DYNAMICAL SYSTEMS

IN THE PAPERS OF THE KISHINEV SEMINAR

ON THE QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS

B. A. SHCHERBAKOV

The topological theory of dynamical systems arose in the 1880s after the works on celestial mechanics by the founder of “qualitative” methods for the study of differential equations, A. Poincaré. The principal foundation of this theory was the well-known book of G. D. Birkhoff [1], in which the main problems and methods of investigation were formulated. The range of questions considered in it was intensively developed in subsequent years and received broad development in the works of Soviet and foreign mathematicians [2, 3]. A significant contribution to the development of the topological theory of dynamical systems was made by Moscow mathematicians.

Certain questions of the topological theory of dynamical systems were also considered in the works of participants of the Kishinev seminar on the qualitative theory of differential equations, organized in October 1960 by Docent K. S. Sibirsky. Below is a brief survey of these works, a list of which is placed at the end of the article [13—35].

In the present survey, which reflects only the main points of the problems considered, emphasis is placed on the formulation of results, and the difficulties that were sometimes encountered on the way to obtaining them are not discussed.

As is known [2, 3], a dynamical system is a collection \((R, J, f)\) consisting of a metric space \(R\), the set of all real numbers \(J\), and a continuous mapping \(f\) of the topological product \(R \times J\) into \(R\), satisfying the axioms of identity and homomorphism. The basic concepts of the theory of dynamical systems, the definitions of which are not given in the present article, are formulated in [2, 4].

1. Classification of Poisson-stable motions [13—19]. Poisson-stable motions were first considered by A. Poincaré [5] in connection with certain problems of celestial mechanics. In the class of these motions, whose characteristic property is “recurrence,” there exists an uncountable set of topologically distinct classes of motions. Therefore it makes sense to classify Poisson-stable motions according to the “frequency” of return of any point of a trajectory to an arbitrary neighborhood of its initial position. The classification constructed according to this principle in [13, 14] singles out the known classes of Poisson-stable motions introduced earlier from various considerations and, moreover, reveals two new classes, one of which is called the class of pseud-

of recurrent motions, and the other with the class of quasi-recurrent motions.

The following criteria hold:

Theorem 1. The motion \(f(p,t)\) is pseudo-recurrent if and only if, for any pair of positive numbers \(\varepsilon\) and \(l\), there exists an \(L>0\) such that every arc of a trajectory \(f(p,J)\) of temporal length \(L\) approximates, with accuracy up to \(\varepsilon\), the arcs adjacent to it of temporal length \(l\).

Theorem 2. In order that a motion \(f(p,t)\) stable in the sense of Lagrange be pseudo-recurrent, it is necessary and sufficient that \(f(p,J)\) consist of trajectories stable in the sense of Poisson in one and the same direction.

It follows from Theorem 1 that the class of pseudo-recurrent motions contains both the class of recurrent motions and the class of uniformly Poisson-stable motions. On the other hand, the “independence” of the class of pseudo-recurrent motions has been proved, in the sense that it does not coincide with any of the known classes. In this same sense the independence has been proved of the class of quasi-recurrent motions, which, as it turned out, is the intersection of the classes of almost recurrent and pseudo-recurrent motions.

Some properties of closures of trajectories of pseudo-recurrent and quasi-recurrent motions have been studied. For example, the following propositions hold:

Theorem 3. The closure of a trajectory of any pseudo-recurrent motion consists of trajectories of pseudo-recurrent motions.

Theorem 4. The closure of a trajectory of any quasi-recurrent motion is a bounded minimal set consisting of trajectories of quasi-recurrent motions.

The last theorems show that pseudo-recurrent and quasi-recurrent motions possess a number of properties similar to those by which recurrent motions are characterized.

All known classes of motions stable in the sense of Poisson (stationary, periodic, almost periodic, uniformly Poisson-stable, recurrent and almost recurrent), as well as the two new classes mentioned above, are called basic classes of Poisson-stable motions. Each of the basic classes is nonempty, and no two of them coincide. However, there exist Poisson-stable motions that belong simultaneously to several basic classes, and also some that do not belong to any of them. In connection with this, the following problem is of interest: to construct a partition of the set of Poisson-stable motions containing the minimal number of classes, in such a way that every basic class is the union of some classes of this partition. The formulated problem is also of interest in questions connected with the study of minimal sets. Some of these questions are noted below.

