第四章：周期轨迹和庞加莱回归映射

4.1 冲击响应下的自治系统(Autonomous Systems with Impulse Effects)

一个冲击响应下的自治系统由三个部分组成：

an autonomous ordinary differential equation, $\dot{x}(t) = f\left(x(t)\right)$, defined on some state space $\mathcal{X}$;
a hyper surface $\mathcal{S}$ at which solutions of the differential equation undergo a discrete transition that is modeled as an instantaneous reinitialization of the differential equation;
a rule $\Delta : \mathcal{S} \to \mathcal{X}$ that specifies the new initial condition as a function of the point at which the solution impacts $\mathcal{S}$.

这样的系统可以用如下的符号表示： $$ \begin{equation}\label{f4.1} \Sigma : \begin{cases} \dot{x}(t) = f\left(x(t)\right) & x^-(t) \notin \mathcal{S} \\ x^+(t) = \Delta\left(x^-(t)\right) & x^-(t) \in \mathcal{S} \end{cases} \end{equation} $$ $\mathcal{S}$被称为冲击平面(impact surface)，$\Delta$被称为冲击映射(impact map)。有时也称他们为switching surface和reset map。

公式($\ref{f4.1}$)的一个解的形式化描述为$\varphi(t)$ is developed on the basis of solutions to the associated ordinary differential equation: $$ \dot{x} = f(x) $$