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On the Sequences Realizing Perron and Lyapunov Exponents of Discrete Linear Time-Varying Systems.

1. Introduction

Consider a discrete time-varying system:

x(n + 1) = A(n) x (n), n [greater than or equal to] 0, (1)

where A = [(A(n)).sub.n[member of]N] is a bounded sequence of invertible s-by-s real matrices such that ([A.sup.-1][(n)).sub.n[member of]N] is bounded. For the coefficient matrices, denote the transition matrices

[PHI] (m, n) = A (m -1) ... A (n) for m > n (2)

and [PHI](n, n) = I, where I is the identity matrix. For an initial condition [x.sub.0] [member of] [R.sub.s], the solution of (1) is denoted by [(x(n,x (n, [x.sub.0]).sub.n[member of]N]; that is,

x (n, [x.sub.0]) = [PHI] (n, 0) [x.sub.0]. (3)

If a = [(a(n)).sub.n[member of]N] is a sequence of real numbers, then the Perron exponent n(a) and the Lyapunov exponent [lambda](a) of a are defined in the following ways:

[mathematical expression not reproducible]. (4)

By [parallel] x [parallel], denote the Euclidean norm in [R.sub.s] and the induced operator norm. For an initial condition [x.sub.0] [member of] [R.sub.s], the Perron [pi]([x.sub.0]) and the Lyapunov [lambda]([x.sub.0]) exponents of the solution [(x(n, [x.sub.0])).sub.n[member of]N] of system (1) are defined (see [1]) as

[mathematical expression not reproducible]. (5)

It means that the Perron and Lyapunov exponents of the solution [(x(n, [x.sub.0])).sub.n[member of]N] are the Perron and Lyapunov exponents of the sequence [([parallel]x(n, [x.sub.0])[parallel]).sub.n[member of]N], respectively

To characterize many properties of system (1) characteristics exponents, for example the Lyapunov, Perron, Bohl, general exponents maybe used. These quantities describe the different types of stability and trajectories growth rate. For interesting summary on main properties of the Lyapunov, Perron, Bohl, general exponents of the discrete time-varying linear system, and relations between these exponents and different types of stability of the considered system see [2].

The Lyapunov [3-15] and the Perron [16-23] exponents are one of the most commonly used numerical characteristics of dynamical systems. They describe, inter alia, such important properties like stability (exponential and Poisson). Numerical calculation of them is related to two main problems. The first one is that they are very sensitive to inaccuracies in the coefficients (they are not even continuous functions of the coefficients; see [24-29]). The second problem is that these quantities are defined by the partial limits (upper and lower one), and we do not know in advance what time sequence they are achieved on; therefore, an a priori one would need to look into all increasing sequences of natural numbers [30, 31].

This paper is linked to the second problem. In the paper, we try to describe a smaller class of all growing sequences of natural numbers with the property that the Lyapunov or Perron exponents are achieved on one of the sequences in this class.

The paper is organized in the following way: in the next paragraph, we establish certain properties of partial limits of real sequences, in particular their Perron and Lyapunov exponents. In the third section, containing the main results of the work, the theorems from the second section are applied to obtain properties of the Perron and Lyapunov exponents of the solutions of system (1). The work ends with a paragraph containing conclusions and suggestions for further research.

2. Preliminaries

Denote by S the set of all sequences of positive real numbers a = [(an).sub.n[member of]N] such that there exist constants [c.sub.1], [c.sub.2] (in general depending on the sequence a) such that

[c.sub.1] [less than or equal to] a(n + 1)/a(n) [less than or equal to] [c.sub.2], n [member of] N. (6)

By C we denote the set of all increasing sequences of natural numbers. If b = [([b.sub.n]).sub.n[member of]N] is any sequence of real numbers and m = [([m.sub.l]).sub.l[member of]N] [member of] C are such that there exists a finite limit

[mathematical expression not reproducible], (7)

then the number [beta] we will be called a partial limit or limit point of sequence b and we will say that it is achieved on the sequence m.

