UDC 517.917

MATHEMATICS

S. Kh. ARANSON

SYSTEMS OF THE FIRST DEGREE OF NON-STRUCTURAL STABILITY ON THE TORUS

(Presented by Academician L. S. Pontryagin, 19 III 1965)

Systems of the “first degree of non-structural stability,” which are relatively structurally stable among the set of non-structurally stable systems, have been studied in detail on the plane and on the cylinder in works \((^{1-3})\). It is natural to transfer the consideration of such dynamical systems to other surfaces as well. Whereas on an orientable surface of genus \(p = 0\) (on the sphere) the situation is analogous to the planar case, on surfaces of genus \(p \geq 1\) fundamental differences arise.

In the present paper we consider dynamical systems of the first degree of non-structural stability*

\[ \dot{x}=P(x,y),\qquad \dot{y}=Q(x,y), \tag{1} \]

defined on the torus \((p = 1)\). What is new (in comparison with propositions completely analogous to the theory of systems of the first degree of non-structural stability on the plane) in the theory of systems of the first degree of non-structural stability on surfaces, in particular on the torus, are propositions on the absence in these systems of trajectories winding from a double limit cycle or from a separatrix issuing from a saddle to the same saddle, and winding again onto that same limit cycle or separatrix issuing from a saddle to the same saddle, as well as the absence of Poisson-stable nonclosed semitrajectories. Whereas the proof of the first proposition presents no difficulties, the proposition on the absence in systems of the first degree of non-structural stability of Poisson-stable nonclosed semitrajectories requires additional considerations. Below a proof of the latter proposition is given by using the “rotation of the field”

\[ \dot{x}=P(x,y)-\mu Q(x,y),\qquad \dot{y}=Q(x,y)+\mu P(x,y). \tag{2} \]

Considering the torus unfolded onto the plane \((x,y)\), we shall call points congruent if their coordinates coincide modulo \(2\pi\). As is known \((^4)\), the following proposition is valid:

Lemma 1. Suppose that on the torus there is a Poisson-stable nonclosed semitrajectory. Then, whatever \(\varepsilon > 0\) is taken, it is always possible on the plane \((x,y)\) to find a triple of trajectories \(L_1, L, L_2\), corresponding on the torus to nonclosed trajectories Poisson-stable in both directions, such that \(L_1\) is above \(L\), \(L_2\) is below \(L\), and \(L_1, L_2\) belong to the \(\varepsilon\)-neighborhood of \(L\).

Lemma 2. Suppose that for sufficiently small \(\varepsilon > 0\) on the plane \((x,y)\) there exists a triple of trajectories \(L_1, L, L_2\) of system (1), such that \(L\) corresponds on the torus to a nonclosed trajectory Poisson-stable in both directions; \(L_1, L_2\) have values of the phase coordinates \(x(t), y(t)\) tending to infinity as \(t \to \pm\infty\); \(L_1\) lies above \(L\), \(L_2\) below \(L\); \(L_1, L_2\) belong to the \(\varepsilon\)-neighborhood of the trajectory \(L\), and between \(L_1, L_2\) there are neither equilibrium states nor limit cycles of system (2).

Then, whatever sufficiently small \(\mu^* > 0\) is taken, in the interval \(\mu \in (-\mu^*, \mu^*)\) there will be found two countable sequences: \(\mu_1, \mu_2, \ldots, \mu_r, \ldots\) \((0 < \cdots < \mu_r < \mu_{r-1} < \cdots < \mu_1 < \mu^*)\), \(\mu_1', \mu_2', \mu_r', \ldots\) \((-\mu^* < \mu_1' <\)

* The functions \(P(x,y)\), \(Q(x,y)\) are \(2\pi\)-periodic in \(x,y\), and belong to class \(C^n\) or to the analytic class.

\(< \mu_2' < \cdots < \mu_r' < \cdots < 0)\), for which there exist nonhomologous to zero closed trajectories of system (2) with pairwise distinct rotation numbers \((^{5,6})\)*.

