Differential Equations on Time Scales
This chapter focuses on differential equations on time scales, including ordinary differential equations and functional differential equations. Section 7.1 briefly introduces the theory of time scales and then fundamental theory of ordinary differential equations on time scales is presented in Section 7.2. Functional differential equations on time scales are discussed in detail in Section 7.3. First, the equation model is formulated in Subsection 7.3.1, and then basic theory about existence and uniqueness of solutions are presented in Subsection 7.3.2. In Subsections 7.3.3 and 7.3.4, various stability results are discussed, such as (uniform) stability, uniform asymptotic stability, and exponential stability. To demonstrate these results, several examples are provided in Subsection 7.3.5 along with numerical simulations.