The next theorem shows that each number between the Perron and Lyapunov exponents of the sequence a = [([a.sub.n]).sub.n[member of]N] [member of]N] [member of] S is a partial limit of the sequence ((1/n) ln a[(n)).sub.n[member of]N].

Theorem 1. For each sequence a [member of] S and each number [alpha] [member of] [[pi](a), [lambda](a)] there exists sequence [([m.sub.l]).sub.l[member of] N] [member of] C such that

[mathematical expression not reproducible]. (8)

Proof. If [alpha] = [pi](a) or [alpha] = [lambda](a) then the conclusion follows from the properties of the upper and lower limits. Suppose that [alpha] [member of] ([pi](a), [lambda](a)). Let us define sequence [([n.sub.l]).sub.i[member of]N] in the following way:

[mathematical expression not reproducible]. (9)

By the inequality [pi](a) < [alpha] < [lambda](a) and by the definition of the upper and lower limits, it follows that the definition of [([n.sub.l]).sub.l[member of]N] is correct, the sequence [([n.sub.l]).sub.l[member of]N] is increasing, and

[mathematical expression not reproducible]. (10)

From the above two inequalities we get

[mathematical expression not reproducible], (11)

[mathematical expression not reproducible]. (12)

Since a [member of] S, therefore there exists a constant [c.sub.2] [member of] R, [c.sub.2] > 0 such that

a([n.sub.2l+1]) < [c.sub.2] a ([n.sub.2l+1] -1), 1 = 1,2, .... (13)

By the last two inequalities, we obtain

a([n.sub.2l+1]) < [c.sub.2] exp (a([n.sub.2l+1] - l)), l = 1,2, ... (14)

[mathematical expression not reproducible]. (15)

Passing to the upper limit and taking into account that [lim.sub.l[right arrow] [n.sub.2l+1] = [infinity], we have

[mathematical expression not reproducible]. (16)

Inequalities (11) and (16) imply that

[mathematical expression not reproducible]. (17)

It means that the sequence [m.sub.l] = [n.sub.2l+1], l = 1,2, ... is that one from the theorem's thesis. The proof is completed.

It is easy to construct an example showing that the theorem is no longer true without the assumption that a [member of] S.

Example 2. Let us define sequence a = [(a(n)).sub.n[member of]N] in the following way:

[mathematical expression not reproducible]. (18)

It is clear that a [not member of] S. Moreover,

[mathematical expression not reproducible]. (19)

Therefore, each convergent subsequence of the sequence ((1 /n) ln a[(n)).sub.n[member of]N] may have as a limit only 1 or -1.

Theorem 1 maybe generalized as follows.

Theorem 3. If [([T.sub.n]).sub.n[member of]N] [member of] C is such that

[mathematical expression not reproducible], (20)

then for each sequence a [member of] S and each number [alpha] [member of] [[pi](a),[lambda](a)] there exists sequence [([n.sub.l]).sub.l[member of]N] [member of] C such that

[mathematical expression not reproducible]. (21)

Proof. This theorem may be obtained from the general fact from the functional analysis as it was shown in [32, Lemma 7.5]. Repeating the construction from the proof of Theorem 1, we obtain instead of inequality (15) the following one:

[mathematical expression not reproducible]. (22)

From this inequality and by assumption (20) the thesis follows. The proof is completed.

Let us now introduce certain relation in the set C (Definition 4). It will appear to be an equivalence relation (Theorem 5) and if two sequences belong to the same equivalence class, then corresponding to them subsequences of [(a(n)).sub.n[member of]N] have the same exponents (Theorem 6).

Definition 4. We say that the sequence m = [([m.sub.l]).sub.i[member of]N] [member of] C is close to the sequence n = [([n.sub.l]).sub.l[member of]N] [member of] C if

[mathematical expression not reproducible]. (23)

This fact will be denoted in the following way m ~ n.

