📂Dynamics

Slow-Fast Systems

Definition ¹

Slow-Fast System

Assuming an element of an open subset $W \subset \mathbb{R}^{m + n}$ of the Euclidean space is expressed as $\left( \mathbf{x}, \mathbf{y} \right)$. For $k \in \mathbb{N}$, two functions $\mathbf{f} : W \times [0,1] \to \mathbb{R}^{m}$ and $\mathbf{g} : W \times [0,1] \to \mathbb{R}^{n}$, in addition to $t = \xi \tau$, are given for $\xi \in (0,1)$ to satisfy the following $C^{k}$ class, $$\begin{align*} {{ d \mathbf{x} } \over { d \tau }} =& \mathbf{f} \left( \mathbf{x} , \mathbf{y} , \xi \right) \\ {{ d \mathbf{y} } \over { d \tau }} =& \xi \mathbf{g} \left( \mathbf{x} , \mathbf{y} , \xi \right) \end{align*}$$ is called the Fast System, $$\begin{align*} \xi {{ d \mathbf{x} } \over { d t }} =& \mathbf{f} \left( \mathbf{x} , \mathbf{y} , \xi \right) \\ {{ d \mathbf{y} } \over { d t }} =& \mathbf{g} \left( \mathbf{x} , \mathbf{y} , \xi \right) \end{align*}$$ is called the Slow System. Such a coupled dynamic system is called a $(m,n)$-Slow-Fast System.

Layer System

The system obtained by applying $\xi \to 0$ to the Fast System is called the Layer System. $$\begin{align*} {{ d \mathbf{x} } \over { d \tau }} =& \mathbf{f} \left( \mathbf{x} , \mathbf{y} , 0 \right) \\ {{ d \mathbf{y} } \over { d \tau }} =& 0 \end{align*}$$

Reduced System

The system obtained by applying $\xi \to 0$ to the Slow System is called the Reduced System. $$\begin{align*} 0 =& \mathbf{f} \left( \mathbf{x} , \mathbf{y} , 0 \right) \\ {{ d \mathbf{y} } \over { d t }} =& \mathbf{g} \left( \mathbf{x} , \mathbf{y} , 0 \right) \end{align*}$$

Description

As intuitively understood from its definition, the Slow-Fast System is a single system with a clear coupling, yet a dynamic system with two types of temporal flow.