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UDC 517.933
ASYMPTOTIC TRAJECTORIES OF NON-AUTONOMOUS DYNAMICAL SYSTEMS
V. A. BELOV
Some basic facts of the topological theory of dynamical systems can be extended to non-autonomous systems of differential equations if, following the suggestion of V. V. Nemytskii, the set of solutions of the system is supplemented by so-called limiting solutions and qualitative considerations are then carried out with respect to this new augmented set. The first results in this direction were obtained by V. M. Millionshchikov [3].
Following V. M. Millionshchikov, in the present paper, in particular, we extend the theorem on necessary and sufficient conditions for the minimality of the \(\Omega\)-limit sets of positively Lagrange-stable solutions of a dynamical system (Theorem 40, p. 427, [1]) and the theorem on sufficient conditions for the set of \(\omega\)-limit points of a positively Lagrange-stable solution of a dynamical system to be a minimal set of almost periodic solutions (Theorem 41, p. 428, [1]).
Consider the non-autonomous system of differential equations
\[ \frac{dx}{dt}=f(x,t), \tag{1} \]
where \(\|f(x,t)\|\leq M\); \(f(x,t)\) is a continuous function of all variables in some cylinder \(\{G,t\geq t_G\}\); \(G\) is some domain (possibly unbounded) in the \(n\)-dimensional Euclidean space \(R_n=\{x\}\); \(t_G\geq -\infty\).
Definition 1. ([3], Definition 1). By a limiting solution of system (1) we shall mean a solution \(\bar{x}(q,t_q;t)\) that is the uniform limit on arbitrary intervals \([t_i^{(1)},t_i^{(2)}]\) of the \(t\)-axis of an infinite sequence of solutions of system (1), or of their shifts in the argument \(t\), \(x(x,t_x;t+t_n)\), as the number \(n\) of the term of the sequence tends to infinity:
\[ \{x(x,t_x;t+t_n)\}\underset{[t_i^{(1)},\,t_i^{(2)}]}{\longrightarrow}\bar{x}(q,t_q;t). \]
If the solutions of system (1) are defined for \(-\infty<t<+\infty\), then the totality of solutions of system (1) and limiting solutions will be called a non-autonomous dynamical system.
A non-autonomous dynamical system does not possess the uniqueness property, i.e., through each point \((q,t_q)\in\{R_n,t\}\) there may pass an infinite set of limiting solutions. In the subsequent discussion we shall consider only such limiting solutions as are constructed on positive arcs; moreover, for this purpose we use only such shifts that, for \(\{x(x,t_x;t+t_n)\}\), \(|t_n|<|t_{n+1}|\), \(t_n\to+\infty\) as \(n\to\infty\). If through the point \((x,t_x)\in\{R_n,t\}\) there pass several solutions or limiting ...
solutions, the arguments are carried out for one of them. We shall regard the set of trajectory points of solutions with initial data \(\{x,0\}\) as belonging to the space \(R_n^0=\{x\}\). For clarity, where this is required, the distance between the projections of the points \((q,t_q)\) and \((s,t_s)\), \(q\) and \(s\in R_n\), onto the phase space \(R_n\) will be denoted by: \(\rho_{R_n}[(q,t_q),(s,t_s)]\). The restriction \(\|f(x,t)\|\le M\) on system (1) can be removed by a change of the time \(t\) (see [3]).
We introduce Definition 2, as well as the subsequent Definitions 3 and 4, analogously to the corresponding definitions from [1] (pp. 358, 349, 400).
Definition 2. A point \(q\in R_n\) will be called an \(\omega\)-limit point of the trajectory \(x(x_0,t_0;t)\) of a nonautonomous dynamical system if there exists a sequence of numbers \(t_1,t_2,\ldots,t_n,\ldots\) such that
\[ \lim_{t_n\to+\infty}\rho[x(x_0,t_0;t_n),q]=0. \]
We shall denote the totality of \(\omega\)-limit points of the trajectory \(x(x_0,t_0;t)\) by \(\Omega_{x_0,t_0}\). Analogously, for the negative semitrajectory of a solution, the definitions of \(\alpha\)-limit points and of the \(A\)-limit set are introduced.
