We are considering an autonomous equation
$$\dot x=f(x),$$
where $f: \mathbb{R}^n\rightarrow\mathbb{R}^n$ is a $C^1$ vectorfield. Let $\phi^t$ be the corresponding flow. Is it true that
$$D\phi^t(x)f(x)=f(\phi^t(x)),$$
why? $D$ stands for the Jacobian matrix of $\phi^t(x)$.
This is a question I encountered while I was reading researching articles:
Coomes, B. A., Koçak, H., Palmer, K. J. (1995). A shadowing theorem for ordinary differential equations. Zeitschrift für angewandte Mathematik und Physik ZAMP, 46(1), 85-106.
K. J. Palmer (1996) Shadowing and Silnikov Chaos. Nonlinear Analysis, Theory, Methods & Applications, 27(9), 1075-1093.
Palmer, K. J. (2008). Transversal periodic-to-periodic homoclinic orbits. Handbook of Differential Equations: Ordinary Differential Equations, 4, 365-439.
The authors simply state this is a fact on Page 96, 1077, 382 of the articles above, but without any derivation process. So I think this might come from some fundamental theory in dynamical systems. So I calculate like this:
$$D\phi^t(x(s))f(x(s))=D\phi^t(x(s))\dot x(s)=\frac{d}{ds}[\phi^t(x(s))]=f(\phi^t(x)).$$
I'm not sure if this is correct or in a standard way. Any help will be deeply appreciated.
Best Answer
This is a combination of the chain rule and the group property of the flow. Consider the $s$-derivative at $s=0$ of $$ϕ^{t+s}(x)=ϕ^t(ϕ^s(x)).$$