((j>0)\) must vanish. Hence it is easy to establish that they must have the form
\[ v_j(2\pi,\sigma)=a\theta\widetilde{\Theta}_j^{(6)}+a\theta\eta\widetilde{\Theta}_j^{(5)}+a\theta\omega\Theta_j^{(4)}+a\nu\Theta_j^{(3)}+\xi\Theta_j^{(2)}+\lambda\Theta_j^{(1)}, \]
where \(\Theta_j^{(2)}\) do not contain \(\lambda\); \(\Theta_j^{(3)}\) do not contain \(\lambda\) and \(\xi\); \(\Theta_j^{(4)}\) do not contain \(\lambda,\xi\), and \(\nu\); \(\widetilde{\Theta}_j^{(5)}\) do not contain \(\lambda,\xi,\nu\), and \(\omega\), while \(\widetilde{\Theta}_j^{(6)}\) do not contain \(\lambda,\xi,\nu,\omega\), and \(\eta\).
To obtain formula (7), it now suffices to show that \(\widetilde{\Theta}_j^{(6)}\) and \(\widetilde{\Theta}_j^{(5)}\) contain the factor \(a\). For this purpose it is enough to restrict ourselves to the case \(\lambda=\xi=\nu=\omega=0\) and to show that the expansion of \(\rho(2\pi,\sigma)-\rho_0\) in powers of \(a\) for the system
\[ dx/dt=-y-\theta x^3+(3\mu-\eta)x^2y+3\theta xy^2-\mu y^3+a(x^3-2xy^2), \]
\[ dy/dt=x+\mu x^3+3\theta x^2y+(\eta-3\mu)xy^2-\theta y^3+a(2x^2y-y^3) \tag{8} \]
does not contain terms below the second degree.
Denoting
\[ H(x,y)=\frac12(x^2+y^2)+\frac14\mu x^4+\theta x^3y+\frac12(\eta-3\mu)x^2y^2-\theta xy^3+\frac14\mu y^4, \]
\[ p(x,y)=x^3-2xy^2,\qquad q(x,y)=2x^2y-y^3, \]
system (8) can be rewritten in the form
\[ dx/dt=-H_y'(x,y)+ap(x,y);\qquad dy/dt=H_x'(x,y)+aq(x,y). \tag{9} \]
For sufficiently small \(h\), consider the family of closed curves \(C_h\) with equation \(H(x,y)=h\). Let \(C_{h_0}\) be one of these curves. In its neighborhood introduce new coordinates by the relations \(H(x,y)=h_0+\xi\), \(M(x,y;s)=0\), the latter of which, for each \(s\), determines an arc without contact, coinciding for \(s=0\) with a segment of the half-line \(\varphi=0\), with \(s\) a cyclic coordinate subject to the condition that \(ds/dt=1\) on the curve \(C_{h_0}\).
In the new coordinates, system (9) can be written as the equation \(d\xi/ds=R(\xi,s,a)\), whose solution we seek in the form of a series in powers of \(a\) and of the initial value \(\xi_0=\xi(0)\):
\[ \xi=c_{10}(s)\xi_0+c_{01}(s)a+c_{20}(s)\xi_0^2+c_{11}(s)\xi_0a+c_{02}(s)a^2+\cdots \]
Putting here \(s=\tau\), where \(\tau\) is the period on the curve \(C_{h_0}\), we obtain, on some segment of the half-line \(\varphi=0\), the function
\[ \xi=\xi_0+c_{01}(\tau)a+c_{11}(\tau)\xi_0a+c_{02}(\tau)a^2+\cdots . \]
It is not difficult to show that the requirement that the expansion of \(\rho(2\pi,\sigma)-\rho_0\) in powers of \(a\) for system (8) contain no terms below the second degree is equivalent to the fulfillment of the equality \(c_{01}(\tau)=0\), independently of the choice of the curve \(C_{h_0}\).
Let us show that this condition is fulfilled. Indeed,
\[ c_{01}(\tau)=\int_0^\tau (p\dot y-q\dot x)\,ds =\int_{C_{h_0}} p\,dy-q\,dx =\int_{C_{h_0}}(x^3-2xy^2)\,dy+(y^3-2x^2y)\,dx . \]
In this integral, replacing \(x\) by \(y\) and \(y\) by \(-x\) entails a change of sign of the integrand. At the same time, from the equation of the curve \(C_{h_0}\) \((H(x,y)=h_0)\) it is clear that, independently of the choice of \(h_0\), if the point with coordinates \((x,y)\) belongs to this curve, then the point \((y,-x)\) also belongs to it.
lies on \(C_{h_0}\). Hence it follows immediately that \(C_{01}(\tau)=0\) independently of the choice of the curve \(C_{h_0}\) from the family \(H(x,y)=h\). Thus it has been proved that \(v_j(2\pi,\sigma)\) for all \(j>0\) have the form (7).
