As we know (1) the flow $\phi(t,\bf{x})$ of a vector field $\bf{f}$ is the general solution of the autonomous ODE $\dot{\bf{x}} = \bf{f(x)}$.
Also from linear systems we know system’s state trajectory is the general solution of the linear autonomous ODE $\dot{\bf{x}} = \bf{Ax}$, given by
$$\bf{x}(t) = \phi(t, t_0)\bf{x}(0) $$
where $\phi(t, s)$ is the state transition matrix from s to t
So my question is what is the the relation between flow and state transition matrix?
Best Answer
Note that your "state transition matrix" is exactly the flow of the general system, with $f(x)=Ax.$ The flow of the linear system is the map $e^{tA}:\mathbb{R}^n\rightarrow\mathbb{R}^n$. Let's write $\varphi_t=e^{tA}.$ Note that this family of maps has the properties $\varphi_0(x)=x,$ $\varphi_s\circ\varphi_t(x)=\varphi(s+t)(x)$ for all $s,t\in\mathbb{R},$ and $\varphi_{-t}\circ\varphi_t(x)=\varphi_{t}\circ\varphi_{-t}(x)=x$ for any $t\in\mathbb{R}.$ This can be generalized as follows:
Let $E\subset\mathbb{R}^n$ is open, and $f\in C^1(E).$ If $x_0\in E,$ let $\varphi(t,x_0)$ be the solution to $$\begin{cases}\dot{x}=f(x)\\ x(0)=x_0\end{cases}$$ defined on the maximal interval of existence. Then, for $t$ in this interval, the mappings $\varphi_t$ defined by $\varphi_t(x_0)=\varphi(t,x_0)$ is called the flow of the differential equation given above.
You can verify that this satisfies all of the listed properties of the linear flow. Hence, this generalizes the concept of flow, and if $f$ is linear, then the definitions coincide. This is all a special case of a one-parameter semigroup.