From the set of all solutions of system (1), their shifts with respect to the argument \(t\), and limiting solutions constructed on positive arcs of solutions, we form a set \(E\).
Definition 3. ([3], Definition 2). A set \(M\subset R_n\) will be called invariant \((E)\) if for every point \(q\in M\) there exists a \(t_q\) such that the trajectory of the solution \(x(q,t_q;t)\), or of the shift of the solution \(x(q,t_q;t+t_n)\), or of the limiting solution \(\bar{x}(q,t_q;t)\), belongs to \(M\) for all \(t\), \(-\infty<t<+\infty\).
Definition 4. A set \(\Sigma\subset R_n\) will be called minimal \((E)\) if it is nonempty, closed, invariant \((E)\), and contains no proper subset possessing the same properties. The characteristic property of a minimal \((E)\) set is the following.
In order that the set \(\Sigma\subset R_n\) be minimal \((E)\), it is necessary and sufficient that for every trajectory of a solution of system (1), or of a limiting solution, contained in \(\Sigma\), the property
\[ \overline{x(q,t_q;t)}=\Sigma \]
be satisfied.
The proof of this property is similar to the proof of the analogous property in [1], p. 402.
The set of \(\omega\)-limit points of a positively Lagrange-stable solution \(x(x_0,t_0;t)\) of system (1) is a connected compact set.
Indeed, let \(q_1,q_2,\ldots,q_n,\ldots\in\Omega_{x_0,t_0}\) and \(\lim_{n\to\infty}q_n=q_0\). We shall show that \(q_0\in\Omega_{x_0,t_0}\). Choose \(\varepsilon>0\) and determine an \(n>N\) such that \(\rho(q_0,q_n)<\varepsilon/2\). Since \(q_n\in\Omega_{x_0,t_0}\), there will be a number \(t=\tau_n\) such that \(\rho[x(x_0,t_0;\tau_n),q_n]<\varepsilon/2\), and hence \(\rho[x(x_0,t_0;\tau_n),q_0]<\varepsilon\), i.e. \(q_0\in\Omega_{x_0,t_0}\). Since the set \(\Omega_{x_0,t_0}\) is bounded, it follows that \(\Omega_{x_0,t_0}\) is compact. Repeating exactly the scheme of the proof of Theorem 13 on p. 361, [1], one can prove the connectedness of the set \(\Omega_{x_0,t_0}\). Since the closure of the positive semitrajectory of a positively Lagrange-stable solution is compact, the set of its \(\omega\)-limit points is nonempty.
Theorem 1. The set of \(\omega\)-limit points \(\Omega_{x_0,t_0}\) of a positively Lagrange-stable solution \(x(x_0,t_0;t)\) of system (1) is invariant \((E)\).
Proof. Consider an arbitrary point \(q\in\Omega_{x_0,t_0}\):
\[ q=\lim_{t_n\to+\infty} x(x_0,t_0;t_n)=\lim_{n\to\infty} x_n . \]
We shift the solution \(x(x_0,t_0;t)\) along the \(t\)-axis so that the points \((x_n,t_n)\) fall into the fixed phase space \(R_n^0\); the corresponding time is \(t=0\). Since, by the general condition,
\[
\left\|\frac{dx}{dt}\right\|\le M,
\]
we have an infinite sequence of uniformly continuous functions, positively stable in the sense of Lagrange. The set of such shifts satisfies Ascoli’s theorem ([2]), according to which it follows that from the infinite sequence of shifts of the solutions in \(t\) one can extract an infinite subsequence of shifts converging uniformly on every finite interval of the \(t\)-axis to some limiting solution. All points of such a limiting solution are \(\omega\)-limit points, which follows from the choice of the uniformly convergent subsequence \((n\to\infty\), where \(n\) is the number of the point \((x_n,t_n))\). Such a construction is possible for any point \(q\in\Omega_{x_0,t_0}\). Thus, the set \(\Omega_{x_0,t_0}\) is invariant \((E)\).
