Abstract
Full Text
UDC 517.917
MATHEMATICS
V. K. DUBOLYAR
DYNAMIC SYSTEMS WITH DELAY
(Presented by Academician I. G. Petrovskii, 23 IV 1968)
In the present note autonomous systems of differential equations with delayed argument are studied from the point of view of the general theory of dynamical systems (¹). This problem was posed in (²–⁴). Below, on the basis of the proposed axiomatization, the principal results in this direction are established, concerning invariant sets, minimal sets, and recurrent motions.
1°. In what follows, by the letter (\rho) we shall denote distance in metric spaces. Let (R) be a metric space; (T) the half-line ([0,+\infty)); (S) a compact set contained in (T); (\Phi) a metric space whose elements are continuous mappings of the compact set (S) into (R), with the metric of uniform convergence on (S); (f) a continuous mapping (\Phi \times T) into (R). The ordered triple ((\Phi,T,f)) will be called a dynamical system with delay if the following axioms are satisfied:
1) (f(\varphi,s)=\varphi(s)) for all (\varphi \in \Phi) and (s \in S);
2) whatever (\varphi \in \Phi) and (\tau \in T) may be, the function (\varphi^\tau), defined by the relation (\varphi^\tau(s)=f(\varphi,\tau+s)) for all (s \in S), belongs to (\Phi);
3) whatever (t \in T), (\tau \in T), and (\varphi \in \Phi) may be, the equality
[
f(\varphi,t+\tau)=f(\varphi^\tau,t).
]
The proposed axiomatization is quite general. Properties 1)—3) are possessed by solutions of the differential equation with delayed argument
[
x'(t)=g(x(t),x(t-\tau)), \qquad \text{where } 0<\tau=\text{const},
]
provided that the right-hand side of the equation ensures existence, uniqueness, continuous dependence on the initial function, and continuability of the solution on ([\tau,+\infty)). Properties 1)—3) are also possessed by solutions of the ordinary differential equation
[
x^{(n)}+p_1(t)x^{(n-1)}+\cdots+p_{n-1}(t)x'+p_n(t)x=0,
]
satisfying the boundary conditions (x(a_k)=A_k,\ a_k \geq 0,\ k=1,2,\ldots,n), provided the conditions of existence, uniqueness, continuous dependence on the boundary conditions, and continuability of solutions for all (t \geq 0) are fulfilled.
Finally, let us give an example of a dynamical system with delay defined in a space of continuous functions. Let (\Psi) be some set of continuous functions defined on the half-line (T) with values in the metric space (R); (S) a compact set contained in (T); (\Phi) the restrictions to (S) of functions of the set (\Psi). We shall regard the space (\Phi) as metric with the metric of uniform convergence on (S). Suppose that (\Psi) satisfies the following conditions: a) if (\psi(t)\in\Psi), then also (\psi(t+\tau)\in\Psi) for any (\tau>0); b) if a sequence ({\psi_n}) from (\Psi) converges uniformly on (S), then it converges uniformly on every segment contained in (T). Define the mapping (f:\Phi\times T\to R) by putting (f(\varphi,t)=\tilde{\varphi}(t)), where (\tilde{\varphi}) is the extension to (T) of the function (\varphi). We note that, by condition b), for each (\varphi\in\Phi) the extension (\tilde{\varphi}) is uniquely determined. Using the conditions
a) and b), it can be shown that the triple ((\Phi,T,f)) constructed above is a dynamical system with delay.
(2^\circ.) Let ((\Phi,T,f)) be an arbitrary dynamical system with delay. Define the mapping (\sigma:\Phi\times T\to\Phi) by setting (\sigma(\varphi,t)=\varphi^t). It is easy to verify that the mapping (\sigma) is continuous, (\sigma(\varphi,0)=\varphi), and (\sigma[\sigma(\varphi,t_1),t_2]=\sigma(\varphi,t_1+t_2)) for any (\varphi\in\Phi,\ t_1\ge0,\ t_2\ge0). Thus it becomes possible to pass from the given dynamical system with delay to the Birkhoff ((^1)) dynamical system (\sigma(\varphi,t)), whose phase space is the set of initial functions (\Phi). Such a passage is useful in a number of cases. The idea of passing from solutions of differential equations with delayed argument to trajectories in the space of initial functions is contained in ((^5)).