Theorem 5. The relation ~ is an equivalence relation in the set C.

Proof. Reflexivity of the relation ~ is obvious. Suppose that m ~ n. For a k [member of] N, denote by l(k) any natural number satisfying the condition

[mathematical expression not reproducible]. (24)

We will show that the set {l(k) : k [member of] N} is infinite. On the contrary, suppose that it is finite and denote its elements by [l.sub.1], ..., [l.sub.p]. Then, there exists infinite set A [subset] N such that

l(k) = [l.sub.i] (25)

for all k [member of] A and certain i = 1, ..., p. For k [member of] A, denote

[mathematical expression not reproducible]. (26)

Then,

[mathematical expression not reproducible]. (27)

The last two equalities are in contradiction with the facts that A is infinite and m tends to infinity. Now we show symmetry of the relation Suppose that m ~ n but the sequence n is not close to the sequence m. Denote

[mathematical expression not reproducible]. (28)

The fact that the sequence n is not close to the sequence m implies that [alpha]' > 0. Let us fix [alpha] [member of] (0, [alpha]'), [alpha] < 1. By the definition of upper limit we know that there exists sequence [(p(k)).sub.k[member of]N] [member of] C such that

[mathematical expression not reproducible]. (29)

It means that

[absolute value of 1 - [m.sub.l]/[n.sub.p(k)]] > [alpha] (30)

or equivalently that

1 - [m.sub.l]/[n.sub.p(k)] > [alpha]

Or 1 - [m.sub.l]/[n.sub.p(k)] < - [alpha] (31)

for all l, k [member of] N. For the fixed I [member of] N, the second inequality may hold only for finite many k [member of] N (in the opposite case, after passing to the limit with k [right arrow] [infinity] we obtain -1 > [alpha]). Moreover, if for certain l,k [member of] N the first inequality holds, then

1 - [m.sub.l]/[n.sub.q] > [alpha] (32)

for all q [member of] N, q [greater than or equal to] p(k). Therefore, for all I [member of] N there exists q(l) [member of] N such that

1 - [m.sub.l]/[n.sub.k] > [alpha] (33)

for all k [member of] N, k [greater than or equal to] q(l). By the definition of the relation the fact that m ~ n and the definition of the limit it follows that there exists [k.sub.1] [member of] N such that

min {[absolute value of 1 - [n.sub.l]/[m.sub.k]] : l [member of] N} < [alpha] (34)

for all k [member of] N, k > [k.sub.1]. The last inequality implies that

[alpha]/1 + [alpha] > 1 - [m.sub.k]/[n.sub.l(k)] (35)

for all k > [k.sub.1]. Finally, notice that [alpha] > [alpha]/(1 + [alpha]). It means that the inequalities (33) and (35) are in contradiction. Therefore, the relation ~ is in fact symmetric.

Now we show the transitivity of the relation ~. Suppose that we have three sequences m,n,p [member of] C such that n ~ m and m ~ p. From the fact n ~ m, we conclude that

[mathematical expression not reproducible], (36)

where [l.sub.1] (k) is any natural number satisfying the condition

[mathematical expression not reproducible]. (37)

Let us fix an arbitrary [xi] [member of] (0,1). By the definition of the limit and the equality (36), it follows that there exists [k.sub.1] [member of] N such that

[mathematical expression not reproducible] (38)

for all k [member of] N, k [greater than or equal to] [k.sub.1]. The last inequality implies that

[mathematical expression not reproducible]. (39)

From the fact that m ~ p, it follows that

[mathematical expression not reproducible]. (40)

To obtain the last equality nondecreaseness of the sequence [([l.sub.1](k)).sub.k[member of]N] is necessary. If this is not the case, then we can choose the nondecreasing subsequence from [([l.sub.1](k)).sub.k[member of]N] and the further reasoning lead for it. Denote by [l.sub.2](k) any natural number such that

[mathematical expression not reproducible]. (41)

