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Limit Cycles of Infinite Multiplicity
V. P. Zakharov
In the present article the concept is introduced of limit cycles of infinite multiplicity and of differential equations having such cycles. Consider the system of differential equations
\[ \frac{dx}{dt}=P(x,y),\qquad \frac{dy}{dt}=Q(x,y), \tag{1} \]
where the functions \(P(x,y)\) and \(Q(x,y)\) are not analytic, but ensure the existence and uniqueness of a particular solution for arbitrary initial data.
The multiplicity of a limit cycle of the system of differential equations (1), as is known ([1], p. 442), is defined as the multiplicity of the corresponding root \(s_0\) of the equation
\[ \psi(s)=f(s)-s=0, \tag{2} \]
where \(f(s)\) is the Poincaré succession function.
By analogy with this definition, we shall call a limit cycle of infinite multiplicity any isolated closed phase trajectory of the system of differential equations (1) such that
\[ \frac{d^k\psi(s_0)}{ds^k}=0\quad (k=0,1,2,3,\ldots). \tag{3} \]
It should be noted that limit cycles of infinite multiplicity cannot occur in the case of analytic right-hand sides of the system of differential equations (1) ([1], pp. 443–444).
Limit cycles of infinite multiplicity (as also limit cycles of finite multiplicity), depending on the disposition of neighboring phase trajectories and on the motion along them of the representing point as \(t\) varies, we shall subdivide into stable, unstable, and semistable.
We shall show that limit cycles of infinite multiplicity exist. Consider the system of differential equations:
\[ \frac{dx}{dt}= \begin{cases} -y\sqrt{x^2+y^2}+x\exp\left[-\left(\sqrt{x^2+y^2}-r_0\right)^{-k}\right], & \text{for } x^2+y^2\ne r_0^2,\\ -y\sqrt{x^2+y^2}, & \text{for } x^2+y^2=r_0^2; \end{cases} \tag{4} \]
\[ \frac{dy}{dt}= \begin{cases} x\sqrt{x^2+y^2}+y\exp\left[-\left(\sqrt{x^2+y^2}-r_0\right)^{-k}\right], & \text{for } x^2+y^2\ne r_0^2,\\ x\sqrt{x^2+y^2}, & \text{for } x^2+y^2=r_0^2. \end{cases} \]
\[ (k \text{ is a natural number},\ r_0>0). \]
For greater clarity of exposition, let us write the differential equation satisfied by the phase trajectories in polar coordinates \(r,\varphi\) (\(r\) is the polar radius, \(\varphi\) the polar angle):
\[ \frac{dr}{d\varphi}=R(r), \tag{5} \]
where
\[ R(r)= \begin{cases} \exp[-(r-r_0)^{-k}] & \text{for } r\ne r_0,\\ 0 & \text{for } r=r_0. \end{cases} \]
The continuity of the function \(R(r)\) and of its derivatives (and also the existence of the latter) follows from the arguments given below.
If \(r\ne r_0\), then
\[ R(r)=\exp[-(r-r_0)^{-k}], \]
\[ \frac{dR(r)}{dr}=k(r-r_0)^{-k-1}\exp[-(r-r_0)^{-k}], \]
\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots \]
\[ \frac{d^k R(r)}{dr^k} = T_k\!\left(\frac{1}{r-r_0}\right) \exp[-(r-r_0)^{-k}], \]
\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots \]
We note that \(T_k\!\left(\dfrac{1}{r-r_0}\right)\) here is a polynomial of some degree in
\[ \frac{1}{r-r_0}. \]
If, however, \(r=r_0\), then
\[ R(r_0)=0, \]
\[ \frac{dR(r_0)}{dr} = \lim_{\substack{r\to r_0\\ r\ne r_0}} \frac{R(r)-R(r_0)}{r-r_0} =0, \]
\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots \]
\[ \frac{d^{k+1}R(r_0)}{dr^{k+1}} = \lim_{\substack{r\to r_0\\ r\ne r_0}} \frac{ \left( \dfrac{d^kR(r)}{dr^k} - \dfrac{d^kR(r_0)}{dr^k} \right) }{r-r_0} =0. \]
\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots \]
Thus we have proved that all derivatives of the function \(R(r)\) at \(r=r_0\) are equal to zero, i.e.
\[ \frac{dR(r_0)}{dr} = \frac{d^2R(r_0)}{dr^2} = \cdots = 0. \tag{6} \]
For equation (5), the circle \(r=r_0\) is a limit cycle ([2], pp. 169–171). Since in this case equation (2) has the form
\[ R(r)=0, \tag{7} \]
we have proved that this limit cycle of infinite multiplicity is semistable.
Let us give an example of a stable limit cycle of infinite multiplicity:
\[ \frac{dx}{dt}= \begin{cases} -y\sqrt{x^{2}+y^{2}}+X\operatorname{sign}(r_{0}-r)\exp\left[-\left(\sqrt{x^{2}+y^{2}}-r_{0}\right)^{-k}\right], & \text{for } x^{2}+y^{2}\ne r_{0}^{2},\\ -y\sqrt{x^{2}+y^{2}}, & \text{for } x^{2}+y^{2}=r_{0}^{2}, \end{cases} \]
\[ \frac{dy}{dt}= \begin{cases} x\sqrt{x^{2}+y^{2}}+y\operatorname{sign}(r_{0}-r)\exp\left[-\left(\sqrt{x^{2}+y^{2}}-r_{0}\right)^{-k}\right], & \text{for } x^{2}+y^{2}\ne r_{0}^{2},\\ x\sqrt{x^{2}+y^{2}}, & \text{for } x^{2}+y^{2}=r_{0}^{2}. \end{cases} \]
\[ (k \text{ is a natural number},\ r_{0}>0). \]
If in the last example, instead of \(\operatorname{sign}(r_{0}-r)\), one takes \(\operatorname{sign}(r-r_{0})\), then one obtains an example with an unstable limit cycle of infinite multiplicity.
References
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Andronov A. A., Vitt A. A., Khaikin S. E. Theory of Oscillations. Fizmatgiz, Moscow, 1959.
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Almukhamedov M. I. Izv. Fiz.-Matem. Obshchestva i Nauchno-Issledovatel’skogo In-ta Matem. i Mekh. pri Kazanskom Universitete, 3rd series, 11, 1938, pp. 161–179.
Received by the editors
December 21, 1964
Chuvash Pedagogical Institute
named after I. Ya. Yakovlev