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UDC 517.916 : 517.917
ON A CLASS OF DYNAMICAL SYSTEMS WITH STABLE STRUCTURES
N. F. OTROKOV
In the present article we consider a system of differential equations of the form
\[ \frac{dx}{dt}=P(x,y), \qquad \frac{dy}{dt}=Q(x,y), \tag{1} \]
where \(P\) and \(Q\) are rational functions, continuous in the \(x,y\)-plane and normalized by means of the condition
\[ \rho(P,Q)=\sup\bigl[(P^2+Q^2)(x^2+y^2+1)^{-n}\bigr]^{\frac12}<K. \tag{2} \]
Here \(K\) is a constant, and \(n\) is a fixed natural number \((n=2,3,\ldots)\).
It is proved that, under transition to nearby equations of the same class
\[ \frac{dx}{dt}=P+p, \qquad \frac{dy}{dt}=Q+q, \tag{3} \]
system (1) preserves the structure of curves in the projective plane if and only if conditions analogous to the known criteria of roughness are satisfied [1].
§ 1. Singular points
We shall assume that the plane has been transformed by means of central projection onto a sphere of unit radius tangent to the plane at a certain point taken as the initial point. Then the curves of system (1) pass into curves on the sphere, symmetrically located with respect to its center. The behavior of the curves at the points of the equator \(L\) and in its neighborhood will be determined by means of the differential equation
\[ z\sum_{k=0}^{\infty} B_k z^k\,d\varphi-\sum_{k=0}^{\infty} A_k z^k\,dz=0, \tag{4} \]
into which system (1) is transformed by the substitution
\[ x=\frac{\cos\varphi}{z}, \qquad y=\frac{\sin\varphi}{z}. \]
Here \(A_k\) and \(B_k\) are rational functions of \(\sin\varphi\) and \(\cos\varphi\); in particular,
\[ B_0=P_n\cos\varphi+Q_n\sin\varphi,\qquad A_0=P_n\sin\varphi-Q_n\cos\varphi, \tag{5} \]
\[ P_n=\left(z^n P\left(\frac{\cos\varphi}{z},\frac{\sin\varphi}{z}\right)\right)_{z=0}, \qquad Q_n=\left(z^n Q\left(\frac{\cos\varphi}{z},\frac{\sin\varphi}{z}\right)\right)_{z=0}. \tag{6} \]
The system (1) and the equation (4) closing it determine a decomposition of the sphere into trajectories such that through each of its points, with the exception of a finite number of singular points, there passes one and only one integral curve. The set of all singular points of the system (1) on the complete sphere consists of “proper” points, whose coordinates make the right-hand sides of equations (1) vanish, and of singular points on the curve \(L\) \((z=0)\), “adjoined” by the projective transformation. Their angular coordinates are found from the equation\(^*\)
\[ A_0(\varphi)=0. \tag{7} \]
Define at points \(a\) of the sphere \(\sigma\) the function \(\Delta(a)\):
\[ \Delta(a)=P'_xQ'_y-P'_yQ'_x,\quad a\notin L;\qquad \Delta(a)=B_0\frac{dA_0}{d\varphi},\quad a\in L, \tag{8} \]
and consider certain special cases, needed below, of the splitting of complex singular points for which \(\Delta(a)=0\) into simple ones under perturbations of equations (1) and (3) by polynomials of the form
\[ p=\sum_{i+k=0}^{n}\alpha_{ik}x^iy^k,\qquad q=\sum_{i+k=0}^{n}\beta_{ik}x^iy^k. \tag{9} \]
- The closing equation corresponding to the system (3) is written in the form
\[ z\sum_{k=0}^{\infty}(B_k+b_k)z^k\,d\varphi - \sum_{k=0}^{\infty}(A_k+a_k)z^k\,dz=0, \tag{10} \]
\[ b_k=p_{n-k}\cos\varphi+q_{n-k}\sin\varphi,\qquad a_k=p_{n-k}\sin\varphi-q_{n-k}\cos\varphi \]
\[ (k=0,\ldots,n), \]
\[ b_k=a_k=0\qquad (k=n+1,\ldots), \]
where \(p_{n-k}\) and \(q_{n-k}\) are the collection of terms of degree \(n-k\) in the expansions of \(p\) and \(q\) in powers of \(x\) and \(y\).