Proof. Without loss of generality we may assume that the trajectory \(L\) of system (1) has the form

\[ x=\varphi(t), \qquad y=\psi(t), \qquad \varphi(0)=\psi(0)=0 \tag{3} \]

and that the value \(P(0,0)>0\). Then for sufficiently small \(\varepsilon>0\), for all points \(M(2\pi m,2\pi n)\) belonging to the domain \(D\) (\(D\) is the domain between \(L_1\) and \(L_2\)), the value \(P(2\pi m,y)>0\), where \(m,n\) are a subset of the numbers \(0,\pm1,\pm2,\ldots\), and \(y\) is any value between the ordinates of the points of intersection of the trajectories \(L_1,L_2\) with the lines \(x=2\pi m\). Let \(y_1,y_2\) be the ordinates of the points of intersection of the trajectories \(L_1,L_2\) with the line \(x=0\). We have: \(D=D_1\cup D_2\cup D_3\cup D_4\), where \(D_1\) is the domain lying above \(L\) to the right of \(x=0\), \(D_2\) is above \(L\) to the left of \(x=0\), \(D_3\) is below \(L\) to the left of \(x=0\), and \(D_4\) is below \(L\) to the right of \(x=0\). Denote by \(L^+\) and \(L^-\) the positive and negative semitrajectories distinguished from \(L\) and having their beginning at the point \(O(0,0)\). On the axis \(x=0\) there must necessarily exist two countable sequences tending to \(O(0,0)\) of points \(G(0,y)\), \(G'(0,y')\), where either \(y\in(0,y_1)\), \(y'\in(0,y_1)\), or \(y\in(y_2,0)\), \(y'\in(y_2,0)\), satisfying the following condition: the positive semitrajectories of system (1) issuing from \(G(0,y)\) pass through the points \(M(2\pi m,2\pi n)\) belonging either to the domain \(D_1\) when \(y\in(0,y_1)\), or to the domain \(D_4\) when \(y\in(y_2,0)\); the negative semitrajectories of system (1) issuing from \(G'(0,y')\) pass through the points \(M'(2\pi m',2\pi n')\) belonging either to the domain \(D_2\) when \(y'\in(0,y_1)\), or to the domain \(D_3\) when \(y'\in(y_2,0)\), with \(m,n,m',n'\) being subsets of the numbers \(0,\pm1,\pm2,\ldots\), and \(m\to+\infty\) as \(y\to0\), \(m'\to-\infty\) as \(y'\to0\). Using the set \(G(0,y)\), considered, for example, for \(y\in(0,y_1)\), and assigning any sufficiently small value \(\mu^*>0\), we construct a sequence \(\mu_1,\mu_2,\ldots,\mu_r,\ldots,\ 0<\cdots<\mu_r<\mu_{r-1}<\cdots<\mu_1<\mu^*\), for which there exist closed trajectories of system (2) with pairwise distinct rotation numbers.

For this purpose consider the trajectory \(L_\mu\) of system (2)

\[ x=\varphi(t,\mu), \qquad y=\psi(t,\mu), \qquad \varphi(0,\mu)=\psi(0,\mu)=0, \qquad \mu\in[0,\mu^*]. \tag{4} \]

Two cases are possible: 1) the trajectory \(L_{\mu^*}\) exits the domain \(D_1\) for some value \(x=2\pi x^*\), \(x^*>0\); 2) the trajectory \(L_{\mu^*}\) does not exit the domain \(D_1\) for any value \(x\in(0,+\infty)\). In both cases, for all \(t>0\) the trajectory \(L_{\mu^*}\) is certainly above the trajectory \(L\).

In the first case one can always choose a value \(2\pi m_1>2\pi x^*\), \(M_1(2\pi m_1,2\pi n_1)\in D_1\), since, according to the choice of the set \(G(0,y)\), in the domain \(D_1\) there exists a countable set of points of the form \(M(2\pi m,2\pi n)\).

