Name of invariance for ODE

If I have a first order ODE $\dot{x} = f(x)$ with $x = (x_1, x_2, \cdots,x_n) \in \mathbb{R}^n$.

What is the property called that if $x$ is a solution then a solution vector given by a permutation of the components of $x$ is also a solution?

Solution 1:

What you are describing is an example of an equivariant dynamical system. Your phase space is equipped with an action of group $S_n$ — a group of all permutations of a set with $n$ elements. It acts on points of phase space by interchanging coordinates: permutation $\sigma \in S_n$ maps point $x = (x_1, x_2, \dots, x_n)$ to point $\sigma(x) = (x_{\sigma(1)}, \; x_{\sigma(2)},\; \dots, \; x_{\sigma(n)})$. (I admit that I slightly abuse notation here) In general permutations don't play nicely with solutions of system $\dot{x} = f(x)$. However, if for any solution $\gamma(t)$ of system and for any permutation $\sigma$ a vector function $\sigma(\gamma(t))$ is also a solution, then system is equivariant with respect to this group action.