If one had a system:
\begin{align} \dot{x} = f(x,y,z)\\ \dot{y} = g(x,y,z)\\ \dot{z}=h(x,y,z) \end{align} Where each function may have parameters. How would one know if the system is dissipative?
I believe fundamental to defining a dissipative system is the Lyapunov function. This function seems have non equilibrium solutions crossing lower and lower level sets of the Lyapunov function. Is this correct?
There are several notions of a dissipative system, one being the existence of a Lyapunov function with some appropriate properties (that may also vary).
Because of this, but also because you don't say anything else about the particular right-hand side, it is really to much to ask "How would one know if the system is dissipative?". In fact, it is complicated in general even if the right-hand side is known explicitly.
On the other hand, the description that you give in your last paragraph is fair.