ON THE EXISTENCE AND UNIQUENESS OF A LIMIT CYCLE OF MULTIDIMENSIONAL AUTONOMOUS SYSTEMS

V. V. Korolev

The question of the existence and uniqueness of a limit cycle in the plane has been studied by many authors. In recent years, investigation has begun of the problem of the existence of a limit cycle for multidimensional systems [1—7]. In the recently published monograph of V. A. Pliss [8], the question of the uniqueness of a limit cycle is also studied.

In the present note we give some sufficient criteria for the existence and uniqueness of a limit cycle for the system

\[ \dot{x}_i=F_i(x_1,\ldots,x_n),\qquad i=1,\ldots,n. \tag{1} \]

Concerning the functions \(F_i\), we assume that they admit continuous partial derivatives of first order with respect to all variables in a sufficiently large domain of the space \(E^n\) (\(E^n\) is \(n\)-dimensional Euclidean space).

Let the following conditions be satisfied:

A) There exist numbers \(x_3^*,\ldots,x_n^*\) such that the vectors of the field determined by the system

\[ \dot{x}_1=F_1(x_1,x_2,x_3^*,\ldots,x_n^*), \]

\[ \dot{x}_2=F_2(x_1,x_2,x_3^*,\ldots,x_n^*), \tag{2} \]

in the plane \(x_1x_2\), at the points of a contour \(C\) enclosing the origin, are directed toward the region \(K\) exterior to \(C\); with respect to the contour \(C\) we assume that the radius vector of any point of the contour \(C\) is directed into \(K\). Consider the region \(V\), which is geometrically the topological product \(K\times I_3\times\cdots\times I_n\), where \(I_s:[x_s^*,+\infty)\), \(s=3,\ldots,n\); the boundary of the region \(V\) consists of the following hypersurfaces:

\[ S=C\times I_3\times\cdots\times I_n,\qquad S_3=K\times x_3^*\times I_4\times\cdots\times I_n, \]

\[ S_r=K\times I_3\times\cdots\times I_{r-1}\times x_r^*\times I_{r+1}\times\cdots\times I_n,\qquad r=4,\ldots,n-1, \]

\[ S_n=K\times I_3\times\cdots\times I_{n-1}\times x_n^*. \]

B)

\[ \frac{x_1F_2-x_2F_1}{x_1^2+x_2^2}\ge \alpha>0 \quad \text{for } x=(x_1,\ldots,x_n)\in V. \]

C)

\[ F_k>0 \quad \text{for } x\in S_k,\quad k=3,\ldots,n. \]

D)

\[ \frac{\partial F_2}{\partial x_k}F_1-\frac{\partial F_1}{\partial x_k}F_2\le 0 \quad \text{for } x\in S,\quad k=3,\ldots,n. \]

D) There exist numbers \(\alpha_k>0,\ k=1,\ldots,n-1\), such that the inequality

\[ \int_0^{2\pi}\mu(\theta)\,d\theta<0,\qquad \text{where}\quad \mu(\theta)=\max_{1\le k\le n-1}\{\mu_k(\theta)\}, \]

\[ \mu_k(\theta)=\sup\left\{ \frac{1}{\alpha_k}\left( \alpha_k\frac{\partial H_k}{\partial u_k} + \sum_{\substack{i=1\\(i\ne k)}}^{n-1} \alpha_i\left|\frac{\partial H_k}{\partial u_i}\right| \right)\right\} \]

holds on the section of the domain \(V\) by the hyperplane
\[ \Pi(\theta):\theta=\arctg\frac{x_2}{x_1}=\operatorname{const}; \]

\[ H_1=\frac{u_1(x_1F_1+x_2F_2)}{x_1F_2-x_2F_1},\qquad H_s=\frac{u_1^2F_{s+1}}{x_1F_2-x_2F_1},\qquad s=2,\ldots,n-1, \]

\[ u_1=\sqrt{x_1^2+x_2^2},\qquad u_i=x_{i+1},\qquad i=2,\ldots,n-1. \]

Theorem. If conditions A)—D) are satisfied, then system (1) has inside \(V\) exactly one limiting cycle, stable in the positive direction.