In [16, 18] it is established that the required partition consists of the following eleven classes, called constituent classes of Poisson-stable motions: \(\Pi_1\)—the class of stationary motions; \(\Pi_2\)—the class of periodic motions that are not stationary; \(\Pi_3\)—the class of almost periodic motions that are not periodic; \(\Pi_4\)—the class of recurrent motions that are uniformly Poisson-stable but not almost periodic; \(\Pi_5\)—the class of recurrent motions that are not uniformly Poisson-stable; \(\Pi_6\)—the class of almost recurrent motions that are uniformly Poisson-stable but not recurrent; \(\Pi_7\)—the class

quasi-recurrent motions that are not recurrent and not uniformly stable in the sense of Poisson; \(\Pi_8\)—the class of almost recurrent motions that are not pseudorecurrent; \(\Pi_9\)—the class of uniformly stable in the sense of Poisson motions that are not almost recurrent; \(\Pi_{10}\)—the class of pseudorecurrent motions that are neither almost recurrent nor uniformly stable in the sense of Poisson; \(\Pi_{11}\)—the class of stable in the sense of Poisson motions that are neither almost recurrent nor pseudorecurrent.

Let us note that the closure of the trajectory of any motion belonging to one of the classes \(\Pi_5\), \(\Pi_7\), and \(\Pi_8\) is a minimal set that is not pseudominimal. The question of the existence of minimal sets of this kind was posed by V. V. Nemytskii (see [4], p. 151, item 6). On the other hand, the closure of the trajectory of any motion belonging to one of the classes \(\Pi_1\), \(\Pi_2\), \(\Pi_3\), \(\Pi_4\), \(\Pi_6\) may serve as an example of a minimal set that is simultaneously pseudominimal.

A number of properties of the component classes have been studied. For example, the following has been proved.

Theorem 5. The component classes of motions stable in the sense of Poisson are invariant under equivariant mappings of dynamical systems.

It follows from this theorem that the component classes are invariant under homeomorphic mappings of compact dynamical systems.

In [16, 18] the question of the classification of motions stable in the sense of Poisson in linear dynamical systems [6] is also considered. This question requires separate consideration, since linear dynamical systems satisfy certain additional conditions that can narrow the diversity of motions. Until recently, the existence in linear dynamical systems of only six classes of motions stable in the sense of Poisson was known. With the aid of Theorem 5 it has been established that in linear dynamical systems there exist motions of all eleven component classes.

2. The study of minimal sets [20—25].

According to the well-known theorems of Birkhoff, every compact minimal set consists of trajectories of recurrent motions, and the closure of the trajectory of a recurrent motion situated in the complete space is a compact minimal set. Thus, Birkhoff’s theorems, on the one hand, establish the structure of compact minimal sets, and, on the other, single out the class of motions whose trajectory closures in complete spaces are compact minimal sets. Naturally, the problem arises of studying these questions for arbitrary minimal sets situated in an arbitrary metric space. Such a problem is considered in [24].

By \(\Pi_0\) we shall denote the class of motions that are stable in the sense of Poisson in one direction and departing in the other. The following is true.

Theorem 6. Let \(M\) be an arbitrary minimal set. Then one and only one of the following three cases occurs:

1) the set \(M\) consists of trajectories of motions of one and only one class of motions \(\Pi_k\) \((k=0, 1, \ldots, 11)\);

2) the set \(M\) contains both trajectories of motions of class \(\Pi_0\) and trajectories of motions of class \(\Pi_{11}\), and consists only of such trajectories;

3) the set \(M\) coincides with the trajectory of a departing motion.

A motion whose trajectory closure is a minimal set will be called minimal. It turns out that every

a motion belonging to one of the classes \(\Pi_k\) \((k=1,2,\ldots,8)\), as well as every outgoing motion, is minimal. As for the classes of motions \(\Pi_k\) \((k=0,9,10,11)\), in each of them there exist both minimal motions and motions that are not minimal. A number of criteria for the minimality of motions have been established. For example, the following has been proved.

Theorem 7. If a motion \(f(p,t)\), Poisson stable in the positive direction, and its positive semitrajectory is uniformly Lyapunov stable with respect to the set of its points in the negative direction, then \(\overline{f(p,J)}\) is a minimal set.

Along with the consideration of arbitrary minimal sets, homogeneous minimal sets were also studied. In [25] the following criterion was established:

Theorem 8. In order that a compact minimal set \(M\) be homogeneous, it is necessary and sufficient that from the condition \(f(p,t_n)\to p\) it follow that \(f(q,t_n)\to q\), whatever the point \(q\in M\).

For the case in which the phase space of the dynamical system is locally compact, a number of criteria for the almost periodicity of a motion \(f(p,t)\) (and consequently also for the homogeneity of the minimal set \(\overline{f(p,J)}\)) were established in [22, 23] in connection with the solution of a problem posed by A. A. Markov in [7].

