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300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-18 Volume Contraction.md

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+Lorenz system is dissapative. This means:
+- Volume in phase space contracts with flow?
+This introduces some questions... How do volumes evolve?
+
+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:
+$$\dot{V} = \int_V (\nabla \cdot \vec f )dV $$