The fact that \(\Theta_j^{(j+l)}=0\) for all \(l>1\), while \(\Theta_j^{(j+1)}\) are constants, follows from Lemma 2. Computations show that
\[ \Theta_1^{(2)}=\pi/4,\qquad \Theta_2^{(3)}=-5\pi/4,\qquad \Theta_3^{(4)}=25\pi/8,\qquad \Theta_4^{(5)}=5\pi/32, \]
\[ \Theta_5^{(6)}=-5\pi/96. \]
Thus Lemma 3 is proved.
\(4^\circ\). On the basis of Lemma 3, the function of the sequence (6) may be written as follows:
\[ \begin{aligned} \rho(2\pi,\sigma)-\rho_0 ={}&2\pi\lambda(1+\lambda\Phi+\rho_0\Psi_0)\rho_0 +\Theta_1^{(2)}\xi(1+\rho_0\Psi_1)\rho_0^3\\ &+\Theta_2^{(3)}a\nu(1+\rho_0\Psi_2)\rho_0^5 +\Theta_3^{(4)}a\theta\omega(1+\rho_0\Psi_3)\rho_0^7\\ &+\Theta_4^{(5)}a^2\theta\eta(1+\rho_0\Psi_4)\rho_0^9 +\Theta_5^{(6)}a^2\theta\chi(1+\rho_0\Psi_5)\rho_0^{11}, \end{aligned} \]
where \(\Phi,\Psi_j\) are series in powers of \(\rho_0\) with coefficients in the form of entire functions of the coordinates of the point \(\sigma\).
Following the arguments of N. N. Bautin \((^2)\), it is now easy to establish that, whatever the point \(\sigma\in \widetilde E\) with coordinates \(\lambda_0,a_0,\mu_0,\nu_0,\xi_0,\eta_0,\theta_0,\omega_0\), one can always specify such \(\varepsilon_0>0\) and \(\delta_0>0\) that for all \(\sigma\in S(\sigma_0,\varepsilon_0)\cap E_f\) the difference \(\rho(2\pi,\sigma)-\rho_0\) has no more than 5 roots inside \((0,\delta_0)\). If, moreover, \(\lambda_0=\xi_0=\nu_0=\omega_0=\eta_0=0\), \(a\theta_0(4\mu_0^2+4\theta_0^2-a_0^2)\ne 0\), then, whatever positive numbers \(\varepsilon<\varepsilon_0\) and \(\delta<\delta_0\), in \(S(\sigma_0,\varepsilon)\) there exists a point \(\sigma\) for which system (3) has 5 limit cycles in \(S(O,\delta)\). The cyclicity of the point \(\sigma_0\) is equal to 4 if \(\lambda_0=\xi_0=\nu_0=\omega_0=0,\ a_0\theta_0\eta_0\ne0\); is equal to 3 if \(\lambda_0=\xi_0=\nu_0=0,\ a_0\theta_0\omega_0\ne0\); is equal to 2 if \(\lambda_0=\xi_0=0,\ a_0\nu_0\ne0\); is equal to one if \(\lambda_0=0,\ \xi_0\ne0\), and is equal to zero if \(\lambda_0\ne0\). This completes the proof of Theorem 1.
Let us also note that it follows from Lemma 3 that the fulfillment of at least one of the three series of conditions (4) is necessary for the existence at the origin of a center of system (3). Using Lemma 1 and the rotation invariants of the phase plane (7), it is then easy to arrive at the conclusion that the following holds.
Theorem 2. For the existence at the origin of a center of the equation
\[ \frac{dy}{dx} = -\frac{x+(\mu+\nu)x^3+(3\psi+3\theta+\alpha)x^2y+(\chi+\beta)xy^2+(\psi-\theta)y^3} {y+(\psi+\theta)x^3+(\chi-\beta)x^2y+(3\psi-3\theta-\gamma)xy^2+(\mu-\nu)y^3} \tag{10} \]
it is necessary and sufficient that at least one of the following two series of conditions be satisfied:
\[ \begin{aligned} 1)\quad& \alpha+\gamma=\alpha\nu+\alpha\beta+4\beta\psi=\theta\alpha^2+(\chi-\mu)\alpha\beta-4\theta\beta^2=0;\\ 2)\quad& \alpha+\gamma=\chi+3\mu=\alpha+5\psi=\beta+5\nu=\psi^2+4(\nu^2-\mu^2-\theta^2)=0. \end{aligned} \]
Thus we arrive at center conditions equivalent to the conditions of K. E. Malkin \((^6)\), who first established that not all cases of the existence of a center for equation (10) had been found by N. A. Sakharnikov \((^8)\), as well as by M. I. Almukhamedov \((^9)\).
Institute of Mathematics with Computing Center
Academy of Sciences of the Moldavian SSR
Received
10 X 1964
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