How are "cyclic" first-order linear differential equations called?

Question

I'm interested in first-order linear differential equations which form a cycle with the variables. An example would be $$\dot{x} = y \hspace{1cm} \dot{y} = z \hspace{1cm} \dot{z} = x$$ Does these types of differential equations have a name ? Because I'm interested in finding literature and in understanding if these kinds of ODE always diverge, or if there are solutions which converge and are stable, except for the trivial initial condition $x_0 = 0, y_0 = 0, z_0 = 0$. My intuition is that they always diverge, since even if the initial condition is set near zero, it will sooner or later build up to bigger and bigger derivatives.

Answer 1

If those equations hold then we must have $\dddot{x}=x$, by finding the roots of it’s characteristic equation you can find all solutions.