Proof. \(1^\circ\). We first prove that the vectors
\(\bar\tau\{F_1,F_2,\ldots,F_n\}\), determined by system (1) on the boundary of \(V\), are directed inward into \(V\). To this end we establish that each of the scalar products
\(\bar n\cdot\bar\tau,\ \bar n_k\cdot\bar\tau,\ k=3,\ldots,n\), where \(\bar n,\bar n_k\) are the unit vectors of the outer (with respect to \(V\)) normal to the hypersurfaces \(S, S_k\), respectively, is negative.

Obviously,
\[ \bar n_k=\{0,0,\ldots,0,-1,0,\ldots,0\} \]
(\(-1\) in the \(k\)-th position), and the vector \(\bar n\) coincides with the unit vector of the outer (with respect to \(K\)) normal to the contour \(C\).

From B) it follows that
\[ \bar n_k\cdot\bar\tau=-F_k<0,\quad \text{if } x\in S_k,\quad k=3,\ldots,n. \]
Consider now the scalar product \(\bar n\cdot\bar\tau\). Decompose the vector
\(\bar\tau\{F_1,\ldots,F_n\}\) into the components
\(\bar\tau_1\{F_1,F_2,0,\ldots,0\}\) and \(\bar\tau_2\{0,0,F_3,\ldots,F_n\}\).
We shall prove that \(\bar n\cdot\bar\tau_1<0\); hence the inequality
\(\bar n\cdot\bar\tau<0\) will follow, since \(\bar n\cdot\bar\tau_2=0\).

Let \(M(x_1,x_2,x_3,\ldots,x_n)\) be an arbitrary point on \(S\). Then
\(M'(x_1,x_2,0,\ldots,0)\in C\), where \(x_1\) and \(x_2\) are the same as at the point \(M\).
At the point \(M'\) construct the vectors

\[ \bar r^{\,0}\left\{\frac{x_1}{u_1},\frac{x_2}{u_1},0,\ldots,0\right\} \quad\text{and}\quad \bar n^{\,0}\left\{-\frac{x_2}{u_1},\frac{x_1}{u_1},0,\ldots,0\right\} \quad \text{(see Fig.).} \]

Place the vector \(\bar\tau_1(x_3,\ldots,x_n)\) (where \(x_1\) and \(x_2\) are regarded as fixed), corresponding to the point \(M\), at the point \(M'\). Denote by
\(\alpha(x_3,\ldots,x_n)\) the angle \((\bar\tau_1,\bar r^{\,0})\), obtained by rotating the vector \(\bar r^{\,0}\) counterclockwise until it coincides with the vector \(\bar\tau_1\).

The vector \(\bar\tau_1(x_3^*,\ldots,x_n^*)\), by virtue of A), enters into \(K\). We shall prove that the vector \(\bar\tau_1(x_3,\ldots,x_n)\), for any \(x_s\in I_s\), is directed inward into \(K\).

For this we first establish the inequality

\[ \cos\alpha(x_3,\ldots,x_n)\ge \cos\alpha(x_3^*,\ldots,x_n^*). \tag{3} \]

Using Lagrange’s theorem, we find

\[ \cos\alpha(x_3,\ldots,x_n)-\cos\alpha(x_3^*,\ldots,x_n^*) = \sum_{k=3}^{n}\frac{\partial}{\partial x_k}(\cos\alpha)(x_k-x_k^*), \tag{4} \]

where the partial derivatives \(\dfrac{\partial}{\partial x_k}(\cos\alpha)\) are evaluated at the point

\[ (x_1,\ x_2,\ x_3^*+\xi(x_3-x_3^*),\ldots,\ x_n^*+\xi(x_n-x_n^*))\in S,\quad 0<\xi<1. \]