We shall agree to denote by \(\sigma(M)\) the equations (3) and (10), as well as the corresponding system of curves on the sphere. We shall regard a pair of functions \(p\) and \(q\), satisfying condition (2), as a point \(M\) of the space of admissible perturbations \(R\). In particular, to the point \(M_0(p=q=0)\) correspond the initial equations (1) and (4). By the \(\delta\)-neighborhood \(S(\delta,M_0)\) of the point \(M_0\) we shall mean rational functions satisfying the inequality \(\rho(p,q)<\delta\).
Let \(S(\varepsilon,a)\) be the \(\varepsilon\)-neighborhood of the point \(a\).
Lemma 1. If the system \(\sigma(M_0)\) has a multiple singular point \(a_1\), then for any \(\varepsilon>0\), \(\delta>0\) there exists a system \(\sigma(M_1)\), \(M_1\in S(\delta,M_0)\), which in the domain \(S(\varepsilon,a_1)\) will have at least two simple singular points, one of which coincides with \(a_1\).
Proof. The lemma is obvious if \(a_1\notin L\). Let \(a_1\in L\). We may suppose that the angular coordinate of this point is \(\varphi_1=0\), since for this
\(^*\) Equality (7) may also be an identity; then all points of the curve \(L\) will be singular. Such a case will always occur if the right-hand sides of the system (1) are polynomials of degree not exceeding \(n-1\).
case every other one is reduced to a rotation of the sphere, which is also equivalent to a certain linear transformation of the parameters of the system.
Case 1: \(A_0(0)=0^*)\). Put
\[ p=\gamma(x^n+yx^{n-1}), \qquad q=(\gamma-\nu)x^{n-1}y, \]
where \(\gamma\) and \(\nu\) are small parameters, not equal to zero. The corresponding equation (10) has two simple singular points, one of which coincides with \(a_1\), while the other, also lying on \(L\), has angular coordinate close to \(\varphi_1=0\) for sufficiently small \(\gamma\) and \(\nu\).
Case 2: \(A_0(0)\ne 0,\quad B_0(0)=0\). A splitting equation (10) with the required properties is obtained by taking
\[ p=\gamma x^n+\nu x^{n-1}, \qquad q=0. \]
In this case one of the singular points that appear does not lie on the curve \(L\).
- For a certain special choice of perturbations, multiple singular points, bypassing the stage of splitting, may pass directly into simple singular points or remain multiple as before**).
Lemma 2. Let \(a_1\) and \(a_2\) be two double singular points of the system \(\sigma(M_0)\). Then, for any \(\varepsilon>0\), one can find a point \(M_1\subset S(\varepsilon,M_0)\) such that \(a_1\) will still be a multiple singular point, and \(a_2\) a simple singular point of the system \(\sigma(M_1)\).
Proof. Three different cases may occur, depending on the position of the points \(a_1\) and \(a_2\) on the sphere.
1st case: \(a_1\subset L,\quad a_2\not\subset L\). We may assume that the coordinates of the singular points are respectively \(a_1(x_1=y_1=0)\), \(a_2(x_2=0,\ y_2=1)\). We obtain the system \(\sigma(M_1)\) of the required form by putting
\[ p=-x\frac{\gamma c_1}{\gamma+c_2}+\delta xy,\qquad q=\gamma(y-y^2), \]
where \(c_1=P_x'(a_1)\), \(c_2=Q_y'(a_2)\), and \(\gamma,\delta\) are small parameters.
2nd case: \(a_1\subset L,\quad a_2\subset L\). We shall assume that \(\varphi_1=0,\ \varphi_2=\pi/2\). Setting \(p=0,\quad q=\gamma x^n\), we obtain the system (3), for which \(a_1\) will be a multiple singular point and \(a_2\) a simple one.
Case 3: \(a_1\not\subset L,\quad a_2\subset L\). Let the coordinates of these points be respectively \(a_1(x_1=y_1=0)\), \(a_2(\varphi_2=0)\). The desired system (3) is obtained if we take \(p=\gamma x^n,\ q=0\).
- Thus, every singular point of the system (1), even if at it the existence or uniqueness of the solution is not violated, is a singular point of the simplest topological type (node, saddle, focus, center) for certain equations of the form (3). This thereby justifies, to a certain extent, the formal definition of singular points as points of indeterminacy of the field of directions.
