# Given a system of functions, find a system of differential equations which describe that system

Inspired by modelling phenomena in biology, I'm wondering whether there has been mathematical study on the following question:

Given some $$\mathbf{X}(t) \in \mathbb{R}^n$$, find $$f = f(X_1, \dots, X_n)$$, where $$f$$ is time invariant so that $$X'(t) = f(X_1, \dots, X_n)$$

To explain, has there been an exposition into the general problem of finding a time invariant differential equation which "best" describes the time evolution of a given system?

• I added the "differential-equations" tag to your post Cheers! – Robert Lewis Dec 1 '18 at 0:42
• Not always possible even for 1 dimension, for example $x(t) = \sin(t)$ has $x(0)=x(\pi)=0$ but $x'(0) \neq x'(\pi)$. So there is no function $f$ for which $x'(t)=f(x(t))$. – Michael Dec 1 '18 at 5:00
• @Michael But one can extend the state space to also include $x_2=x'$. – Kwin van der Veen Dec 1 '18 at 5:33
• So you are asking what kind of research has been done on continuous nonlinear time-invariant system identification? – Kwin van der Veen Dec 1 '18 at 5:37
• I suppose. Do we have any tools to "solve" for the nonlinear time-invariant system? Even in any special cases? – libby Dec 1 '18 at 18:52