Denote by \(A,B,C\), respectively, the points of intersection of \(L_{\mu^*}\) with \(L_1\), of \(L_1\) with \(x=2\pi m_1\), and of \(L\) with \(x=2\pi m_1\). The arcs \(AB\) and \(BC\) are arcs without contacts for the trajectories \(L_\mu\) of system (2) for any \(\mu\in[0,\mu^*]\). Since for \(\mu=0\) \(L_\mu\) passes through \(C\), and for \(\mu=\mu^*\) \(L_\mu\) passes through \(A\), there exists a value \(\mu_1\), \(0<\mu_1<\mu^*\), for which \(L_{\mu_1}\) closes on the torus, since it passes through \(M_1(2\pi m_1,2\pi n_1)\). We shall show that there exists a value \(\mu_2\), \(0<\mu_2<\mu_1<\mu^*\), such that the trajectory \(L_{\mu_2}\) passes through the point \(M_2(2\pi m_2,2\pi n_2)\in D_1\), and moreover \(n_1/m_1\ne n_2/m_2\). Since \(L_{\mu_1}\) passes through the points \(O(0,0)\) and \(M_1(2\pi m_1,2\pi n_1)\), it will also pass through the points \(M(2\pi m_1 k,2\pi n_1 k)\). By virtue of the choice of the set \(G(0,y)\) and by the theorem on continuous dependence on initial conditions for system (1), there will be found a countable set of trajectories \(L'\), congruent to \(L\) and passing through points \(M'(2\pi m',2\pi n')\in D_1\), moreover such that they intersect the line \(CM_1\)

* From Lemma 2 there follows a proof, simpler than in (7), of the theorem on the absence, in Poisson-stable rough dynamical systems on the torus, of nonclosed trajectories.

between the points \(C\) and \(M_1\), i.e., from the moment of reaching the line \(x=2\pi m_1\), remaining between the trajectories \(L\) and \(L_{\mu_1}\). Since the trajectories \(L'\) form a countable set, there exists a countable subset of trajectories \(L'' \subset \{L'\}\) whose points \(M''(2\pi m'',2\pi n'')\), belonging to the domain \(D_1\), lie to the right of the line \(x=2\pi m_1\) and between \(L_{\mu_1}, L\). Therefore \(n''/m'' \ne n_1/m_1\). Fix one of the trajectories \(L''\) and denote \(M''(2\pi m'',2\pi n'')\) by \(M_2(2\pi m_2,2\pi n_2)\), where \(n_2/m_2 \ne n_1/m_1\). Analogously to what was indicated above, one can show that there exists a value \(\mu_2 \in (0,\mu_1)\) for which \(L_{\mu_2}\) passes through \(M_2(2\pi m_2,2\pi n_2)\). In exactly the same way one can show the existence of a countable sequence
\[ \mu_1,\ \mu_2,\ldots,\mu_r,\ldots,\quad 0<\cdots<\mu_r<\mu_{r-1}<\cdots<\mu_1<\mu^*, \]
such that the corresponding trajectories \(L_{\mu_1}, L_{\mu_2},\ldots,L_{\mu_r},\ldots\) pass through the points \(M_1(2\pi m_1,2\pi n_1), M_2(2\pi m_2,2\pi n_2),\ldots, M_r(2\pi m_r,2\pi n_r),\ldots\), and none of the ratios
\[ n_1/m_1,\ n_2/m_2,\ldots,\ n_r/m_r,\ldots \]
is equal to another.