§ 2. LIMIT CYCLES
Among the conditions determining the behavior of trajectories of a dynamical system, there may be some that are easily violated under small changes of the parameters. Closed curves, like singular points, may disappear or break up, forming new limit cycles.
\[ \begin{aligned} &\text{*) This also includes the critical case } A(\varphi)\equiv 0.\\ &\text{**) For the number and character of the singular points born, see [2].} \end{aligned} \]
- If equation (7) has no real roots, then the curve \(L\) will be a closed integral curve without singular points. In this case equation (4) has an integral \(z(\varphi,c)\), regular with respect to the constant \(c\), satisfying the condition \(z(0,c)=c\). The structure of the curves close to \(L\) is completely determined by the successor function
\[ \psi(c)=z(2\pi,c)-z(0,c)=\sum_{k=1}^{\infty}\gamma_k c^k . \tag{11} \]
The coefficients \(\gamma_k\) are found by the usual methods; in particular,
\[ \gamma_1=\exp h_0-1,\qquad h_0=\frac{1}{2\pi}\int_0^{2\pi}\frac{B_0(\varphi)}{A_0(\varphi)}\,d\varphi, \tag{12} \]
where \(h_0\) is the characteristic exponent of the cycle \(L\). The functions \(P_n\) and \(Q_n\) entering into its expression are homogeneous of degree \(n\) with respect to \(x\) and \(y\). In view of this, from (5), taking into account
\[ y\frac{\partial P_n}{\partial y}=nP_n-x\frac{\partial P_n}{\partial x},\qquad x\frac{\partial Q_n}{\partial x}=nQ_n-y\frac{\partial Q_n}{\partial y}, \]
we find
\[ B_0=\frac{1}{n+1}\left(\frac{dA_0}{d\varphi}+2\left(P'_{nx}+Q'_{ny}\right)\right). \]
Substituting this value of \(B_0\) into (12), we obtain
\[ h_0=\frac{1}{\pi(n+1)}\int_0^{2\pi}\frac{P'_{nx}+Q'_{ny}}{A_0}\,d\varphi . \tag{13} \]
- Closed trajectories of system (1) lying in a sufficiently small neighborhood \(S(\varepsilon,L)\) of the curve \(L\), and only they, correspond to real zeros of the function \(\psi(c)\) in the interval \(|c|<\varepsilon\). \(L\) is a simple cycle if \(h_0\ne0\). \(L\) is a multiple cycle if \(h_0=0,\ \gamma_2=0,\ldots,\gamma_{m-1}=0,\ \gamma_m\ne0\). Since the curves on the sphere are arranged symmetrically, the number \(m\) is always odd, and therefore the order of multiplicity of the cycle \(L\) should be taken as
\[ k=\frac{m+1}{2}. \]
In the case \(\psi(c)\equiv0\), all curves close to \(L\) are closed.
Theorem 1. Let \(L\) be a multiple cycle of some finite multiplicity \(k>1\). In that case, whatever \(\varepsilon>0,\ \delta>0\) may be, there exists a point \(M_1\subset S(\delta,M_0)\) such that the corresponding system \(\sigma(M_1)\) will have in \(S(\varepsilon,L)\) at least two limit cycles.
Proof. For sufficiently small values of the coefficients in (9), \(L\) is a closed integral curve of the corresponding equation (10). Its characteristic exponent, according to (13), is equal to
\[ h(M)=\frac{1}{\pi(n+1)}\int_0^{2\pi}\frac{P'_{nx}+Q'_{ny}+c_0}{A_0+a_0}\,d\varphi . \tag{14} \]
Here
\[ c_0=p'_{nx}+q'_{ny},\qquad a_0=yp_n-xq_n \tag{15} \]
are homogeneous polynomials in \(x\) and \(y\) of degrees \(n-1\) and \(n+1\), respectively, in which \(x=\cos\varphi,\ y=\sin\varphi\). By the known properties of the successor function ...
it remains, for the proof of the theorem, to show that in any neighborhood \(S(\delta,M_0)\) there exist points \(M_1\) and \(M_2\) such that
\[ h(M_1)<0,\qquad h(M_2)>0. \tag{16} \]
Let us note that the coefficients of the polynomials \(c_0\) and \(a_0\) may be taken as new independent parameters instead of \(\alpha_{ik}\) and \(\beta_{ik}\) \((i+k=0,\ldots,n)\). Indeed, regarding the equalities (15) as equations with unknowns \(p_n\) and \(q_n\), we find from them
\[ p_n=\int_0^x c_0\,dx-\frac{1}{n+1}\frac{\partial}{\partial y} \left(y\int_0^x c_0\,dx-a_0\right), \]
\[ q_n=\frac{1}{n+1}\frac{\partial}{\partial x} \left(y\int_0^x c_0\,dx-a_0\right). \tag{17} \]
The equalities (17) show that whatever the homogeneous polynomials \(c_0\) and \(a_0\) of degrees respectively \(n-1\) and \(n+1\) may be, the functions \(p_n\) and \(q_n\) are homogeneous polynomials of degree \(n\). Putting in (14) \(a_0\equiv 0\) and \(c_0=\lambda x^{n-1}\), where \(\lambda\) is a small parameter, we shall have from (16) \(h(\lambda)=\lambda k\). The existence of points \(M_1\) and \(M_2\) satisfying the inequalities (16) follows from the fact that the constant \(k\ne 0\).