Theorem 1. For any (\varepsilon>0,\ l>0), and (\varphi\in\Phi), there exists a (\delta>0) such that if (\psi\in\Phi) and (\rho(\varphi,\psi)<\delta), then (\rho(f(\varphi,t),f(\psi,t))<\varepsilon) for all (0\le t\le l).
Theorem 1 is a generalization of the property of integral continuity known in the general theory of dynamical systems.
A set (F\subseteq\Phi) is called invariant in (\Phi) if, for any (\varphi\in F) and (t\ge0), the relation (\varphi^t\in F) holds. Using the system (\sigma(\varphi,t)) defined above, one can show that the union (intersection) of any collection of sets invariant in (\Phi), as well as the closure of any set invariant in (\Phi), is an invariant set in (\Phi).
The function (f(\varphi,t)), for fixed (\varphi\in\Phi), will be called a motion, and the set of points ({f(\varphi,t):t\ge0}) the trajectory of this motion. The trajectory of the motion (f(\varphi,t)) will be denoted by the symbol (f(\varphi,T)). Note that every finite arc of the trajectory (f(\varphi,T)), as the continuous image of some segment from (T), is compact.
A set (M\subseteq R) is called invariant in (R) if there exists a set (\Psi\subseteq\Phi) such that
[
M=\bigcup_{\psi\in\Psi} f(\psi,T),
]
i.e., if (M) consists of whole trajectories.
Theorem 2. In order that a set (M\subseteq R) be invariant in (R), it is necessary and sufficient that there exist a set (F), invariant in (\Phi), such that (M=f(F,0)).
We outline the proof. If the set (M) is invariant in (R), then
[
M=\bigcup f(\psi,T),
]
where (\Psi) is some set from (\Phi). The set
[
{\psi^t:\psi\in\Psi,\ t\ge0}
]
is invariant in (\Phi). In this case (M=f(F,0)). Conversely, if (M=f(F,0)), where (F) is a set invariant in (\Phi), then
[
M=\bigcup_{\varphi\in F} f(\varphi,T),
]
i.e., (M) is invariant in (R).
From the equality
[
\bigcup f(F_\alpha,t)=f\left(\bigcup F_\alpha,t\right),
]
valid for any (t\in T) and (F_\alpha\subseteq\Phi), and from Theorem 2, it follows that the union of any collection of sets invariant in (R) is an invariant set in (R). However, there are examples showing that the intersection of invariant sets in (R) need not be an invariant set in (R). One may also give an example of a set invariant in (R) whose closure no longer has this property. Nevertheless, the following assertion is valid: if (M\subseteq R) and (F\subseteq\Phi) is a set such that
[
\bigcup_{\varphi\in F} f(\varphi,T)\subseteq M,
]
then
[
\bigcup_{\varphi\in \overline{F}} f(\varphi,T)\subseteq \overline{M}.
]
In particular, if (M) is invariant in (R), i.e. if
[
M=\bigcup_{\varphi\in F} f(\varphi,T),
]
where (F) is some set from (\Phi), then
[
\bigcup_{\varphi\in \overline{F}} f(\varphi,T)\subseteq \overline{M}.
]
What was noted above shows that, in their properties, dynamical systems with delay differ substantially from Birkhoff dynamical systems.
A dynamical system with delay ((\Phi,T,f)) will be called regular if the following conditions are satisfied: 1) the compact (S) consists-
contains zero; 2) from every sequence of initial functions ({\varphi_n}) from (\Phi), for which the corresponding sequence of points ({\varphi_n(0)}) of the space (R) converges, one can extract a subsequence converging in (\Phi). In this definition condition 2) is essential. It is satisfied, for example, if the space (\Phi) is compact or when (\Phi) consists of constant functions. In particular, the regularity conditions are satisfied if ((\Phi, T, f)) is a Birkhoff dynamical system (i.e., (\Phi) consists of constant functions, and the compact set (S) consists of a single point, coinciding with zero).