Since

\[ \cos\alpha = \frac{\bar r^{0}\,\bar\tau_1}{|\bar\tau_1|} = \frac{x_1F_1+x_2F_2}{u_1\sqrt{F_1^2+F_2^2}}, \]

differentiating, we obtain

\[ \frac{\partial}{\partial x_k}(\cos\alpha) = \frac{-(x_1F_2-x_2F_1)}{u_1(F_1^2+F_2^2)^{3/2}} \times \left( \frac{\partial F_2}{\partial x_k}F_1 - \frac{\partial F_1}{\partial x_k}F_2 \right). \tag{5} \]

From B), Г), (5), and (4) we obtain (3). We next find:

\[ \sin\alpha = \frac{\bar\tau_1\,\bar n^{0}}{|\bar\tau_1|} = \frac{x_1F_2-x_2F_1}{u_1\sqrt{F_1^2+F_2^2}}>0 \quad\text{according to B).} \]

Hence, and from (3), we obtain
\(\alpha(x_3,\ldots,x_n)\leq \alpha(x_3^*,\ldots,x_n^*)\).
The last inequality means that the vector \(\bar\tau_1,(x_3,\ldots,x_n)\), just as the vector
\(\bar\tau_1(x_3^*,\ldots,x_n^*)\), is directed into \(K\), and therefore
\(\bar\tau_1\cdot\bar n<0\) and \(\bar\tau\cdot\bar n<0\).

Thus it has been established that the positive semitrajectories of system (1) beginning at points of the boundary of \(V\) are contained inside \(V\).

\(2^\circ\). Let \(S_0\) be the section of the domain \(V\) by the hyperplane (half-hyperplane)
\(\Pi:x_2=0,\ x_1>0\), and let \(A^0\in S_0\).

From \(A^0\) emit the positive semitrajectory \(f(A^0,t)\). Along \(f(A^0,t)\) we have

\[ \frac{d\theta}{dt} = \frac{x_1^2}{x_1^2+x_2^2}\, \frac{\dot x_2x_1-\dot x_1x_2}{x_1^2} = \frac{x_1F_2-x_2F_1}{u_1^2}. \tag{6} \]

From (6) and B) we find \(\dfrac{d\theta}{dt}>\alpha\).

Consequently, the semitrajectory \(f(A^0,t)\), beginning at the point \(A^0\) of the section \(S_0\), after a finite interval of time will again meet (for the first time) the section \(S_0\) at some point \(A\). This means that, by means of the positive semitrajectories beginning at points of the section \(S_0\), a certain mapping \(T:TA^0=A\) of the section \(S_0\) into itself is generated. Note that \(S_0\) is a complete metric space.

\(3^\circ\). We shall prove that the mapping \(T\) is a contraction.

Introduce in \(E^n\) cylindrical coordinates \(\theta,u_1,\ldots,u_{n-1}\), where
\(x_1=u_1\cos\theta,\ x_2=u_1\sin\theta,\ x_{k+1}=u_k,\ k=2,\ldots,n-1\).

The trajectories of system (1) coincide with the integral curves of the system

\[ \frac{du_k}{d\theta}=H_k(\theta,u_1,\ldots,u_{n-1}),\qquad k=1,\ldots,n-1, \tag{7} \]

where the \(H_k\) are defined as in D).

Indeed, using (6), we find

\[ u_1\frac{du_1}{d\theta} = u_1\frac{du_1}{dt}\frac{dt}{d\theta} = (x_1\dot x_1+x_2\dot x_2)\frac{u_1^2}{x_1F_2-x_2F_1} = \frac{u_1^2(x_1F_1+x_2F_2)}{x_1F_2-x_2F_1}, \]

whence

\[ \frac{du_1}{d\theta}=H_1. \]

Next we obtain

\[ \frac{du_i}{d\theta} = \frac{dx_{i+1}}{dt}\frac{dt}{d\theta} = \frac{u_1^2F_{i+1}}{x_1F_2-x_2F_1} = H_i,\qquad i=2,\ldots,n-1. \]