Applying the introduced notation and the definition of the limit, we conclude that there exists [k.sub.2] [member of] N such that

[mathematical expression not reproducible] (42)

for all k [member of] N, k [greater than or equal to] [k.sub.2]. The last inequality implies that

[mathematical expression not reproducible]. (43)

From inequalities (39) and (43), we get

[mathematical expression not reproducible] (44)

for k [member of] N, k [greater than or equal to] max{[k.sub.1], [k.sub.sub.2]}; that is,

[mathematical expression not reproducible]. (45)

Due to arbitrariness of selection of [epsilon] [member of] (0,1), the last inequality means that

[mathematical expression not reproducible]. (46)

However, since

[mathematical expression not reproducible], (47)

then

[mathematical expression not reproducible], (48)

that is, n ~ p. The proof of transitivity of the relation ~ is finished.

Theorem 6. If [(a(n)).sub.n[member of]N] [member of] S, [([m.sub.l]).sub.l[member of]N], [([n.sub.l]).sub.l [member of] N] [member of], and [([m.sub.l]).sub.l[member of]N] ~ [([n.sub.l]).sub.l[member of]N] and there exists the limit

[mathematical expression not reproducible], (49)

then there exists the limit

[mathematical expression not reproducible] (50)

and the limits are equal.

Proof. Since [(a(n)).sub.n[member of]N] [member of] S, then there exists a constant c such that

a(n)/a(m) < [c.sup.[absolute value of n-m]]. (51)

For k [member of] N, denote by l(k) any natural number satisfying the condition

[absolute value of [m.sub.k] - [n.sub.l(k)]] = min {[absolute value of [m.sub.k] - [n.sub.l]] : l [member of] N}. (52)

We have

[mathematical expression not reproducible]. (53)

To obtain the last inequality, we used inequality (51). Since

[mathematical expression not reproducible] (54)

and there exists the limit

[mathematical expression not reproducible], (55)

then inequality (53) implies the thesis of the theorem. The proof is completed.

Denote by [x] the greatest integer no greater than x. The next theorem describes [lambda](a) and [pi](a) by the partial limits of a which correspond to time subsequences of the form [([[[theta].sup.n]]).sub.n[member of]N], where [theta] > 1, [theta] [member of] R.

Theorem 7. For any sequence [(a(n)).sub.n[member of]N] [member of] S, the following equalities hold:

[mathematical expression not reproducible], (56)

[mathematical expression not reproducible]. (57)

Proof. Let [([n.sub.l]).sub.l[member of]N] [member of] C be such that

[mathematical expression not reproducible]. (58)

Without loss of generality, for further consideration, we may assume that, for fixed [theta] > 1 in each interval [[[[theta].sup.n]], [[[theta].sup.n+1]]), n [member of] N, there are no more than one element of the sequence [([n.sub.l]).sub.l[member of]N]. For l [member of] N, denote by m(l) [member of] N such a number that

[n.sub.l] [member of][[[[theta].sup.m(l)]], [[[theta].sup.m(l)+1]]). (59)

Additionally, denote by f: (1, [infinity]) [right arrow] R a function given by

[mathematical expression not reproducible]. (60)

Since [(a(n)).sub.n[member of]N] [member of] S, then there exists a constant c, such that

a(n)/a(m) < [c.sup.(n-m)] for n,m [member of] N, n [greater than or equal to] m. (61)

In particular, taking [mathematical expression not reproducible], we get

[mathematical expression not reproducible]. (62)

Using this inequality, the introduced notation, and the definition of upper limit, we have

[mathematical expression not reproducible]. (63)

Denoting

[r.sub.[theta]] (l) = [[[theta].sup.m(l)]]/[n.sub.l] (64)

we get

[mathematical expression not reproducible], (65)

where

[mathematical expression not reproducible]. (66)

From inequalities (63) and (65), we have

f([theta]) [less than or equal to] [lambda](a) [less than or equal to] (1 - [theta]) ln c + [r.sub.[theta]] f([theta]). (67)

Passing in the last inequality to upper limit with [theta] [right arrow] [1.sup.+] and taking into account that

[mathematical expression not reproducible] (68)

we get

[mathematical expression not reproducible]. (69a)

Analogically, passing to the lower limit with [theta] [right arrow] [1.sup.+], we obtain

[mathematical expression not reproducible]. (69b)

Equality (56) follows from the equalities (69a) and (69b). In the same way, one can prove (57). The proof is completed.