Another problem, also connected with the consideration of minimal sets, was posed by V. V. Nemytskii [8]. It consists in studying the character of the recurrence of trajectories in a neighborhood of their dynamical limit set and its influence on the properties of motions in this set. An important result in this area is a theorem of V. V. Nemytskii according to which the set \(\Omega_p\) of all \(\omega\)-limit points of a motion \(f(p,t)\) that is positively Lagrange stable is minimal if and only if \(f(p,J^+)\) uniformly approximates \(\Omega_p\). This result was generalized in [20, 21], where the following was proved.

Theorem 9. The semitrajectory \(f(p,J^+)\) of a motion \(f(p,t)\) positively Lagrange stable uniformly approximates some subset \(Q\subseteq \Omega_p\) if and only if \(f(Q,J)\) is the unique minimal set in \(\Omega_p\).

It was also proved that

Theorem 10. The set \(\Omega_p\) of all \(\omega\)-limit points of a motion \(f(p,t)\) positively Lagrange stable is a trajectory of a periodic motion if and only if, for every point \(q\in \Omega_p\), there exists a relatively dense sequence of positive numbers \(t_n\) such that \(f(p,t_n)\to q\).

In addition, criteria for the almost periodicity of motions in the \(\omega\)-limit set were established. At the same time, one theorem of V. V. Nemytskii was refined; it is formulated as follows: in order that the \(\omega\)-limit set \(\Omega_p\) of a motion \(f(p,t)\) positively Lagrange stable be a minimal set consisting of trajectories of almost periodic motions, it is sufficient that the semitrajectory \(f(p,J^+)\) be uniformly positively Lyapunov stable with respect to the set of its points and that \(f(p,J^+)\) uniformly approximate \(\Omega_p\). It turned out that in this assertion the requirement of uniform approximation of the set \(\Omega_p\) is superfluous.

3. Generalizations of Birkhoff dynamical systems [26—35]. Along with the ordinary dynamical systems discussed above, certain generalizations were considered. One such generalization is the notion of a dispersed dynamical system (without

uniqueness) introduced by E. A. Barbashin [9]. In a dispersive dynamical system \((R, J, f)\), the values of the function \(f\) are bicompact subsets of the space \(R\). In [30—33], using results from the theory of multivalued mappings of a topological space, dynamical systems without uniqueness \((R, S, f)\) are studied, where \(R\) is a topological space and \(S\) is a topological semigroup. The transition to a semigroup is natural, since in the theory of dispersive dynamical systems one in fact considers separately the properties of the semigroup of mappings \(f(p,t)\) for \(t \geq 0\) and, in parallel, the properties of the semigroup of mappings \(f(p,t)\) for \(t \leq 0\). Under sufficiently general assumptions it was proved that a dynamical system without uniqueness, defined on the semigroup of nonnegative numbers \(J^{+}\), can be extended in a unique way to the entire group of real numbers.

In dynamical systems \((R,S,f)\), four types of invariant sets were studied, defined as follows: a set \(A \subseteq R\) is called invariant, semi-invariant, quasi-invariant, pseudo-invariant (with respect to the semigroup \(S\)) if, for every \(s \in S\), respectively,
\(f(A,s)=A\), \(f(A,s)\subseteq A\), \(f(A,s)\supseteq A\), \(f(p,s)\cap A \ne \Lambda\) for \(p \in A\). Corresponding to these types of invariant sets, four types of minimal sets were introduced. A set \(M \subseteq R\) is called a minimal invariant set if it is nonempty, closed, invariant, and contains no proper subset possessing these three properties. The other types of minimal sets are defined analogously.

Some properties and the connection between the sets thus introduced were studied. For example, the following was proved.

Theorem 11. If the semigroup \(S\) is commutative, then a bicompact set is minimal invariant if and only if it is minimal semi-invariant.

For dynamical systems \((R,J^{+},f)\), the notions of minimal centers of attraction were introduced and their properties studied.

In systems \((R,S,f)\), certain properties of the funnels \(f(p,S)\) were considered. In this connection, various notions were introduced which generalize the corresponding notions of ordinary dynamical systems. In what follows we shall assume that \(R\) is a uniform separable space.

A point \(p \in R\) is called a point of nonwandering if, for any \(s \in S\) and any open set \(U\) in \(R\) intersecting \(f(p,s)\), there exists a neighborhood \(V(p)\) such that \(f(q,s)\cap U \ne \Lambda\) for any point \(q \in V(p)\).

A point \(p \in R\) is called recurrent if, for any entourage \(\alpha\) of the uniform structure of the space \(R\), there exists a bicompact set \(K \subseteq S\) such that
\(f(p,S)\subseteq \alpha[f(q,K)]\), whatever the point
\(q \in \overline{f(p,S)}\).

