1. Introduction
A new type of oscillation, called an unpredictable trajectory, was introduced in the paper [
1]. An unpredictable trajectory is necessarily positively Poisson stable, and one of its distinctive features is the emergence of chaos in the corresponding quasi-minimal set. The type of chaos based on the presence of an unpredictable trajectory is called Poincaré chaos [
1]. Symbolic dynamics, logistic map, and Hénon map are some examples that possess unpredictable motions [
1,
2]. Instead of interaction of several motions, which is a requirement in other chaos types [
3,
4,
5], it is enough to check the existence of a single unpredictable motion to verify Poincaré chaos. Thus, the connection of unpredictable oscillations with Poincaré chaos is based on Poisson stable motions. The basics of the Poisson stable motion can be seen in [
6]. In the theory of differential equations, the class of Poisson stable solutions has been intensively studied [
7,
8,
9,
10].
The theoretical basics of the present research lies in the theory of dynamical systems, which was founded H. Poincaré and G. Birkhoff [
6,
11]. It was the French genius, who learned that underneath of chaotic dynamics is the Poisson stable motion. The line of different types of oscillations was maintained by the notion of an unpredictable point in paper [
1]. Let
be a metric space and
, where
is the set of non-negative real numbers, be a semiflow on
X. A point
and the trajectory through it are unpredictable if there exists a positive number
and sequences
both of which diverge to infinity, such that
and
for each
. In paper [
1], it was proved that a Poisson stable motion equipped with the unpredictability property determines sensitivity in the quasi-minimal set. It is usual that dynamical properties generate functional ones. For example, recurrent or almost periodic motions allow us to define recurrent and almost periodic functions, respectively. Similarly, issuing from the unpredictable point introduced in [
1], in paper [
12] the unpredictable function was defined as a point of the Bebutov dynamics, where the state space is a set of functions. In the present research we replace the metrical state space of the dynamics with the B-topology [
13], where a convergence of piecewise continuous functions on compact subsets of the real axis is utilized. This makes our suggestions more universal and effective in applications, since constructive verifiable conditions are proposed. In this study it is applied for linear impulsive systems.
Unpredictable points and unpredictable functions are becoming increasingly applicable in the study of chaos theory. For instance, some topological properties of Poincaré chaos were considered by Miller [
14], and Thakur and Das [
15]. Various types of differential equations with unpredictable solutions were considered in the papers [
2,
12,
16,
17,
18] and in the book [
19]. Recently it is proved in paper [
20] that unpredictable motions take place also in random processes.
Alongside the theory of chaos, impulsive differential equations have also played an essential role from both theoretical and practical points of view over the past few decades [
13,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30]. Such differential equations describe dynamics of real world phenomena in which abrupt interruptions of continuous processes are present, and they play a crucial role in various fields such as mechanics, electronics, medicine, neural networks, communication systems, and population dynamics [
31,
32,
33,
34,
35,
36,
37,
38]. Chaos in the sense of Li-Yorke and the presence of period-doubling route to chaos in impulsive systems were investigated in the studies [
39,
40] by means of the replication of chaos technique.
In the present study, we show that unpredictable perturbations can lead to the presence of discontinuous unpredictable motions in the dynamics of linear impulsive systems. In the absence of the perturbation term, the linear system under investigation has a simple dynamics such that it admits an asymptotically stable regular solution. However, our results reveal that the presence of an unpredictable term dominates the behavior of the resulting dynamics and a discontinuous unpredictable solution occurs. In the present study, we also investigate the case of the presence of unpredictability in the impulsive moments. Due to the nature of impulsive systems, the concept of B-topology [
13] is required for the theoretical investigation of discontinuous unpredictable solutions.
We have already obtained the essential results for ordinary linear and quasilinear systems, as well as discrete equations [
17,
18]. One of the main contributions of the present study is the presentation of the novel definitions of discontinuous unpredictable function and unpredictable discrete set. Another main contribution is the demonstration of the existence and uniqueness of unpredictable solutions for linear impulsive systems. Systems with both non-unpredictable and unpredictable impulsive actions are under investigation.
Benefiting from the techniques introduced in [
2,
12,
16,
19] and results on the theory of impulsive differential equations [
13,
21], the existence, uniqueness, and stability of discontinuous unpredictable solutions of linear impulsive systems are investigated in this study. To construct an unpredictable function, an unpredictable sequence resulting from a randomly defined discrete Bernoulli process [
20] is utilized.
2. Preliminaries
Throughout the paper,
,
, and
respectively stand for the sets of real numbers, natural numbers, and integers. Moreover, we make use of the usual Euclidean norm for vectors and the spectral norm for square matrices [
41].
