Constructing on ODE given a set of fixed points

Question

Is it possible to construct an $N$ dimensional ODE $\dot{x}=g(x)$, where $g:R^n \to R^n$, given a finite set of desirable fixed points $X^*$?

Answer 1

Very interesting question!

Assume that you have no requirement about the stability of the fixed point (e.g. it could be an attractor, repellor, etc.) Then you could simply do this

1. construct a $n$ dimen function that has 0 value at the given fixed points. For example, a product of distance to all the fixed points forms a landscape that has local minima at each fixed point. $$ h(x)=\prod_i\|x-x^*_i\|^2 $$ Then the gradient of this landscape form an $n$ dimensional function that has $0$ value at the fixed point. $$ \nabla h(x)=0,\forall x\in X^* $$

Then the flow according to the gradient of $h(x)$ has fixed points at $X^*$. $$ \dot x = \nabla h(x) $$

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Edited according to the suggestion of @Lutz! Thanks twice!