ALMOST PERIODIC SKEW-SYMMETRIC DIFFERENTIAL SYSTEMS

We analyse solutions of almost periodic skew-symmetric
homogeneous linear differential systems. We prove that in any
neighbourhood of such a system there exists an almost periodic
skew-symmetric system which does not possess any non-trivial
almost periodic solution.


Introduction
We study solutions of almost periodic linear differential systems.This field is called the Favard theory what is based on the famous Favard result in [10] (see, e.g., [3,Theorem 1.2] or [28,Theorem 1]).It is a well-known corollary of the Favard (and the Floquet) theory that any bounded solution of a periodic linear differential system is almost periodic (see [12,Corollary 6.5] and [13] for a generalization in the homogeneous case).This result is no longer valid for almost periodic systems.There exist systems whose all solutions are bounded and none of them is almost periodic (see [18,31]).Homogeneous systems have the zero solution which is almost periodic.But they do not need to have any non-zero almost periodic solution.The existence of a homogeneous system, which has bounded solutions (separated from zero) and, at the same time, all systems from some neighbourhood of it do not possess non-trivial almost periodic solutions, is proved in [33].
In this paper, we consider almost periodic skew-symmetric homogeneous linear differential systems.The basic motivation of our research is paper [38], where skew-Hermitian systems are analysed.The main result of [38] says that, in an arbitrary neighbourhood of a skew-Hermitian system, there exists another skew-Hermitian system which does not possess an almost periodic solution other than the trivial one (not only with a fundamental matrix which is not almost periodic-this problem is discussed in [34]).Our aim is to prove the corresponding result for real skew-symmetric systems.Note that the process from [38] cannot be applied in the real case.
We use a recurrent method for constructing almost periodic functions.For non-almost periodic solutions of homogeneous linear differential equations, we refer to [27] (and [26]), where a method of constructions of minimal cocycles, which one gets as solutions of recurrent homogeneous linear differential systems, is mentioned.Special constructions of almost periodic homogeneous linear differential systems with given properties can be found in [19,23,24] as well.A method to construct fundamental matrices for almost periodic homogeneous linear systems is introduced in [30].
The importance of skew-symmetric systems may be illustrated by the Cameron-Johnson theorem which states that any almost periodic homogeneous linear differential system can be reduced by a Lyapunov transformation to a skew-symmetric system if all solutions of the given system and all of its limit equations are bounded (see [4]).Further, it is known (see [32]) that the skew-symmetric systems, all of whose solutions are almost periodic, form a dense subset in the space of all skew-symmetric systems (special cases are considered in [20,21] and the corresponding result about unitary difference systems is mentioned in [36]).This fact also motivates the study of skew-symmetric systems without almost periodic solutions.
More precisely, it is proved in [32] that, in any neighbourhood of an almost periodic skew-symmetric system with frequency module F , there exists a system with a frequency module contained in the rational hull of F possessing all almost periodic solutions with frequencies belonging to the rational hull of F as well.From [35,Theorem 1] it follows that a neighbourhood of an almost periodic skew-symmetric system with frequency module F may not contain a system with almost periodic solutions and frequency module F .
In addition (see [34]), the systems with k-dimensional frequency basis, having solutions which are not almost periodic, form a subset of the second category in the space of all systems with k-dimensional frequency basis.Thus, it is known (see also [32, Corollary 1]) that the systems with k-dimensional frequency basis and with an almost periodic fundamental matrix form a dense subset of the first category in the space of all considered systems with k-dimensional frequency basis.For more details concerning the frequency modules and bases of almost periodic linear differential systems and their solutions, we refer to monograph [12,Chapters 4,6] or to articles [28,38].
Let us give a short literature overview about almost periodic solutions of almost periodic linear differential equations.Sufficient conditions for the existence of almost periodic solutions are mentioned in [5,9,17] (for generalizations and supplements, see [8,16,22]).Certain sufficient conditions, under which homogeneous systems that have non-trivial bounded solutions also have non-trivial almost periodic solutions, are given in [29].Concerning known basic results about skew-symmetric systems and their fundamental matrices, we refer to [2,11,25].For the general theory of almost periodicity in connection with differential equations, see [7].We add that the elements of the theory of almost periodicity can be found in many classical books, e.g., [1,6].EJQTDE, 2012 No. 72, p. 2