From the set of shifts in \(t\) of the solution \(x(x_0,t_0;t)\) of system (1), and of limiting solutions constructed on its positive arcs, we form the set \(E_{x_0,t_0}\). It follows from the proof of Theorem 1 that the set of \(\omega\)-limit points of the solution \(x(x_0,t_0;t)\), positively stable in the sense of Lagrange, of system (1) is invariant \((E_{x_0,t_0})\).
Theorem 2. In order that the set \(\Omega_{x_0,t_0}\) of \(\omega\)-limit points of the solution \(x(x_0,t_0;t)\) of system (1), positively stable in the sense of Lagrange, be minimal \((E_{x_0,t_0})\), it is necessary and sufficient that the positive semi-trajectory \(x(x_0,t_0;I^+)\) uniformly approximate the set \(\Omega_{x_0,t_0}\).
Proof. Necessity. Let \(\Omega_{x_0,t_0}\) be minimal \((E_{x_0,t_0})\). Suppose that the semi-trajectory \(x(x_0,t_0;I^+)\) does not uniformly approximate the set \(\Omega_{x_0,t_0}\). Then there exist a number \(\alpha>0\), a sequence of intervals
\[
(t_1,t_1'),\ (t_2,t_2'),\ldots,\ (t_n,t_n'),\ldots;\qquad
t_n>0;\quad t_n'-t_n\to\infty\quad \text{as } n\to\infty
\tag{2}
\]
and a sequence of points \(q_n\in\Omega_{x_0,t_0}\) such that
\[
\rho[x(x_0,t_0;t_n,t_n'),q_n]>\alpha .
\]
In the compact set \(\Omega_{x_0,t_0}\) the sequence \(\{q_n\}\) contains a subsequence converging to a point \(q_0\in\Omega_{x_0,t_0}\). For simplicity of notation, let
\[
q_0=\lim_{n\to\infty} q_n;\qquad \text{for } n>N,\quad \rho[q_0,q_n]<\frac{\alpha}{3}.
\]
Then for \(n>N\)
\[
\rho[x(x_0,t_0;t_n,t_n'),q_0]>\frac{2\alpha}{3}.
\tag{3}
\]
Consider the sequence of points:
\[
x_n=x\left(x_0,t_0;t_n+\frac{t_n'-t_n}{2}\right)=x(x_0,t_0;\tau_n),
\]
\[
t_n>0,\qquad t_n'-t_n\to+\infty,\qquad \tau_n\to+\infty\quad \text{as } n\to\infty .
\]
By virtue of positive stability in the sense of Lagrange, the sequence \(\{x_n\}\) contains a subsequence converging to some point \(P\in\Omega_{x_0,t_0}\). For simplicity of notation, let the sequence itself converge to the point \(P\): \(P=\lim_{n\to\infty}x_n\). Just as in the proof of Theorem 1, we construct a limiting solution on the positive arcs of the solution \(x(x_0,t_0;t)\), pro-
passing with its trajectory through the point \(P:\bar x(P,0;t);\ (P,0)\in R_n^0\). By virtue of the minimality of the set \(E_{x_0,t_0}\), the set \(\Omega_{x_0,t_0}\) of trajectories of the limiting solution \(\bar x(P,0;t)\) can be made arbitrarily close to the point \(q_0\), i.e., there will be a point \(\bar x(P,0;t^*)\) such that
\[
\rho[\bar x(P,0;t^*),q_0]<\frac{\alpha}{3}.
\]
In the \(\frac{\alpha}{3}\)-tube of the limiting arc \(\bar x(P,0;0,t^*)\) there is contained some shift of the arc of the solution \(x(x_0,t_0;t)\), of time length \(|t^*|\).
Let \(|t^*|=t'_{n_1}-t_{n_1}\), where \(t'_{n_1}\) and \(t_{n_1}\) are taken from the sequence (2). We write this shift as
\[
x_1(x_0,t_0;t_{i_1}+t_{n_1},\,t_{i_1}+t'_{n_1}).