Theorem 3. Suppose that the dynamical system ((\Phi, T, f)) is regular. Then the closure of every invariant set in (R) is invariant.
We give the scheme of the proof. If (M) is invariant in (R), then on the basis of Theorem 2 the equality (M = f(F, 0)) holds, where (F) is some invariant set in (\Phi). In this case (\overline{M} = f(\overline{F}, 0)), whence the invariance of the closure (\overline{M}) follows.
(3^\circ). A set (F \subseteq \Phi) is called minimal in (\Phi) if it is nonempty, closed, invariant, and contains no proper subset possessing these three properties.
Theorem 4. In order that a nonempty invariant set (F \subseteq \Phi) be minimal in (\Phi), it is necessary and sufficient that, for every function (\varphi \in F), the closure in (\Phi) of the set ({\varphi^t : t \geq 0}) coincide with (F).
Theorem 5. The following assertions hold: 1) every nonempty invariant closed compact set (F \subseteq \Phi) contains some minimal set in (\Phi); 2) if the space (\Phi) is compact, then it contains some minimal set in (\Phi).
The proof of Theorems 4 and 5 is carried out with the aid of the Birkhoff dynamical system (\sigma(\varphi, t)) constructed in (2^\circ). Its properties and its connection with the dynamical system with delay are used.
The motion (f(\varphi, t)) is called recurrent if for every (\varepsilon > 0) there exists (l > 0) such that, whatever (t \geq 0) may be, on every interval of length (l) there is a (\tau) for which
[
\rho(f(\varphi, t+s), f(\varphi, t+s+\tau)) < \varepsilon
]
for all (s \in S).
Theorem 6. If (\varphi) belongs to a compact and minimal set in (\Phi), then the motion (f(\varphi, t)) is recurrent.
Theorem 7. Suppose that the space (R) is complete. Then, if the motion (f(\varphi, t)) is recurrent, the closure in the space (\Phi) of the set ({\varphi^t : t \geq 0}) is a compact and minimal set in (\Phi).
The last two theorems are a generalization of the fundamental results of the general theory of dynamical systems known as Birkhoff’s theorems (see ((^1)), pp. 402–404).
In the study of dynamical systems with delay, along with sets minimal in (\Phi), it makes sense also to consider sets minimal in (R). In the case of Birkhoff systems these two notions coincide.
A set (M \subseteq R) is called minimal in (R) if it is nonempty, closed, invariant, and contains no proper subset possessing the same properties.
Theorem 8. Suppose that the dynamical system with delay ((\Phi, T, f)) is regular. Then, in order that a nonempty invariant set (M \subseteq R) be minimal in (R), it is necessary and sufficient that, for any trajectory (f(\varphi, T)) contained in (M), the equality
[
\overline{f(\varphi, T)} = M
]
hold.
The proof of this theorem is similar to the proof of the analogous assertion from ((^1)).
Theorem 9. Suppose that the dynamical system with delay ((\Phi, T, f)) is regular, and that the set (M \subseteq R) is compact and minimal
in (R). Then there is a function (\varphi \in \Phi) such that (\tilde f(\varphi, T)=M), and the motion (f(\varphi,t)) is recurrent.
The proof is based on Theorems 5, 6, and 8. We note that in Theorems 8 and 9 the regularity condition for the system ((\Phi, T, \tilde f)) is essential.
In conclusion I express my deep gratitude to B. A. Shcherbakov for his advice and constant attention to the present work.
Kishinev State
University
Received
19 IV 1968
REFERENCES
- V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, Moscow—Leningrad, 1949.
- A. Halanay, Mathematics Collection, 10, 5, 85 (1966).
- L. E. Elsgolts, Abstracts of Reports, All-Union Interuniversity Conference on the Theory and Applications of Differential Equations with Deviating Argument, Chernivtsi, 1965.
- L. E. Elsgolts, International Congress of Mathematicians, 1966, Abstracts of Short Scientific Communications, Section 6, p. 54.
- N. N. Krasovskii, Some Problems in the Theory of Stability of Motion, Moscow, 1959.