Abstract
Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.
Highlights
In 1843, Hamilton initiated the concept of quaternions that extends the complex numbers to fourdimensional space [1]
The impulsive dynamic equations play a vital role in describing the natural phenomena with sudden changes; it is a hot topic of the research of impulsive dynamic systems since the instantaneous change caused by impulses has a very significant meaning in explaining and mastering the change rule of the object
There are no research results related to the Cauchy matrix and Liouville formula for the theory of quaternion impulsive dynamic equations on time scales, which will lead to many difficulties in studying the quaternion impulsive dynamic equations on complex hybrid domains
Summary
In 1843, Hamilton initiated the concept of quaternions that extends the complex numbers to fourdimensional space [1]. The impulsive dynamic equations play a vital role in describing the natural phenomena with sudden changes; it is a hot topic of the research of impulsive dynamic systems since the instantaneous change caused by impulses has a very significant meaning in explaining and mastering the change rule of the object Due to this reason, there have been many literature in this field [15,16,17,18]. There are no research results related to the Cauchy matrix and Liouville formula for the theory of quaternion impulsive dynamic equations on time scales, which will lead to many difficulties in studying the quaternion impulsive dynamic equations on complex hybrid domains Several concrete examples and counter examples are provided to analyze the feasibility of our obtained results
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