Take two arbitrary points \(A_1^0, A_2^0\in S_0\) and consider the positive semitrajectories \(f(A_1^0,t)\) and \(f(A_2^0,t)\). Let us investigate how the distance \(\rho(\theta)=\rho(A_1,A_2)\) between the points \(A_1(\theta,u_1',\ldots,u_{n-1}')\) and \(A_2(\theta,u_1'',\ldots,u_{n-1}'')\), the intersections of the curves \(f(A_1^0,t)\) and \(f(A_2^0,t)\) with the hyperplane \(\Pi(\theta):\theta=\mathrm{const}\), changes as \(\theta\) increases in the interval \(0\le \theta\le 2\pi\). From (7) we find

\[ \frac{du_k'}{d\theta} = H_k(\theta,u_1',\ldots,u_{n-1}'),\qquad \frac{du_k''}{d\theta} = H_k(\theta,u_1'',\ldots,u_{n-1}''). \]

Subtracting, applying Lagrange’s theorem, and denoting \(u_k''-u_k'=v_k\), we obtain

\[ \frac{dv_k}{d\theta} = \frac{\partial H_k}{\partial u_1}v_1+\ldots+ \frac{\partial H_k}{\partial u_k}v_k+\ldots+ \frac{\partial H_k}{\partial u_{n-1}}v_{n-1}, \tag{8} \]

\[ k=1,\ldots,n-1. \]

Define the distance \(\rho(\theta)\) by the relation

\[ \rho(\theta)= \max_{1\le i\le n-1} \left\{\frac{|v_i(\theta)|}{a_i}\right\}. \]

Suppose that \(\rho(\theta)=\dfrac{|v_k(\theta)|}{a_k}\) on some interval \([\theta',\theta'']\subset[0,2\pi]\). From (8) we find

\[ \frac{dv_k\,a_k}{a_k v_k} = \left( \frac{\partial H_k}{\partial u_1}\frac{v_1a_ka_1}{a_1v_ka_k} +\ldots+ \frac{\partial H_k}{\partial u_k} +\ldots+ \frac{\partial H_k}{\partial u_{n-1}}\frac{v_{n-1}a_ka_{n-1}}{a_{n-1}v_ka_k} \right)d\theta \]

(\(v_k\ne0\) by virtue of the uniqueness of the solution of system (7), and therefore division by \(v_k\) is possible). Using D) and the inequalities

\[ \left|\frac{v_i a_k}{a_i v_k}\right|\le 1,\qquad i=1,\ldots,n-1;\quad i\ne k, \]

we obtain

\[ \frac{d\left(\dfrac{v_k}{\alpha_k}\right)}{\dfrac{v_k}{\alpha_k}} \leq \frac{1}{\rho_k} \left( \left|\frac{\partial H_k}{\partial u_1}\right|\alpha_1+\cdots+ \left|\frac{\partial H_k}{\partial u_k}\right|\alpha_k+\cdots+ \left|\frac{\partial H_k}{\partial u_{n-1}}\right|\alpha_{n-1} \right)d\theta \leq \mu_k(\theta)d\theta \leq \mu(\theta)d\theta . \]

Integrating over the interval \([\theta',\theta'']\), we find

\[ \frac{|v_k(\theta'')|}{\alpha_k} \leq \frac{|v_k(\theta')|}{\alpha_k} \exp\left(\int_{\theta'}^{\theta''}\mu(\theta)d\theta\right) \]

or

\[ \rho(\theta'') \leq \rho(\theta')\exp\left(\int_{\theta'}^{\theta''}\mu(\theta)d\theta\right). \]

Using the method of induction, we find

\[ \rho(2\pi)\leq \rho(0)\exp\left(\int_0^{2\pi}\mu(\theta)d\theta\right) \]

and

\[ \rho(TA_1^0,\ TA_2^0)\leq \rho(A_1^0,\ A_2^0)\exp\left(\int_0^{2\pi}\mu(\theta)d\theta\right). \]

Hence, and from D), we conclude that the mapping \(T\) is a contraction and, by the Banach principle [9], has exactly one fixed point, to which there corresponds the unique limiting cycle of system (1). The stability of the limiting cycle follows from the condition

\[ \frac{d\theta}{dt}=\frac{x_1F_2-x_2F_1}{u_1^2}>0. \]

The theorem is proved.