The following proposition was proved, generalizing the known Birkhoff theorems to dynamical systems without uniqueness.

Theorem 12. If \(M\) is a bicompact minimal semi-invariant set, all of whose points are points of nonwandering, then \(M\) consists of recurrent points. Conversely, if a point \(p \in R\) is recurrent and the space \(R\) is complete, then \(\overline{f(p,S)}\) is a bicompact minimal semi-invariant set.

In dynamical systems \((R,S,f)\), a number of other properties of points of the space \(R\) were also considered, each of which, in the case of ordinary dynamical systems, means recurrence.

The notions of Poisson stability, orbital stability, and transitivity were also introduced, and the connections between them were studied.

A point \(p\in R\) is called Poisson stable if, for every neighborhood \(U(p)\) and every point \(q\in f(p,S)\), there exists an element \(s\in S\) such that
\[ f(q,s)\cap U(p)\ne \Lambda. \]

A point \(p\in R\) is called orbitally stable if, for every entourage \(\alpha\) of the uniform structure of the space \(R\), there exists an entourage \(\beta\) such that from \(q\in\beta(p)\) it follows that
\[ f(q,S)\subseteq \alpha[f(p,S)]. \]

A set \(A\subseteq R\) is called transitive if
\[ \overline{A}\subseteq \overline{f(p,S)} \]
for every point \(p\in A\).

Theorem 13. If all points of the funnel \(f(p,S)\) are simultaneously points of continuity and orbital stability, and the point \(p\) is Poisson stable, then the funnel \(f(p,S)\) is a transitive set. In a transitive set all points are Poisson stable.

In addition to the concepts mentioned above, various notions of periodicity and almost periodicity have been studied. In this connection some results of M. I. Minkevich [10] were refined.

Also considered were the partially ordered dynamical systems introduced by E. A. Barbashin [11], i.e. dynamical systems of the form \((R,G,f)\), in which \(G\) is a partially ordered group. In [26—28] certain results concerning the connection between recurrence, almost periodicity, and Lyapunov stability were transferred to such systems.

In [34] some limiting properties of partially ordered dispersive dynamical systems were considered. In particular, questions of invariance of dynamically limiting sets of such systems were studied.

Let \((R,G,f)\) be a partially ordered dispersive dynamical system given in the metric space \(R\). By \(G^{+}\) and \(G^{-}\) we shall denote, respectively, the semigroups of positive and negative elements of the group \(G\).

A point \(q\in R\) is called an \(\omega\)-limit point of the flow \(f(p,g)\) if, for every neighborhood \(U(q)\) and every \(g_0\in G^{+}\), there exists an element \(g>g_0\) such that
\[ f(p,g)\cap U(q)\ne \Lambda. \]

The domain of positive attraction of a point \(q\in R\) is the set of all points \(p\in R\) for which \(q\in\Omega_p\), where \(\Omega_p\) is the set of all \(\omega\)-limit points of the flow \(f(p,g)\).

Theorem 14. The set \(\Omega_p\) is pseudoinvariant and quasi-invariant with respect to each of the semigroups \(G^{+}\) and \(G^{-}\).

Theorem 15. The domain of positive attraction of an arbitrary point \(q\in R\) is quasi-invariant with respect to \(G^{+}\) and semi-invariant with respect to \(G^{-}\).

The following criterion of Lagrange stability was established:

Theorem 16. In order that the funnel \(f(p,G)\) be Lagrange stable in the positive direction, it is necessary and sufficient that the following conditions be satisfied:

1) the set \(\Omega_p\) is nonempty and compact;

2) for every \(\varepsilon>0\) there exists \(g_\varepsilon\in G^{+}\) such that
\[ f(p,g)\subseteq S(\Omega_p,\varepsilon) \]
for all \(g>g_\varepsilon\);

3) whatever \(g_0\in G^{+}\) may be, the union of the sets \(f(p,g)\) over all elements \(g\in G^{+}\) not exceeding \(g_0\) is a compact set in \(R\).

The last assertion generalizes one of the criteria for Lagrange stability of funnels proved for dispersive dynamical systems [29].

Let us note that in considering partially ordered dynamical systems [11, 12] it was assumed that in the group \(G\) there exists an \(\omega\)-sequence, i.e. a sequence of elements
\[ g_1<g_2<\cdots<g<n\cdots \]
such that for every \(g\in G\) there exists an element \(g_n\) of this sequence satisfying the inequality \(g<g_n\). In [34] this restriction was removed.

References

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Received by the editors
5 September 1964

Institute of Mathematics with the Computing Center of the Academy of Sciences
of the Moldavian SSR