\tag{4}
\]
On this shift of the arc there is a point \(x_1(x_0,t_0;t_{i_1}+t_{n_1}+\tau_0)\), \(0<\tau_0<t'_{n_1}-t_{n_1}\), such that
\[
\rho[x_1(x_0,t_0;t_{i_1}+t_{n_1}+\tau_0),q_0]
<\rho[x_1(x_0,t_0;t_{i_1}+t_{n_1}+\tau_0),
\]
\[
\bar x(P,0;t^*)]+\rho[\bar x(P,0;t^*),q_0]
<\frac{\alpha}{3}+\frac{\alpha}{3}.
\tag{5}
\]
The shift (4) corresponds to the arc of the solution \(x(x_0,t_0;t):x(x_0,t_0;t_{n_1},t'_{n_1})\), for which, by assumption, inequality (3) is valid, with \(n_1>N\). Since shifting a solution in \(t\) does not change the trajectory, inequality (5) contradicts (3).
Sufficiency. The set \(\Omega_{x_0,t_0}\) is invariant \((E_{x_0,t_0})\). Let \(q\) be any point of the set \(\Omega_{x_0,t_0}\). From the characteristic property of the minimal \((E_{x_0,t_0})\) set, it is enough to prove that if \(x(q,t_q;t)\in E_{x_0,t_0}\) and \(x(q,t_q;t)\subset \Omega_{x_0,t_0}\) by its trajectory, then \(\overline{x(q,t_q;t)}=\Omega_{x_0,t_0}\). If the solution \(x(x_0,t'_0;t)\in E_{x_0,t_0}\) is contained in \(\Omega_{x_0,t_0}\) entirely by its trajectory, then for it always
\[
\overline{x(x_0,t'_0;t)}=\Omega_{x_0,t_0}.
\]
We now consider a limiting solution passing with its trajectory through the point \(q\). Let the limiting solution \(x(q,t_q;t)\in E_{x_0,t_0}\) be such that \(\overline{x(q,t_q;t)}\subset \Omega_{x_0,t_0}\), i.e. the set \(\Omega_{x_0,t_0}\) is not a minimal \((E_{x_0,t_0})\). Then there exist a point \(r\in\Omega_{x_0,t_0}\) and a sphere \(s(r,\alpha)\), \(\alpha>0\), such that
\[
s(r,\alpha)\cap \overline{x(q,t_q;t)}=\varnothing.
\]
From the condition of uniform approximation, for \(\frac{\alpha}{2}>0\) we find the number \(T\!\left(\frac{\alpha}{2}\right)\). In any \(\varepsilon\)-neighborhood of the closure \(\overline{x(q,t_q;t)}\) there will be contained an arc of some shift of the solution \(x(x_0,t_0;t)\) of time length \(T\!\left(\frac{\alpha}{2}\right)\). Therefore
\[
\rho[r,\overline{x(q,t_q;t)}]\leq \varepsilon+\frac{\alpha}{2}.
\]
By virtue of the arbitrariness of \(\varepsilon>0\) we have a contradiction with the assumption.
Definition 5. The solution \(x(x_0,t_0;t)\) has property \(L_1\) if, for any \(\varepsilon>0\) and any \(T\), there exist numbers \(\delta>0\) and \(t^*\) such that, if
\[
\rho_{R_n}[x(x_0,t_0;t'),\,x(x_0,t_0;t'')]<\delta;\qquad t',t''\geq t^*,
\]
then
\[
\rho_{R_n}[x(x_0,t_0;t'+t),\,x(x_0,t_0;t''+t)]<\varepsilon
\]
for all \(t\geq T\).