It is known that solutions of an impulsive system are piecewise continuous functions, which have discontinuities of the first kind. A difficulty of investigation of impulsive system is that, in general, discontinuity moments of distinct solutions do not coincide. For that reason the concept of
B-topology is required in our investigations [
13].
Denote by the set of all functions defined on the real axis. They are continuous except countable sets of points. At the points the, functions admit one-sided limits. The sets of points do not necessarily coincide, if functions are different. The sets of points do not have finite accumulation points and are unbounded on both sides.
Two functions and from , are said to be -equivalent on an interval if the points of discontinuity of the functions and in J can be respectively numerated and , , such that for each , and for each , except those between and for each i. In the case that F and G are -equivalent on J, we also say that the functions are in -neighborhoods of each other. The topology defined with the aid of such neighborhoods is called the B-topology.
In what follows we will denote by , the interval , if and the interval , if .
Let , , be a sequence of real numbers such that for some positive numbers , , and as .
Definition 1. A piecewise continuous and bounded function with the set of discontinuity points , , satisfying for each is called discontinuous unpredictable function (d.u.f.) if there exist positive numbers , σ, sequences of real numbers and sequences of integers all of which diverge to infinity such that
- (a)
as on each bounded interval of integers and for each natural number n;
- (b)
for every positive number ϵ, there exists a positive number δ such that whenever the points and belong to the same interval of continuity and ;
- (c)
as in B-topology on each bounded interval;
- (d)
for each natural number n there exists an interval which does not contain any point of discontinuity of and , and for each .
In Definition 1, we call the property conditional uniform continuity of , the property Poisson stability of , and the property unpredictability of .
Definition 2. Suppose that is a piecewise continuous and bounded function with the set of discontinuity points , , satisfying and , , is a bounded sequence in . The couple is called unpredictable if there exist positive numbers , σ, sequences of real numbers and sequences of integers all of which diverge to infinity such that
- (a)
as on each bounded interval of integers and for each natural number n;
- (b)
for every positive number ϵ there exists a positive number δ such that whenever the points and belong to the same interval of continuity and ;
- (c)
as in B-topology on each bounded interval;
- (d)
for each natural number n there exists an interval which does not contain any point of discontinuity of and , and for each ;
- (e)
as for each i in bounded intervals of integers and for each natural number n.
Similarly to Definition 1, in Definition 2 we call the property conditional uniform continuity of , the property Poisson stability of , and the property unpredictability of .
The sequence , is said to be an unpredictable discrete set if the condition is satisfied.
It is clear that if the couple is unpredictable in the sense of Definition 2, then is a discontinuous unpredictable function in the sense of Definition 1.
Obviously, Definition 1 does not follow from the Definition 2, since one cannot obtain the former just by diminishing the terms . The sequence of zeros is not an unpredictable sequence. Consequently, both definitions are needed in the paper.
The definition of the unpredictable sequence is as follows.
Definition 3 ([
16])
. A bounded sequence , in is called unpredictable if there exist a positive number and sequences , , , of positive integers both of which diverge to infinity such that as for each i in bounded intervals of integers and for each . According to the purpose of the present study, we specify the discontinuity moments of the impulsive systems that will be investigated as
where
,
, is a sequence of real numbers which is unpredictable in the sense of Definition 3 and
is a number such that
for some number
.
Since , is an unpredictable sequence, there exists a positive number and sequences , , both of which diverge to infinity such that as for each i in bounded intervals of integers and for each natural number
Let us show that the sequence
,
, is unpredictable discrete set. More precisely, we will demonstrate that property
mentioned in Definition 1 is valid for
,
, with
,
, and
for each natural number
n. By these choices of the sequences
, and
, we have that
Therefore,
as
for each
i in bounded intervals of integers. On the other hand,
for each natural number
Additionally, one can confirm that
,
, defined by (
1) satisfies the inequality
with
and
.
3. Linear Systems with Non-Unpredictable Impulses
The main object of the present section linear impulsive system,
where
, the matrices
and
commute, the sequence
,
, of discontinuity moments is defined by Equation (
1), and
is a d.u.f. in the sense of Definition 1. We suppose that
, where
I is the
identity matrix.
Let us denote by
the Cauchy matrix of the following linear impulsive system associated with (
2),
Since the matrices
A and
B commute, we have for
that
where
denotes the number of the terms of the sequence
,
, which belong to the interval
, and
[
21].
Let us denote by , , the eigenvalues of the matrix .
The following condition on the system (
2) is required.
, where is the real part of for each .
In consequence of (
4), under the condition
, there exist numbers
and
such that
for
[
13,
21].
Let us prove the following auxiliary assertion.