The second case, when the trajectory \(L_\mu\) does not leave the domain \(D_1\) for any value \(x\in(0,+\infty)\), can be reduced to the first case by small perturbations. Considering the interval \(\mu\in(-\mu^*,0)\) and the set \(G'(0,y')\), \(y'\in(0,y_1)\), we obtain a sequence
\[ \mu_1',\ \mu_2',\ldots,\mu_r',\ldots,\quad -\mu^*<\mu_1'<\cdots<\mu_{r-1}'<\mu_r'<\cdots<0, \]
satisfying the requirements of Lemma 2. The case when in the set of points \(G(0,y)\), \(G'(0,y')\) one has \(y\in(y_2,0)\), \(y'\in(y_2,0)\), is completely analogous to the case when \(y\in(0,y_1)\), \(y'\in(0,y_1)\). The lemma is proved.

Theorem 1. In a dynamical system of the first degree of non-roughness on the torus there exists only a finite number of equilibrium states and limit cycles, and no combination of two non-rough trajectories is possible: equilibrium states (saddle-nodes, compound foci of first order), double limit cycles, or separatrices going from saddle to saddle.

Theorem 2. In a dynamical system of the first degree of non-roughness on the torus there cannot be Poisson-stable nonclosed semitrajectories.

Proof. Suppose the contrary. Then system (1) has on the torus a Poisson-stable nonclosed semitrajectory. Therefore the possible closed trajectories or closed contours composed of equilibrium states and their separatrices are homologous to zero. Using Lemma 1 and Theorem 1, it is not difficult to verify that the hypotheses of Lemma 2 are satisfied. Choose a number \(\mu^*>0\) from the condition that all non-rough systems (2) in the interval \((-\mu^*,\mu^*)\ni\mu\) have a topological structure analogous to system (1)*. We shall show that the interval \((-\mu^*,\mu^*)\) contains not a single value \(\mu\) for which system (2) is a rough dynamical system on the torus. Otherwise, according to the definition of roughness \((^8)\), there exists an interval \((\alpha,\beta)\ni\tilde\mu\) such that system (2), considered at any value \(\mu\in(\alpha,\beta)\), \(\mu\ne\tilde\mu\), has a decomposition of the torus into trajectories topologically equivalent to the decomposition at \(\mu=\tilde\mu\). Moreover, at least one of the endpoints of the interval \((\alpha,\beta)\), which we denote by \(\mu^{**}\), lies inside the interval \((-\mu^*,\mu^*)\) (since for \(\mu=0\) system (2) is a non-rough system on the torus), corresponds to an obviously non-rough system (2), and, moreover, system (2) at \(\mu=\mu^{**}\) has a Poisson-stable nonclosed semitrajectory.

Then, by virtue of Lemma 2, it is always possible to find a value of the parameter
\[ \mu\in(\alpha,\beta)\cap(-\mu^*,\mu^*) \]
such that system (2) at \(\mu=\bar\mu\) has on the torus a closed trajectory not homologous to zero, with rotation number certainly different from the rotation numbers of any of the limit cycles (if they exist) of system (2) at the value \(\mu=\tilde\mu\). However, by the choice of the number \(\mu^*>0\), system (2) at
\[ \mu=\bar\mu\in(-\mu^*,\mu^*) \]
must have the same topological structure as at \(\mu=\tilde\mu\), which is impossible.

* This follows from the definition of systems of the first degree of non-roughness \((^1)\).

Therefore, whatever $\tilde{\mu} \in (-\mu^*, \mu^*)$ we take, system (2) for the value $\mu = \tilde{\mu}$ must be a non-rough dynamical system on the torus and, in accordance with the choice of the number $\mu^* > 0$, must have on it a Poisson-stable nonclosed semitrajectory. But then, by Lemma 2, one can choose a value $\mu^{**} \in (-\mu^*, \mu^*)$, $\mu^{**} \ne \tilde{\mu}$, such that system (2) for $\mu = \mu^{**}$ has on the torus a closed trajectory nonhomologous to zero, which is impossible. The theorem is proved*.

Gorky State University
named after N. I. Lobachevsky

Received
4 III 1965

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* The assertion of Theorem 2 is not violated if one considers dynamical systems in the class of “polynomials” of the given system $N$ (7).