Floquet theory based on new periodicity concept for hybrid systems involving q -difference equations

Adıvar, Murat; Koyuncuoğlu, Halis Can

doi:10.1016/j.amc.2015.08.124

Cited by 9 publications

(4 citation statements)

References 21 publications

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“…DaCunha [27] extend the Floquet theory to a more general case of an arbitrary periodic time scale which unifies discrete, continuous, and hybrid periodic cases. Adivar and Koyuncuoglu [28] constructs a unified Floquet theory for homogeneous and nonhomogeneous hybrid periodic systems on domains having continuous, discrete or hybrid structure using the new periodicity concept based on shifts. It is known that Floquet multipliers (characteristic multipliers) play great role in the Floquet theory, and Floquet multipliers determine the stability of the periodic equation.…”

Section: Introduction 1historymentioning

confidence: 99%

Floquet multipliers and the stability of periodic linear differential equations: a unified algorithm and its computer realization

Wu¹,

Xia²,

Zhao³

2022

Preprint

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Motivated by the work on the unified Floquet theory on time scales (DaCunha and Davis [1] J. Differential Equations, 2011), in this paper, we provide a unified algorithm to determine the stability of the second order periodic linear differential equations on periodic time scales. Our approach is based on calculating the value of A and B (see Theorem 3.1), which are the sum and product of all Floquet multipliers (characteristic multipliers) of the system, respectively. Furthermore, a Matlab program is designed to realize our algorithm. We obtain an explicit expression of A (see Theorem 4.1) by the method of variation and approximation theory and an explicit expression of B by Liouville's formula. In particular, on an arbitrary discrete periodic time scale, we can do a finite number of calculations to get the explicit value of A (see Theorem 4.2). Finally, in Section 6, several examples are presented to illustrate the effectiveness of our algorithm.

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Section: Introduction 1historymentioning

confidence: 99%

Floquet multipliers and the stability of periodic linear differential equations: a unified algorithm and its computer realization

Wu¹,

Xia²,

Zhao³

2022

Preprint

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“…Since the theory of quantum calculus has important applications in quantum theory (see Kac and Cheung [1]), it has received much attention. For example, since Bohner and Chieochan [2] introduced the concept of periodicity for functions defined on the quantum time scale, quite a few authors have devoted themselves to the study of periodicity for dynamic equations on the quantum time scale ( [3,4,5,6]).…”

Section: Introductionmentioning

confidence: 99%

“…For example, since Bohner and Chieochan [2] introduced the concept of the periodicity for functions defined on the quantum time scale, quite a few authors have devoted themselves to the study of the periodicity for dynamic equations on the quantum time scale [3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

Almost Automorphic Functions on the Quantum Time Scale and Applications

2017

Discrete Dynamics in Nature and Society

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We first propose two types of concepts of almost automorphic functions on the quantum time scale. Secondly, we study some basic properties of almost automorphic functions on the quantum time scale. Then, we introduce a transformation between functions defined on the quantum time scale and functions defined on the set of generalized integer numbers; by using this transformation we give equivalent definitions of almost automorphic functions on the quantum time scale; following the idea of the transformation, we also give a concept of almost automorphic functions on more general time scales that can unify the concepts of almost automorphic functions on almost periodic time scales and on the quantum time scale. Finally, as an application of our results, we establish the existence of almost automorphic solutions of linear and semilinear dynamic equations on the quantum time scale.

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“…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.…”

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confidence: 99%

Floquet Theory for Quaternion-Valued Differential Equations

Dong

Kou

Xia

2020

Qual. Theory Dyn. Syst.

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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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Floquet theory based on new periodicity concept for hybrid systems involving q -difference equations

Cited by 9 publications

References 21 publications

Floquet multipliers and the stability of periodic linear differential equations: a unified algorithm and its computer realization

Floquet multipliers and the stability of periodic linear differential equations: a unified algorithm and its computer realization

Almost Automorphic Functions on the Quantum Time Scale and Applications

Floquet Theory for Quaternion-Valued Differential Equations

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