Reports of the Academy of Sciences of the USSR
1962. Vol. 146, No. 2

MATHEMATICS

B. A. SHCHERBAKOV

CLASSIFICATION OF POISSON-STABLE MOTIONS. PSEUDORECURRENT MOTIONS

(Presented by Academician P. S. Aleksandrov on 7 IV 1962)

In the general theory of dynamical systems one distinguishes a number of types of Poisson-stable motions, the classification of which is indicated in \((^1)\).

In the present note, proceeding from one general definition, various particular cases of which exhaust the definitions of all known types of Poisson-stable motions, a new class of motions is revealed, and its properties and connection with known classes are established.

\(1^\circ\). Let \(f(p,t)\) be a dynamical system defined in an arbitrary metric space \(R\) \((^2)\). In what follows an essential role will be played by a certain function \(T = T(\varepsilon,t,l)\) of the variables \(\varepsilon\), \(t\), and \(l\), varying respectively over the sets \((0,+\infty)\), \((-\infty,+\infty)\), and \([0,+\infty)\). In order to distinguish those cases when \(T\) does not depend on some of its arguments, we introduce into consideration the set \(\{E\}\) of all subsets of the set \(\{t,l\}\), whose elements are the variables \(t\) and \(l\).

Definition 1. We shall assign the motion \(f(p,t)\) to the set of motions \(\Omega^E\) \((A^E)\), if there exists a nonnegative function \(T = T(\varepsilon,t,l)\), defined for all \(\varepsilon > 0\), \(t \in (-\infty,+\infty)\), and \(l \geq 0\), such that:

1) for any triple of numbers \(\varepsilon\), \(t\), and \(l\), which are values of the corresponding variables, on the interval \([l,l+T]\) \(([ -l-T,-l])\) there exists a number \(\tau\) such that

\[ \rho [f(p,t+\tau), f(p,t)] \leq \varepsilon; \tag{1} \]

2) the function \(T\) does not depend on the variables from \(E\). The number \(\tau\) satisfying inequality (1) is called an \(\varepsilon\)-shift of the point \(f(p,t)\).

Let us consider all possible sets \(\Omega^E\) and \(A^E\) for possible \(E\), \(E \subseteq \{t,l\}\). (As is easy to see, there will be 8 of them.) It turns out that each of the introduced sets, with the exception of \(\Omega^t = A^t\), coincides with one or another known class of Poisson-stable motions. This connection is indicated in the following table.

Motions belonging to the set \(\Omega^t\) will be called pseudorecurrent. As will be shown below, the set of pseudorecurrent motions contains the set of all uniformly Poisson-stable motions, without coinciding with the latter. Thus, special (i.e., periodic and stationary) and almost periodic motions belong to the set \(\Omega^{t,l}\) together with recurrent motions, while uniformly Poisson-stable motions belong to the set \(\Omega^t\) together with pseudorecurrent motions. However, if the quantity \(\tau\) occurring in Definition 1 is regarded as a function

variables \(\varepsilon, t\), and \(l\), then one can obtain a more detailed classification, in which special and almost periodic motions will be separated out from the set of recurrent motions, and uniformly Poisson-stable motions from the set of pseudorecurrent motions. In this case, no new types of Poisson-stable motions will be found.

\(2^\circ\). Let us consider in more detail the set \(\Omega^t\) of pseudorecurrent motions. As is easy to verify, the definition of a pseudorecurrent motion is equivalent to the following. A motion \(f(p,t)\) is called pseudorecurrent if, for every pair of positive numbers \(\varepsilon\) and \(l\), there exists a number \(L,\ L \ge l\), such that on the interval \([l,L]\) there is an \(\varepsilon\)-shift of every point of the trajectory \(f(p,I)\).

It follows from Definition 1 that \(\Omega^{t,l} \subseteq \Omega^t \subseteq \Omega^\Lambda \cap A^\Lambda\), i.e. every recurrent motion is pseudorecurrent, and every pseudorecurrent motion is Poisson-stable (in both directions).