- Suppose now that the system (1) has a non-rough periodic solution
\[ x=\varphi(t),\qquad y=\psi(t). \tag{18} \]
The splitting of a multiple limit cycle of multiplicity \(k\geqslant 3\) into no fewer than two simple cycles is possible whenever, in the space of nearby systems, the functionals
\[ g_0(p,q)=\int_0^T \mu(Qp-Pq)\,dt,\qquad g_1(p,q)=\int_0^T \frac{(\mu p)'_x+(\mu q)'_y}{\mu}\,dt \tag{19} \]
turn out to be linearly independent [3]. The integrals here are taken along the given curve (18), and \(\mu\) is an integrating factor of the system (1). For \(k=2\), as well as in general for even \(k\), in order to split the cycle it is sufficient that \(g_0(p,q)\ne 0\). All these conditions are fulfilled if the perturbations are taken in the form of rational functions of the form [4]
\[ p=\sum_{i=1}^{2}\sigma_i\lambda_i\bigl[(A_i x+B_i y+c_i)^2+\nu_i^2\bigr]^{-1}, \]
\[ q=\sum_{i=1}^{2}\delta_i\lambda_i\bigl[(A_i x+B_i y+c_i)^2+\nu_i^2\bigr]^{-1}. \tag{20} \]
In the case considered here these functions are admissible, since \(n\geqslant 2\). Hence it follows
Theorem 2. Let \(l\) be a limit cycle of finite multiplicity \(k\geqslant 2\) of the system \(\sigma(M_0)\). In that case, whatever \(\varepsilon>0\) and \(\delta>0\) may be, there will be found a point \(M_1\subset R,\ M_1\subset S(\delta,M_0)\), such that the corresponding system \(\sigma(M_1)\) will have at least two limit cycles of odd multiplicity in \(S(\varepsilon,l)\).
§ 3. STABLE STRUCTURES
We shall call the system of differential equations (1) rough, or structurally stable, at the point \(M_0 \subset R\), if the corresponding decomposition \(\sigma(M_0)\) of the Poincaré hemisphere, in the sense of distance measured along an arc of a great circle\(^*\), satisfies the general definition of roughness [5].
Theorem 3. A structurally stable system (1) is characterized by the following:
A) it has a finite number of singular points, and moreover such that the characteristic equation has no roots equal to zero;
B) it has a finite number of separatrices, and a separatrix cannot go from saddle to saddle unless it is an arc of the equator;
C) it has a finite number of closed integral curves, each of which is a simple limit cycle;
D) it cannot have singular points of the type of a compound focus.
Proof. It is obvious that the conditions A), B), C), D) are sufficient, since \(R\) is the true part of the set of all functions of class \(C_1\) [6, 9], while the closed hemisphere is a compact invariant set [7, 8, 10]. It remains to prove necessity\({}^{**}\).
Necessity of condition A). Let \(\sigma(M_0)\) be a stable system, and let \(V(M_0)\) be the number of its simple singular points on the hemisphere, i.e., those for which \(\Delta(a) \ne 0\). \(V(M_0)\) is a finite number; therefore there is an \(\varepsilon_0 > 0\) such that for all \(M \subset S(\varepsilon_0, M_0)\) the system \(\sigma(M)\) will have no fewer than \(V(M_0)\) singular points of the kind under consideration. To show that \(\sigma(M_0)\) has no singular points for which \(\Delta = 0\), suppose the contrary, and let \(W(M_0)\) be the set of such points.
Case 1: \(W(M_0)\) consists of one point \((W(M_0)=1)\). On the basis of Lemma 1 of § 1, for any \(\varepsilon < \varepsilon_0\) there is a point \(M_1 \subset S(\varepsilon, M_0)\) such that the system \(\sigma(M_1)\) will have simple singular points whose number \(V(M_1) \geq V(M_0)+2\), i.e., greater than the number of all singular points of the system \(\sigma(M_0)\). But in systems with the same structure this is impossible.