Preliminaries
Let m ∈ N \ {1} be arbitrarily given as the dimension of systems under consideration.Throughout this paper, we will use the following notations: Mat (R, m) for the set of all m × m matrices with real elements, SO(m) ⊂ Mat (R, m) for the set of all orthogonal matrices with determinant 1, so(m) ⊂ Mat (R, m) for the set of all skew-symmetric (i.e., antisymmetric) matrices, I ∈ SO(m) for the identity matrix, O ∈ so(m) for the zero matrix.We remark that the Lie algebra associated to the Lie group SO(m) consists of the skew-symmetric m × m matrices (i.e., this Lie algebra is so(m) and it is sometimes called the special orthogonal Lie algebra).
For the reader's convenience, we recall the definition of almost periodicity and basic properties of almost periodic functions which we will need later.Since we have to consider the almost periodicity of vector valued and, at the same time, matrix valued functions, we formulate the definition and the properties for functions with values in an arbitrary metric space X with a metric µ.Definition 1.A continuous function ψ : R → X is almost periodic if for any ε > 0, there exists a number l(ε) > 0 with the property that any interval of length l(ε) of the real line contains at least one point s satisfying Theorem 2. Let ψ : R → X be a continuous function.Then, ψ is almost periodic if and only if from any sequence of the form {ψ(t + s n )} n∈N , where s n are real numbers, one can extract a subsequence {ψ (t + r n )} n∈N satisfying the Cauchy uniform convergence condition on R; i.e., for any ε > 0, there exists n(ε) ∈ N with the property that Proof.See, e.g., [38,Theorem 2.4].
Let us consider systems of m homogeneous linear differential equations of the form where A : R → so(m) is an almost periodic function.Let S denote the set of all systems (1).We can identify the function A with the system (1) which is determined by A. Especially, we will write A ∈ S. Let X S = X S (t) denote the principal fundamental matrix of S ∈ S satisfying X S (0) = I.
In the vector space R m , we will use the Euclidean norm • 2 (one can also replace it by the absolute norm or the maximum norm).Let • be the corresponding matrix EJQTDE, 2012 No. 72, p. 3 norm in Mat (R, m) and let ̺ be the metric given by • .Considering that every almost periodic function is bounded (see directly the definition of almost periodicity), the distance between two systems A, B ∈ S is defined by the norm of the matrix valued functions A, B, uniformly on R; i.e., we introduce the metric For ε > 0, the symbol O σ ε (A) will denote the ε-neighbourhood of a system A in S and O ̺ ε (M) the ε-neighbourhood of a matrix M in a given subset of Mat (R, m).Now we can repeat the above mentioned result (see Introduction) in a more explicit form.