Concerning the closure of the trajectory of a pseudorecurrent motion, one can state the following:

Theorem 1. If a point \(q\) belongs to the closure of the trajectory of a pseudorecurrent motion, then the motion \(f(q,t)\) is pseudorecurrent.

Theorem 2. A motion \(f(p,t)\) of a compact dynamical system is pseudorecurrent if and only if, for every point \(q\) belonging to the closure of the trajectory \(\overline{f(p,I)}\), the motion \(f(q,t)\) is stable in the positive direction in the sense of Poisson.

\(3^\circ\). Let us construct an example of a pseudorecurrent motion. Put

\[ l_1=1;\qquad l_{n+1}=(4n+5)l_n\quad (n=1,2,\ldots). \tag{2} \]

The segment \([(2i-1)l_n,(2i+1)l_n]\), where \(i\) is an integer and \(n\) a natural number, will be denoted by \(\sigma_i^n\). It follows from (2) that the segment \(\sigma_i^{n+1}\) contains an odd number, namely \(4n+5\), of segments of the form \(\sigma_k^n\). Fix an arbitrary \(n\) and consider any segment \(\sigma_i^{n+1}\). Write down all the segments \(\sigma_k^n\) contained in \(\sigma_i^{n+1}\), in the order in which they occur:

\[ \sigma_{k_i+1}^n,\quad \sigma_{k_i+2}^n,\ldots,\quad \sigma_{k_i+2n+2}^n,\quad \sigma_{k_i+2n+3}^n,\ldots,\quad \sigma_{k_i+4n+5}^n, \]

where \(k_i\) is an integer which can be determined from the equation

\[ (2k_i+1)l_n=(2i-1)l_{n+1}. \tag{3} \]

The rightmost segment \(\sigma_{k_i+4n+5}^n\) and the segment \(\sigma_{k_i+2n+2}^n\) preceding the middle one will be called special segments, and the remaining ones ordinary. If \(\sigma_{k_i+s}^n\) is a special segment, then, as is easy to see, \(\sigma_{k_j+s}^n\) is also special, while \(\sigma_{k_j+s+1}^n\) is ordinary, for every integer \(j\).

Now on \((-\infty,+\infty)\) define a function \(\varphi(x)\) so that \(\varphi(x)=-1-|x|\) on \(\sigma_0^1\). Suppose that \(\varphi(x)\) is defined on \(\sigma_0^n\). Then on \(\sigma_0^{n+1}\setminus \sigma_1^n\) define \(\varphi(x)\) by the formula:

\[ \varphi(x)= \begin{cases} \dfrac{n+2-i}{n+2}\,\varphi(x-2il_n) & \text{on } \sigma_i^n \text{ for } i=1,2,\ldots,n+1;\\[6pt] 0 & \text{on } \sigma_i^n \text{ for } i=n+2,n+3,\ldots,2n+2;\\[6pt] \varphi(x+l_n+l_{n+1}) & \text{on } \sigma_i^n \text{ for } i=-2n-2,-2n-1,\ldots,-1. \end{cases} \]

Thus, on \((-\infty,+\infty)\), the function \(\varphi(x)\) has been constructed by induction. From the definition of this function there follow its properties:

1) \(\varphi(x)\) is uniformly continuous and \(0 \le \varphi(x) \le 1\) on \((-\infty,+\infty)\);

2) if the segment \(\sigma_i^n\) is special, then \(\varphi(x)\equiv 0\) on \(\sigma_{i-1}^n \cup \sigma_i^n\);

3) if the segment \(\sigma_i^n\) is ordinary and \(x\in \sigma_i^n\), then
\[ |\varphi(x+2l_n)-\varphi(x)|<\frac{1}{n}. \]

In the dynamical system of M. V. Bebutov \((^{1,3})\) consider the motion \(f(\varphi,t)\), determined by the function \(\varphi(x)\).