We shall prove that if, on the positive arcs of the solution \(x(x_0,t_0;t)\) possessing property \(L_1\), one constructs a limiting solution, then the limiting solution will possess property \(L_2\):
for any \(\varepsilon>0\) and \(T\) there exists such a \(\delta>0\) that if
\[ \rho_{R_n}\left[\bar x(q,t_q;t'),\ \bar x(q,t_q;t'')\right]<\delta, \]
then
\[ \rho_{R_n}\left[\bar x(q,t_q;t'+t),\ \bar x(q,t_q;t''+t)\right]<\varepsilon \quad \text{for } t\geq T . \]
Fix numbers \(\varepsilon>0\) and \(T\). For them, from property \(L_1\) of the solution \(x(x_0,t_0;t)\), we determine the numbers \(\delta\left(\dfrac{\varepsilon}{2},T\right)\) and \(t^*\left(\dfrac{\varepsilon}{2},T\right)\). Suppose that the limiting solution does not possess property \(L_2\). Then, for the chosen \(\varepsilon>0\) and \(T\), no matter how small \(\delta_1>0\), \(\delta_1<\delta\), we take, there will always be found at least two points \(\bar x(q_1,t_q;t'_{\delta_1})\) and \(\bar x(q,t_q;t''_{\delta_1})\) such that
\[ \rho_{R_n}\left[\bar x(q_1,t_q;t'_{\delta_1}),\ \bar x(q,t_q;t''_{\delta_1})\right]<\delta_1, \]
but
\[ \rho_{R_n}\left[\bar x(q,t'_q;t'_{\delta_1}+t),\ x(q,t'_q;t''_{\delta_1}+t)\right]>\varepsilon \quad \text{for } t\geq T . \]
Let \(t''_{\delta_1}>t'_{\delta_1}\). In any \(\alpha\)-tube of the limiting arc
\(\bar x[q,t'_q;\ \min(t'_{\delta_1},\, t'_{\delta_1}+t),\ \max(t''_{\delta_1}+t,\, t''_{\delta_1})]\)
there is contained an arc of some shift of the solution \(x(x_0,t_0;t)\) of time length
\[
\left|\max(t''_{\delta_1},\, t'_{\delta_1}+t)-\min(t'_{\delta_1},\, t'_{\delta_1}+t)\right|,
\quad t\geq T .
\]
For this shift
\[ \rho_{R_n}\left[x(x_0,t_0;t_n+t'_n),\ x(x_0,t_0;t_n+t''_n)\right]<\delta_1, \]
\[ \rho_{R_n}\left[x(x_0,t_0;t'_n),\ \bar x(q,t_q;t'_{\delta_1})\right]\leq \alpha, \]
\[ \rho_{R_n}\left[x(x_0,t_0;t''_n),\ \bar x(q,t_q;t''_{\delta_1})\right]\leq \alpha,\quad t'_n,t''_n\geq t^*, \]
but
\[ \rho_{R_n}\left[x(x_0,t_0;t_n+t'_n+t),\ x(x_0,t_0;t_n+t''_n+t)\right]<\frac{\varepsilon}{2}. \]
In view of the arbitrariness of \(\alpha\), we have a contradiction with the supposition.
Let the solution \(x(x_0,t_0;t)\) not be contained, by its trajectory, in the set of \(\omega\)-limit points \(\Omega_{x_0,t_0}\). Then the following holds.
Theorem 3. In order that the set of \(\omega\)-limit points of a solution \(x(x_0,t_0;t)\) that is positively stable in the sense of Lagrange be a minimal \((E_{x_0,t_0})\) set of almost periodic solutions, it is sufficient that the semitrajectory \(x(x_0,t_0;I^+)\) uniformly approximate the set \(\Omega_{x_0,t_0}\) and that the solution \(x(x_0,t_0;t)\) possess property \(L_1\).
Here by a solution is meant not only a true solution, but also a limiting solution.
Proof. By Theorem 2, the set \(\Omega_{x_0,t_0}\) is minimal \((E_{x_0,t_0})\). We shall prove the almost periodicity of the trajectories contained in it. We shall use the result from [3] (Theorem 2, [3]). The trajectory of the solution \(x(x_0,t_0;t)\), or of a limiting solution constructed on the positive arcs of the solution \(x(x_0,t_0;t)\), and contained entirely in \(\Omega_{x_0,t_0}\), is recurrent if the solution \(x(x_0,t_0;t)\) is positively stable in the sense of Lagrange and its set \(\Omega_{x_0,t_0}\) of \(\omega\)-limit points is minimal \((E_{x_0,t_0})\).
