Floquet Theory for Quaternion-Valued Differential Equations

Abstract

This paper describes the Floquet theory for quaternion-valued differential equations (QDEs). The Floquet normal form of fundamental matrix for linear QDEs with periodic coefficients is presented and the stability of quaternionic periodic systems is accordingly studied. As an important application of Floquet theory, we give a discussion on the stability of quaternion-valued Hill's equation. Examples are presented to illustrate the proposed results. theorem of algebra, Vieta's formulas of quaternions, it is difficult to solve QDEs. In [13,14,15,16], the authors proposed several new methods to construct the fundamental matrices of linear QDEs.As a generalization, QDEs have many properties similar to ODEs. At the same time, for the relatively complicated algebraic structure of quaternion, one may encounter various new difficulties when studying QDEs.1. Factorization theorem and Vieta's formulas (relations between the roots and the coefficients) for quaternionic polynomials are not valid (see e. g. [17,18,19]). 2.A quaternion matrix usually has infinite number of eigenvalues. Besides, the set of all eigenvectors corresponding to a non-real eigenvalue is not a module (see e. g. [20,21]).3. The study of quaternion matrix equations is of intricacy (see e. g. [22,23]).4. Even the quaternionic polynomials are not "regular" (an analogue concept of holomorphic). This fact leads to noticeable difficulties for studying analytical properties of quaternion-valued functions (see e. g. [24,9]).Up to present, the theory of QDEs remains far from systemic. To the best of authors' knowledge, there was virtually nonexistent study about the stability theory of QDEs. Based on this fact, we are motivated to investigate the stability of the linear QDEṡwhere A is a smooth n × n quaternion-matrix-valued function. In particular, we will focus on the important special cases where A is a quaternionic constant or periodic quaternion-valued function. In the real-valued systems, the well-known Floquet theory indicates that the case where A is a periodic matrix-valued function is reducible to the constant case (see e. g. [25,26]). Floquet theory is an effective tool for analyzing the periodic solutions and the stability of dynamic systems. Owing to its importance, Floquet theory has been extended in different directions. Johnson [27] generalized the Floquet theory to the almost-periodic systems. In [28,29,30], the authors extended the Floquet theory to the partial differential equations. Recently, the Floquet theory has been extensively explored for dynamic systems on time scales (see e. g. [31,32,33,34]).As a continuation of [13,14,15], we generalize the Floquet theory to QDEs in this paper. Specifically, the contributions of this paper are summarized as follows.1. We show that the stability of constant coefficient homogeneous linear QDEs is determined by the standard eigenvalues of its coefficient matrix.2. Floquet normal form of the fundamental matrix for linear QDEs with periodic coefficients is presented.3. The monodromy matrix, characteristic...

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