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?

  • $\begingroup$ I added the "differential-equations" tag to your post Cheers! $\endgroup$ – Robert Lewis Dec 1 '18 at 0:42
  • $\begingroup$ 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 $x'(t)=f(x(t))$. $\endgroup$ – Michael Dec 1 '18 at 5:00
  • $\begingroup$ @Michael But one can extend the state space to also include $x_2=x'$. $\endgroup$ – Kwin van der Veen Dec 1 '18 at 5:33
  • $\begingroup$ So you are asking what kind of research has been done on continuous nonlinear time-invariant system identification? $\endgroup$ – Kwin van der Veen Dec 1 '18 at 5:37
  • $\begingroup$ I suppose. Do we have any tools to "solve" for the nonlinear time-invariant system? Even in any special cases? $\endgroup$ – libby Dec 1 '18 at 18:52

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