Results
To prove the announced new result, we need the following lemmas.
Lemma 1.There exist ξ > 0 and a neighbourhood Õ (O) of the zero matrix in so(m) for which the exponential map is a bijection between Õ (O) and are Lipschitz continuous.
We will also use a simple method for constructing almost periodic functions with prescribed values, which is formulated in the next lemma.Note that this lemma is a modification of [38,Theorem 3.1] and that the analogous way, one can generate almost periodic sequences with several given properties, can be found in [39].Lemma 3. If the sequence of non-negative numbers a(i) for i ∈ N has the property that then any continuous function ψ : R → so(m) for which EJQTDE, 2012 No. 72, p. 5 Proof.Let ε > 0 be arbitrarily given and let From Thus (see ( 4)), it is true This inequality implies that we can choose l(ε) := 2 2k(ε) +1 for any ε > 0 (see Definition 1); i.e., the resulting function ψ is almost periodic.
Now we can prove the result that the systems having no non-zero almost periodic solution form an everywhere dense subset of S. Proof.Using Theorem 1, the almost periodicity of A implies that there exist δ ∈ (0, 1/3) and an almost periodic matrix valued function Ã : R → so(m) satisfying Ã ∈ O σ ε/2 (A) and Ã| [k,k+δ] ≡ const.for any k ∈ Z. Indeed, it suffices to define Ã as follows where δ > 0 is sufficiently small.Thus, we will assume without loss of generality that Every almost periodic function is bounded.Hence, there exists η ∈ (0, 1) with the property that X S (t + s) − X S (t) < ξ (5) for any t ∈ R, s ∈ [0, η], and S ∈ O σ ε (A), where ξ > 0 is taken from Lemma 1.We can also assume that δ < η.Further (see again Lemma 1), there exists M ∈ N satisfying where p(ϑ) is taken from Lemma 2.
Since the sum of skew-symmetric matrices is a skew-symmetric matrix and since the sum of two almost periodic functions is almost periodic as well (see Theorem 2), we have EJQTDE, 2012 No. 72, p. 7 A 1 + A 2 ∈ S for any A 1 , A 2 ∈ S. Thus, it suffices to find C ∈ S ∩ O σ ε (O) for which the system A + C does not have any non-zero almost periodic solution.We will construct such a system C (as continuous function) applying Lemma 3 for Let us denote In the first step of the construction, we put 1}, and we define C so that it is linear on intervals In the second step, we put and we define C as linear on intervals At the same time, we define We proceed further in the same way.In the i-th step, we put and we define C as a linear function on intervals we have i.e., there exists ε ∈ (0, ε) with the property that C(t) < ε, t ∈ R. Thus, we obtain an almost periodic (continuous) function C ∈ S ∩ O σ ε (O).We denote In the construction, we can choose constant values C 1 i , . . ., C 2 2n(i) i on 2 2n(i) subintervals of I i , where the length of each one of these intervals is EJQTDE, 2012 No. 72, p. 9 Each value C j i can be chosen arbitrarily from the (ε/2 i )-neighbourhood of a skew-symmetric matrix, which is given by the previous steps of the construction.Further (see (8)), the function C is determined on intervals We repeat that C is linear on the remaining subintervals of I i .These intervals will be denoted by J 1 i , . . ., J 2 2n(i)+1 i , where Especially, we see that the length of each J j i is less than δ/2 2n(i) and that i.e., the total length l k i of all subintervals Let us consider To describe the principal fundamental matrix X S , we define Xi it is valid that (see also (10)) if t ≤ a i + k, k ∈ 1, . . ., 2 2n(i) .Considering S ∈ O σ ε (A) and X S (t) , Xi S (t) ∈ SO(m), t ∈ R, from (11) and ( 12) it follows that there exists N ∈ N satisfying We put X 1 := −I, X 2 = −I, when m is even, and EJQTDE, 2012 No. 72, p. 11 then it is seen that we can use Lemma 2 to choose values C j i on subintervals for which (see ( 5), ( 9) with M ∈ N satisfying (6) and with and the fact that we can choose all matrix C j i from the (ε/2 i )-neighbourhood of a given skew-symmetric matrix arbitrarily.Note that and a i + 4n(i)2 n(i) < b i for sufficiently large i ∈ N, i.e., we can construct the resulting function C with the above mentioned properties on I i for all i ≥ n 0 (see also (14)).Now we use ( 13) and ( 14) in connection with (15).For k ∈ 1, . . ., 4n(i)2 n(i) , where i ≥ n 0 , we have Especially, for all i ≥ n 0 (i ∈ N), we obtain EJQTDE, 2012 No. 72, p. 12 where We recall that we need to prove that any non-trivial solution of S is not almost periodic.By contradiction, suppose that the solution of the Cauchy problem for all i(1), i(2) from an infinite set N 0 ⊆ N.
The presented process can be applied to prove the existence of systems from S with several properties.For example, we mention the following result.Proof.Let a sequence {X k } k∈N ⊂ SO(m) be dense in SO(m).In the proof of Theorem 4, we can replace considered matrices X 1 , X 2 by arbitrary matrices X k , X k+1 .Thus, there is shown the existence of a system S = A + C ∈ O σ ε (A) with the property that (see ( 16)) X S s i j − X j < N • 4n(i) 2 n(i) 2 2n(i)−1 for some s i j ∈ R and all j ∈ {1, . . ., 2n(i)}, i ≥ n 0 .Now it suffices to consider that lim i→∞ N • 4n(i) 2 n(i) 2 2n(i)−1 = 0.
At the end, we remark that the question of generalizations of Theorem 4 concerning other homogeneous linear differential systems, which can have only almost periodic solutions, remains open (contrary to the corresponding discrete case, see [14,37]).

Theorem 1 .
An almost periodic function with values in X is uniformly continuous on the real line.Proof.The theorem can be easily proved by modifying the proof of [6, Theorem 6.2].

Theorem 4 .
Let A ∈ S and ε > 0 be arbitrary.There exists B ∈ O σ ε (A) which does not have an almost periodic solution other than the trivial one.

Theorem 5 .
Let A ∈ S and ε > 0 be arbitrarily given.There exists B ∈ O σ ε (A) with the property that {X B (t); t ∈ R} = SO(m).