In this paper, we obtain some basic results of quaternion algorithms and quaternion calculus on time scales. Based on this, a Liouville formula and some related properties are derived for quaternion dynamic equations on time scales through conjugate transposed matrix algorithms. Moreover, we introduce the quaternion matrix exponential function by homogeneous quaternion matrix dynamic equations. Also a corresponding existence and uniqueness theorem is proved. In addition, the commutativity of quaternion‐matrix–valued functions is investigated and some sufficient and necessary conditions of commutativity and noncommutativity are established on time scales. Also the fundamental solution matrices of some basic quaternion matrix dynamic equations are obtained. Examples are provided to illustrate the results, which are completely new on hybrid domains particularly when the time scales are the quantum case and the discrete case ; , both of which are significant for the study of quaternion q‐dynamic equations and quaternion difference dynamic equations. Finally, we present several applications including multidimensional rotations and transformations of the submarine, the gyroscope, and the planet whose dynamical behaviors are depicted by quaternion dynamics on time scales and the corresponding iteration numerical solution for homogeneous quaternion dynamic equations are provided on various time scales.

中文翻译：

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四元数矩阵值函数和四元数矩阵动力学方程在时间尺度上的可交换性

本文在时间尺度上获得了四元数算法和四元数演算的一些基本结果。在此基础上，通过共轭转置矩阵算法，导出了四元数动力学方程的一个Liouville公式和一些相关性质。此外，我们通过齐次四元数矩阵动力学方程式介绍了四元数矩阵的指数函数。并证明了相应的存在唯一性定理。另外，研究了四元数矩阵值的函数，并在时标上建立了一些可交换性和非可交换性的充分必要条件。还获得了一些基本四元数矩阵动力学方程的基本解矩阵。提供了示例来说明结果，这些结果在混合域上是全新的，尤其是在时标是量子情况和离散情况时；，这对研究四元数q动力学方程和四元数差分动力学方程都是有意义的。最后，我们介绍了几种应用程序，包括潜艇，陀螺仪和行星的多维旋转和变换，其动力学行为在时标上用四元数动力学来描述，并且在各种时标上提供了均相四元数动力学方程的相应迭代数值解。