Case 2: \(W(M_0)\) has at least two points \((W(M_0) \geq 2)\). By virtue of Lemma 2 of § 1, for any \(\varepsilon_1 < \varepsilon_0\) there is a system \(\sigma(M_1)\), \(M_1 \subset S(\varepsilon_1, M_0)\), which will have \(V(M_1)\) simple singular points and \(W(M_1)\) compound ones, with
\[
V(M_1) \geq V(M_0)+2, \qquad W(M_1) \geq 1 .
\]
If \(W(M_1)=1\), the system \(\sigma(M_1)\), equivalent to \(\sigma(M_0)\), satisfies the conditions of the just considered case and, by what has been proved, will be unstable; consequently, \(\sigma(M_0)\) will also be unstable. If \(W(M_1)>1\), then, applying Lemma 2 again, we find \(\varepsilon_2 < \varepsilon_1\) and a point \(M_2 \subset S(\varepsilon_2, M_1)\) such that the system \(\sigma(M_2)\), equivalent to \(\sigma(M_1)\), will have
\[
V(M_2) \geq V(M_0)+4
\]
simple singular points and \(W(M_2)\geq 1\) compound singular points.
Continuing these arguments, applying Lemma 2 each time, we arrive at the conclusion that either \(W(M_k)=1\), or at a system \(\sigma(M_k)\) with the number of simple singular points
\[
V(M_k)\geq V(M_0)+2k .
\]
But this is impossible for arbitrary \(k\), since the number \(V(M_0)\) is finite. The contradiction obtained proves the assertion.
Necessity of condition B). Putting in equations (10) and (3)
\[
p=\lambda Q,\qquad q=-\lambda P,
\]
we obtain an admissible system which, having the same finite singular points as system (1), for sufficiently small \(|\lambda|\), \(\lambda\ne0\), has no separatrices going from a simple saddle to a simple saddle [5]. This
\(^*\) For boundary points \(a_1 \subset L\) and \(b_1 \subset L\), the geodesic distance
\[
\rho(a_1,b_1)=\lim_{\substack{a\to a_1\\ b\to b_1}}\rho(a,b),
\]
where \(a\subset L,\ b\subset L\).
\({}^{**}\) The necessity of these conditions cannot be accepted without proof, since a system that is not rough in some class of admissible equations may turn out to be rough with respect to a certain part of it.
the result is easily extended also to the case when one of the singular points or both are located on the curve \(L\). The exception is formed by the separatrices that are arcs of \(L\). These curves remain separatrices lying on \(L\), and join simple saddles which, for all sufficiently small \(p\) and \(q\), move little along the curve \(L\). Hence condition Б) follows.
Necessity of condition В). Let us first note that a rough system cannot have an infinite set of closed trajectories. Assuming the contrary and taking into account that the number of singular points is finite, we find an infinite sequence of closed curves, nested one inside another, and converging to some limiting continuum \(\Gamma\), consisting of whole trajectories. \(\Gamma\) cannot be a polygon of trajectories, since it would contain separatrices joining saddles and having points in a finite part of the plane, which, as has been proved, is impossible in rough systems. \(\Gamma\) cannot be a closed trajectory without singular points, because the system is analytic. It remains the case when \(\Gamma = L\). But from Theorem 2 it follows that there exist points \(M_1 \subset S(\varepsilon, M_0)\) such that for the corresponding systems the curve \(L\) is a limit cycle. Since \(L\) is the boundary curve of the domain of existence, it follows from this that the systems \(\sigma(M_0)\) and \(\sigma(M_1)\) are not equivalent. The contradiction obtained shows that in rough systems there is only a finite number of closed trajectories, each of which is a limit cycle of finite multiplicity.
Let the system \(\sigma(M_0)\) have \(A(M_0)\) cycles of odd multiplicity and \(B(M_0)\) cycles of even multiplicity. For all \(M \subset S(\varepsilon, M_0)\) one must have
\[
A(M)=A(M_0)\quad\text{and}\quad B(M)=B(M_0).
\]
Now suppose that the system \(\sigma(M_0)\) has a non-rough cycle. By virtue of Theorem 1 or 2, for any \(\varepsilon<\varepsilon_0\) there will be found a point \(M_1 \subset S(\varepsilon, M_0)\) such that the corresponding system \(\sigma(M_1)\) will have \(A(M_1)\) cycles of odd multiplicity, with
\[
A(M_1)\geq A(M_0)+1.
\]
The contradiction obtained proves the assertion.
The necessity of condition Г) follows from item В) and the theorem on the birth of limit cycles from a complex focus [5].
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Received by the editors
December 15, 1965
Gorky University
named after N. I. Lobachevsky