四元数体上微分方程理论已经在微分方程定性与稳定性研究中发挥着重要的作用，并以其丰富的理论思想和复杂的数学技巧应用到数学的各个研究领域之中，本书总结国内外知名学者的研究成果下，作者根据几年来在这方面的研究总结，把一些最新的研究进展和新成果介绍给广大读者，希望读者能进一步了解它。目前国际上没有一本关于四元数体上微分方程的著作。本书内容翔实，适合高等院校数学方向的教师、研究生或相关研究领域的科研人员阅读参考。

Professor Yong-Hui Xia（夏永辉）, Distinguished Professor in Zhejiang Normal University, received his Ph.D. degree in applied mathematics in 2009 from Shanghai Normal University, Shanghai, China. From Mar. 2004 to Sept. 2009, he was with Fuzhou University. From Sept. 2009 to May 2016, he was with Zhejiang Normal University. He was a Min-Jiang-Distinguished Professor in Huaqiao University from Apr. 2016 to May 2018. In Jun. 2018, he rejoined Zhejiang Normal University as a Distinguished Professor, Jinhua, China. From Oct. 2007 to Jan. 2008, he was a Visiting Scholar in the School of Information Systems, York University, Toroto, Canada. From-Jul. 2012 to Aug. 2013, he was a Research Fellow at CAMTP, Maribor Univesity, Slovenia. From 16 Jan. 2015 to 15 Feb. 2015, he was as a visiting Professor in Macau University. From 1 Jul. 2015 to 31 Dec. 2016, he was visiting Macau University as research fellow.

Preface
Athors’biography
Chapter 1 Background of Quaternion and Quaternion-valued Differential Equations
1.1 Background for quaternions
1.2 Background for QDEs
1.2.1 Quaternion Frenet frames in differential geometry
1.2.2 QDEs appears in kinematic modelling and attitude dynamics
1.2.3 QDE appears in fluid mechanics
1.2.4 QDE appears in quantum mechanics
1.3 History and motivation of our research
Chapter 2 Preliminary Concepts and Notations
2.1 Quaternion algebra
2.2 Biquaternion algebra
2.3 Definitions of determinants
2.4 Groups， rings， modules
2.5 Existence and uniqueness of solution to QDEs
Chapter 3 Basic Theory of Linear Homogeneous Quaternion-valued Differential Equations
3.1 Structure of general solutions for 2D QDEs
3.2 Structure of general solutions for any finite dimensional QDEs based on permutation
3.3 Fundamental matrix and solution to QDEs
3.4 Algorithm for computing fundamental matrix
3.4.1 Method 1: using expansion of exp{At}
3.4.2 Method 2: eigenvalue and eigenvector theory
Chapter 4 Algorithm for Linear Homogeneous QDEs when Linear Homogeneous System Has Multiple Eigenvalues
4.1 Motivations
4.2 Solving linear homogenous QDEs when linear homogeneous system has multiple eigenvalues
4.2.1 Multiple eigenvalues with enough eigenvectors
4.2.2 Multiple eigenvalues with fewer eigenvectors
Chapter 5 Floquet Theory of Quaternion-valued Differential Equations
5.1 Preliminary results
5.2 Stability of linear homogeneous QDEs with constant coefficients
5.3 Floquet theory for QDEs
5.4 Quaternion-valued Hill’s equations
Chapter 6 Solve Linear Nonhomogeneous Quaternion-valued Differential Equations
6.1 Notations
6.2 Main results
6.3 Some examples
Chapter 7 Linear Quaternion Dynamic Equations on Time Scale
7.1 Notations and preliminary results
7.1.1 Notations and lemmas
7.1.2 Calculus on time scales
7.2 First order linear QDETS
7.3 Linear systems of QDETS
7.4 Linear QDETS with constant coefficients
Chapter 8 Laplace Transform: a New Approach in Solving Linear Quaternion Differential Equations
8.1 Introduction
8.2 Biquaternion algebra
8.2.1 Biquaternion exponential function
8.2.2 Fundamental theorem of quaternion algebra and factorization theorem revisited
8.3 Definition and properties of the Laplace transform in biquaternion domain
8.4 Using QLT to solve QDEs
Chapter 9 Solving Quaternion Differential Equations with Two-sided Coefficients
9.1 Introduction
9.2 Notations and preliminary results
9.3 Solving QDEs with unilateral coefficients
9.4 Solving QDEs with two-sided coefficients
9.4.1 Homogeneous linear QDEs with two-sided coefficients
9.4.2 Nonhomogeneous linear QDEs with two-sided coefficients
Chapter 10 Controllability and Observability of Linear Quaternionvalued Systems
10.1 Motivations
10.2 Notations and preliminary results
10.3 Main results on the controllability and observability of linear QVS
10.3.1 Controllability
10.3.2 Observability
10.3.3 Duality
Chapter 11 Stability Analysis of Quaternion-valued Neural Networks
11.1 Notations and preliminary results
11.2 Main results
11.3 Examples
Chapter 12 Convex Function Optimization Problems with Quaternion Variables
12.1 Notations and preliminary results
12.1.1 Quaternion algebra analysis
12.1.2 Generalized gradient
12.2 Main results on the convex function optimization problems with quaternion variables
12.3 Examples and simulations
12.4 Proof of the Proposition 12.1.4
Chapter 13 Penalty Method for Constrained Distributed Quaternionvariable Optimization
13.1 Introduction
13.2 Preliminaries
13.3 Main results
13.4 An example
Bibliography