Chapter 2 introduces functional differential equations (FDEs), including delay differential equations (DDEs), neutral FDEs (NFDEs), and mixed FDEs (MFDEs). These equations describe systems where the current state change depends on the system's history or future. DDEs with state-dependent delay are also considered. Section 2.1 discusses infinite dynamical systems generated by time lags. In Newtonian mechanics, the system's state is governed by ordinary differential equations (ODEs). However, in many applications, the system's current state change depends on its history, leading to DDEs of the form: $$ \dot{x}(t)=f(x(t),x(t-\tau)), $$ where $ x(t) $ is the system's state, $ f $ is a given mapping, and $ \tau > 0 $ is a constant delay. This equation arises in population dynamics, where the population density depends on the maturation period. The solution requires specifying the system's history on $ [-\tau, 0] $, and the initial value problem is solvable under certain conditions. The solution process can be extended to larger time intervals, leading to a solution for the entire system. The space of continuous functions $ C_{n,\tau} $, equipped with a supremum norm, serves as the state space for DDEs. This space forms an infinite-dimensional dynamical system, with the semiflow $ [0,\infty] \ni t \mapsto x_{t} \in C_{n,\tau} $. Studying the asymptotic behavior of solutions to DDEs is challenging due to the infinite-dimensionality of the phase space and the generated semiflow, even for scalar equations.Chapter 2 introduces functional differential equations (FDEs), including delay differential equations (DDEs), neutral FDEs (NFDEs), and mixed FDEs (MFDEs). These equations describe systems where the current state change depends on the system's history or future. DDEs with state-dependent delay are also considered. Section 2.1 discusses infinite dynamical systems generated by time lags. In Newtonian mechanics, the system's state is governed by ordinary differential equations (ODEs). However, in many applications, the system's current state change depends on its history, leading to DDEs of the form: $$ \dot{x}(t)=f(x(t),x(t-\tau)), $$ where $ x(t) $ is the system's state, $ f $ is a given mapping, and $ \tau > 0 $ is a constant delay. This equation arises in population dynamics, where the population density depends on the maturation period. The solution requires specifying the system's history on $ [-\tau, 0] $, and the initial value problem is solvable under certain conditions. The solution process can be extended to larger time intervals, leading to a solution for the entire system. The space of continuous functions $ C_{n,\tau} $, equipped with a supremum norm, serves as the state space for DDEs. This space forms an infinite-dimensional dynamical system, with the semiflow $ [0,\infty] \ni t \mapsto x_{t} \in C_{n,\tau} $. Studying the asymptotic behavior of solutions to DDEs is challenging due to the infinite-dimensionality of the phase space and the generated semiflow, even for scalar equations.