Consider such a limiting solution \(\bar x(q,t_q;t)\). It possesses property \(L_2\). For arbitrary \(\varepsilon>0\) and \(T<0\) we determine, from property \(L_2\), the num-
therefore \(\delta\left(\dfrac{\varepsilon}{2}\right)>0\). From the recurrence of \(\bar{x}(q,t_q;t)\) there follows the existence of a relatively dense set of numbers \(\{\tau\}\) satisfying the inequality
\[ \rho_{R_n}\,[\bar{x}(q,t_q;t'),\ \bar{x}(q,t_q;t'+\tau)]<\frac{\delta}{2}. \]
Indeed, there exists \(T\left(\dfrac{\delta}{2}\right)\) such that any arc of time length \(T\left(\dfrac{\delta}{2}\right)\) of the limiting solution \(\bar{x}(q,t_q;t)\) approximates its entire trajectory with accuracy up to \(\delta/2\). We shall show that every \(\tau\) is an \(\varepsilon\)-almost period of the limiting solution \(\bar{x}(q,t_q;t)\). From property \(L_2\) of the limiting solution \(\bar{x}(q,t_q;t)\), for \(\tau\) and \(\delta/2>0\) we determine \(\sigma>0\) such that, if
\[ \rho_{R_n}\,[\bar{x}(q,t_q;t'),\ \bar{x}(q,t_q;t_1)]<\sigma, \]
then
\[ \rho_{R_n}\,[\bar{x}(q,t_q;t'+\tau),\ \bar{x}(q,t_q;t_1+\tau)]<\frac{\delta}{2}. \]
Let \(t\) be any number. By the property of a recurrent solution, there exists \(t_2<t\) such that
\[ \rho_{R_n}\,[\bar{x}(q,t_q;t'),\ \bar{x}(q,t_q;t_2)]<\min(\sigma,\delta). \tag{6} \]
Then, by the definition of \(\sigma\),
\[ \rho_{R_n}\,[\bar{x}(q,t_q;t'+\tau),\ \bar{x}(q,t_q;t_2+\tau)]<\frac{\delta}{2}. \]
Hence, and from the definition of \(\tau\),
\[ \rho_{R_n}\,[\bar{x}(q,t_q;t'),\ \bar{x}(q,t_q;t_2+\tau)]<\delta. \tag{7} \]
Again, using property \(L_2\) of the limiting solution, we have \((t-t_2>0)\), from inequalities (6) and (7):
\[ \rho_{R_n}\,[\bar{x}(q,t_q;t'+t-t_2),\ \bar{x}(q,t_q;t)]<\frac{\varepsilon}{2}, \]
\[ \rho_{R_n}\,[\bar{x}(q,t_q;t'+t-t_2),\ \bar{x}(q,t_q;t+\tau)]<\frac{\varepsilon}{2}. \]
Applying the triangle inequality, we obtain
\[ \rho_{R_n}\,[\bar{x}(q,t_q;t),\ \bar{x}(q,t_q;t+\tau)]<\varepsilon. \]
This theorem, notwithstanding a certain difference between Definition 5 and Definition 3 in [3], can be justified by the results of [3] (Theorem 3 [3]).
In conclusion, I express my gratitude to V. M. Millionshchikov for posing the problem and for his guidance.
References
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Nemytskii V. V., Stepanov V. V. Qualitative Theory of Differential Equations. M., GITTL, 1949.
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Bourbaki N. “Topologie générale.” Fascicule de résultats, 1949, ch. X, § 4, théorème 1. Paris. Hermann, éditeurs; 6, rue de la Sorbonne, 6.
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Millionshchikov V. M. Dokl. Akad. Nauk SSSR, 161, No. 1, 43, 1965.
Received by the editors
December 23, 1965
Moscow State University
named after M. V. Lomonosov