Lemma 1. Assume that the condition is fulfilled, then the following inequalityholds, where . Proof. By using (
4) and (
5), we can show that
for
. □
The following theorem is concerned with the discontinuous unpredictable solution of system (
2).
Theorem 1. Suppose that the condition is valid. If is a d.u.f. in the sense of Definition 1, then system (2) possesses a unique asymptotically stable discontinuous unpredictable solution. Proof. As it is known from the theory of impulsive differential equations [
13,
21], according to the boundedness of the function
, system (
2) admits a unique solution
which is bounded on the real axis and satisfies the equation
One can verify for points of continuity that
where
. Therefore,
is a conditional uniform continuous function. The asymptotic stability of
can be verified in a very similar way to the stability of a bounded solution mentioned in [
13].
Since is a d.u.f., there exist positive numbers , , sequences of real numbers and sequences of integers all of which diverges to infinity such that the properties and in Definition 1 hold for , i.e., when is replaced by f.
Let us continue with the verification of the Poisson stability of , i.e., property (c) in Definition 1.
Fix an arbitrary positive number
and an arbitrary compact interval
, where
. We will show for sufficiently large
n that the inequality
is satisfied for each
t in
Choose numbers
and
such that
and
Let
n be a sufficiently large natural number such that
for
,
, and
for
. We assume without loss of generality that
. Additionally, suppose that
for
.
If
, then we have
Using the inequality (
6), one can obtain
In a similar way to the last inequality, one can deduce that
The inequalities (
9)–(
11) imply that
for
, and therefore,
uniformly on each compact interval in
B-topology.
Next, we will show the existence of a sequence , which diverges to infinity, and positive numbers , such that for .
Corresponding to the Definition 1 for the d.u.f. , the interval , does not admit any points of discontinuity of and . For that reason, the following discussion ignores the presence of a discontinuity moment.
There exists a positive number
such that the inequalities
and
are valid for every
and natural number
n.
Let us fix an arbitrary natural number
n, and suppose that
, where each
,
, is a real valued function. One can confirm that there exists an integer
,
, such that
Therefore, the inequality
is valid for
.
There exist numbers
in the interval
such that
It can be verified by utilizing the inequality (
12) that
Additionally, using the equation
we deduce that
Hence, the inequality
holds.
Suppose that
for some number
. Let us denote
and
Let us choose
such that
Then, for
, we have that
Thus,
for each
t from the intervals
,
. It is clear that the sequence
diverges to infinity. Consequently,
is the unique discontinuous unpredictable solution of system (
2). □
4. Linear Systems with Unpredictable Impulses
Consider the following linear impulsive system,
where
, the matrices
and
commute, the sequence
,
, of discontinuity moments is defined by Equation (
1), and
is an unpredictable couple in the sense of Definition 2. Additionally,
, where
I is the
identity matrix.
It is worth noting that (
14) is a linear impulsive system with unpredictable impulses, and it is not a particular case of system (
2). Indeed, to introduce the perturbations
in the impulsive part, one must not only consider the sequence to be unpredictable but also assume that the sequences
and
proper for the unperturbed system have to be consistent with the new terms.
In the proof of the following theorem, we will again use the notations and as mentioned in the proof of Theorem 1. Moreover, we set .
Theorem 2. Suppose that the condition is valid. If the couple is unpredictable in the sense of Definition 2, then system (14) possesses a unique asymptotically stable discontinuous unpredictable solution. Proof. According to the results of [
13,
21], system (
14) possesses a unique solution
which is bounded on the whole real axis. Moreover,
satisfies the equation
for
. If
t is not one of the discontinuity moments
, then
The last inequality yields the conditional uniform continuity of
. Moreover, the asymptotic stability of
can be verified using the results mentioned in the books [
13,
21].
The asymptotic stability of the solution
can be verified as stability of a bounded solution in [
13].
Since the couple is unpredictable, there exist positive numbers , , sequences of real numbers and sequences of integers all of which diverges to infinity such that the properties , , and in Definition 2 hold for the function and the sequence , i.e., when and are respectively replaced by f and .
Next, we focus on the Poisson stability of . Fix an arbitrary positive number and a compact interval with . We will show that for sufficiently large n the inequality holds for each
Let us fix numbers
and
satisfying the inequalities
and
One can check that if
n is a sufficiently large natural number, then the equation
is valid.
Fix a sufficiently large natural number
n such that
,
whenever
and
for
. Assume without loss of generality that
. For all
, we have
Using the inequality (
6), it can be verified that
On the other hand, one can obtain that
Additionally, the inequality
holds for
. Furthermore, we have
Thus,
for
in accordance with the inequalities (
16)–(
20). Therefore,
uniformly in
B-topology on each compact interval.
The unpredictability of solution
can be proved identically to that for solution
of system (
2). □