Lemma. Let \(n\) be a natural number. Then for every positive \(t\) there is a number \(\tau\), equal to \(2l_n\) or \(2l_{n+1}\), such that
\[ |\varphi(x+t+\tau)-\varphi(x+t)|<\frac{1}{n} \quad \text{for } x\in[0,2l_n]. \tag{4} \]

Let now \(\varepsilon\) and \(l\) be given positive numbers. Choose a natural \(n\) so that \(n\ge 1/\varepsilon\) and \(2l_n\ge l\), and put \(L=2l_{n+1}\). Let \(t\) be any real number. According to the lemma, for the number \(t-l_n\), by choosing \(\tau=2l_n\) or \(\tau=2l_{n+1}\), one can ensure that the inequality
\[ |\varphi(y+t-l_n+\tau)-\varphi(y+t-l_n)|<\varepsilon \]
holds for \(y\in[0,2l_n]\), whence it follows that
\[ |\varphi(x+t+\tau)-\varphi(x+t)|<\varepsilon \]
for \(|x|\le l_n\), and, a fortiori, for \(|x|\le 1/\varepsilon\) (since always \(l_n\ge n\ge 1/\varepsilon\)). Hence
\[ \rho[f(\varphi,t+\tau),f(\varphi,t)]<\varepsilon, \]
with \(\tau\in[2l_n,2l_{n+1}]\subset[l,L]\). According to the definition, the motion \(f(\varphi,t)\) is pseudorecurrent.

\(4^\circ.\) Obviously, every uniformly Poisson-stable motion is pseudorecurrent. However, even in a compact dynamical system there exists a pseudorecurrent motion which is not uniformly Poisson-stable.

Indeed, the motion \(f(\varphi,t)\), constructed in Sec. \(3^\circ\), is Lagrange stable in view of property 1 of the function \(\varphi(x)\). Let us show that this motion is not uniformly Poisson-stable. Let \(\tau\in(1,+\infty)\). Since the sequence \(\{l_n\}\) is strictly increasing, and \(\tau>1\), there is a natural \(n\) such that \(\tau\in\sigma_0^{\,n+1}\setminus\sigma_0^n\). According to the definition of the function \(\varphi(x)\), \(\varphi(0)=1\), while \(\varphi(-\tau)=0\), if
\[ \tau\in\bigcup_{i=1}^{n+1}\sigma_i^n . \]
Consequently, the inequality
\[ |\varphi(x+\tau)-\varphi(x)|\le \tfrac{1}{2} \tag{5} \]
is violated at \(x=-\tau\). If, however,
\[ \tau\in\bigcup_{i=n+2}^{2n+2}\sigma_i^n, \]
then inequality (5) is violated at \(x=0\). Thus it has been shown that the function \(\varphi(x)\) is not Bohr pseudoperiodic and, consequently, the motion \(f(\varphi,t)\) is not uniformly Poisson-stable.

The motion considered above is also not almost recurrent. Indeed, since there exist intervals of arbitrarily large length on which \(\varphi(x)\equiv0\), while \(\varphi(0)=1\), for \(\varepsilon<1\), for the point \(\varphi(x)\in f(\varphi,I)\) there is no relatively dense set of \(\varepsilon\)-translations.

Thus, there exists a pseudorecurrent motion which is not almost recurrent (and hence not recurrent).

The converse is also true: there exists an almost recurrent motion which is not pseudorecurrent. An example of such a motion can be constructed in the dynamical system of M. V. Bebutov.

As was already noted, every pseudorecurrent motion is Poisson-stable. However, even in a compact dynamical system there exists a Poisson-stable motion which is not pseudorecurrent. One can verify this using the example of the dynamical system given in \((^2)\) on page 365, taking into account Theorem 1 of the present note.

Institute of Physics and Mathematics
Academy of Sciences of the MSSR

Received
7 IV 1962

CITED LITERATURE

\(^{1}\) V. V. Nemytskii, UMN, 4, 6 (1949).
\(^{2}\) V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, Moscow—Leningrad, 1949.
\(^{3}\) M. V. Bebutov, Bull. Moscow State Univ., 2, 5 (1941).