An example has been constructed of a system satisfying all the conditions of the theorem.

Remark 1. The assertion of the theorem will also be true in the case when, together with conditions A)—D), the following condition is satisfied: there exist numbers \(\beta_k>0\), \(k=1,\ldots,n-1\), such that

\[ \int_0^{2\pi}\lambda(\theta)d\theta<0, \]

where

\[ \lambda(\theta)=\max_{1<k<n-1}\{\lambda_k(\theta)\}, \]

\[ \lambda_k(\theta)=\sup\left\{ \frac{1}{\beta_k} \left( \beta_k\frac{\partial H_k}{\partial u_k} + \sum_{\substack{i=1\\(i\ne k)}}^{n-1} \beta_i\left|\frac{\partial H_i}{\partial u_k}\right| \right) \right\} \]

on the section of the domain \(V\) by the hyperplane \(\Pi(\theta)\).

Here the distance \(\rho(\theta)\) is defined by the formula

\[ \rho(\theta)=\sum_{k=1}^{n-1}\beta_k|v_k(\theta)|. \]

Remark 2. System (1) will have a unique limit cycle also in the case when conditions A)—D) are satisfied and there exist numbers \(\gamma_k>0,\ k=1,\ldots,n-1,\ h>0\), such that in the domain \(V\) the inequalities
\[ (-1)^k\Delta_k>0,\qquad \frac{\Delta_{n-1}}{(\operatorname{Sp}H)^{\,n-2}}<-h, \]
hold, where \(\Delta_k,\ k=1,\ldots,n-1\), are the principal minors of the matrix
\[ H=\left\|\gamma_i\frac{\partial H_i}{\partial u_j} +\gamma_j\frac{\partial H_j}{\partial u_i}\right\| \qquad (i,j=1,\ldots,n-1). \]

In this case the metric is defined by the formula
\[ \rho^2(\xi)=\sum_{k=1}^{n-1}\gamma_k v_k^2(\theta). \]

The assertion that the operator \(T\) is a contraction in the latter case follows from the results of [10].

Let us also note the following. If the condition in Remark 2 ensures the monotone decrease of the function \(\rho(\theta)\) over the whole interval \([0,2\pi]\), then the conditions of the theorem and of Remark 1 guarantee, generally speaking, only the integral decrease of the function \(\rho(\xi)\) on the interval \([0,2\pi]\).

The conditions of the theorem on the existence and uniqueness of a limit cycle given in [8] require the monotone decrease of \(\rho(\theta)\).

References

1. Friedrichs K. O. On nonlinear vibrations of the third order, Studies in nonlinear vibration theory, New York Univer., 1946.
2. Rauch L. L. Ann. Math. studies, 20, 1950.
3. Nemitskii V. V. On some methods of qualitative investigation “in the large” of multidimensional autonomous systems. Trudy Mosk. matem. o-va, 5, 1956.
4. Shirokorad B. V. Avtomatika i telemekhanika, 19, No. 10, 1958.
5. Pliss V. A. Some problems of the theory of stability of motion as a whole. Izd. OGU, 1958.
6. Vaysbord E. M. Matem. sb., 56 (98), No. 1, 1962.
7. Blinchevskii V. S. Matem. sb., 50 (92), No. 1, 1960.
8. Pliss V. A. Nonlocal problems of the theory of oscillations. Izd. “Nauka”, 1964.
9. Kantorovich L. V. and Akilov G. P. Functional analysis in normed spaces. Fizmatgiz, 1959.
10. Demidovich B. P. Vestnik MGU, No. 6, 1961.

Received by the editors
September 5, 1964

Odessa State University
named after I. I. Mechnikov