Lorenz system is dissapative. This means:
- Volume in phase space contracts with flow?
This introduces some questions... How do volumes evolve?
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Suppose a surface $S(t)$ encloses volume $V(t)$, with normal vectors pointing away from the surface ($\vec{n}$). 

A trajectory starts on S. let them evolve for $dt$. With a flux vector $\vec{f}$, we have 
- $\vec f \cdot \vec n$ - normal, outward component of velocity
In $dt$ time, $dA$ sweeps out a volume. 

Volume: $(\vec f \cdot \vec n dt)dA$
$$V(t+dt) = V(t) + \int_S (\vec f \cdot \vec n dt)dA $$
$$\dot{V} = \int_S (\vec f \cdot \vec n)dA $$
Now we can apply the divergence theorem:
$$\dot{V} = \int_V (\nabla \cdot \vec f )dV $$
If you start with a solid blob of initial conditions, this integral will evaluate down to where things end up. If $\dot V$ is negative, then the system will converge to a stable subspace.
Limiting set will consist of
- fixed points
- limit cycles
- strong attractors
Proving which type something will end up on is much harder. But, repellers will always result in a positive $\dot V$.