Almost Periodic Skew-symmetric Differential Systems

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 This result is no longer valid for almost periodic systems.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.