3. Main Results

Consider a solution [(x(n, [x.sub.0])).sub.n[member of]N] of system (1) and denote by c a common bound for the sequences [([parallel][A.sup. -1](n)[parallel]).sub.n[member of]N] and [([parallel]A(n)[parallel]).sub.n[member of]N]. We have

[mathematical expression not reproducible]. (70)

The two above inequalities show that [(x(n, [x.sub.0])).sub.n[member of]N] [member of] S.

Applying Theorems 1, 3, and 6 to the sequence [([parallel]x(n, [x.sub.0])[parallel]).sub.n[member of]N], we get the following result.

Theorem 8. The set of limit points of the sequence ((1/n) ln [parallel]x[(n, [x.sub.0])[parallel]).sub.n[member of]N] is the interval [[pi]([x.sub.0]), [lambda]([x.sub.0])]. If the sequence [([T.sub.n]).sub.n[member of] N] [member of] C satisfies assumption (20), then for any number [alpha] [member of] there exists a sequence [([n.sub.l]).sub.l[member of]N] [member of] C such that

[mathematical expression not reproducible]. (71)

Moreover, if m = [([m.sub.l]).sub.l[member of]N] [member of] C and n ~ m, then

[mathematical expression not reproducible]. (72)

Notice that each arithmetic sequence satisfies condition (20). Then, taking in the above theorem [alpha] = n([x.sub.0]) or [alpha] = [lambda]([x.sub.0]) we conclude that the Lyapunov and Perron exponents are achieved at a certain subsequence of any arithmetic sequence. We do not know whether the analogous statement is true for geometric sequences. But, applying Theorem 7 to the sequence [([parallel]x(n, [x.sub.0])[parallel]).sub.n[member of]N], we may formulate the following result.

Theorem 9. For any solution [(x(n,[x.sub.0])).sub.n[member of]N] of system (1), we have

[mathematical expression not reproducible]. (73)

4. Conclusions

In the paper for the discrete time-varying linear system, we described the limits points of the sequence [((1/n) ln [parallel]x(n, [x.sub.0])[parallel]).sub.n[member of]N]. This set is equal to the interval [n([x.sub.0]), [lambda]([x.sub.0])]. Moreover, we proved that each partial limit of this sequence is achievable on a certain subsequence of any sequence satisfying condition (20), in particular on certain subsequence of any arithmetic sequence (Theorem 8). Finally, we showed that the Perron and Lyapunov exponents may be approximated by subsequences in certain sense similar to geometric sequences (Theorem 9). The objective of future works will be the investigation of the possibility of omitting limits with [theta] [right arrow] [1.sup.+] in equalities (73).

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research presented here was done as part of the project funded by the National Science Centre in Poland granted according to decision DEC-2012/07/N/ST7/03236. Moreover, the Article Processing Charge was covered by funds of the Silesian University of Technology Rector's Habilitation Grant no. 02/010/RGH16/0048.

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http://dx.doi.org/10.1155/2016/1487824

Michal Niezabitowski

Faculty of Automatic Control, Electronics and Computer Science, Silesian University of Technology, Akademicka 16 Street, 44-100 Gliwice, Poland

Correspondence should be addressed to Michal Niezabitowski; michal.niezabitowski@polsl.pl

Received 29 July 2016; Accepted 12 October 2016

Academic Editor: Sotiris K. Ntouyas
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