16 lines
684 B
Markdown
16 lines
684 B
Markdown
Lorenz system is dissapative. This means:
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- Volume in phase space contracts with flow?
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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 ($\vec{n}$).
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A trajectory starts on S. let them evolve for $dt$. With a flux vector $\vec{f}$, we have
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- $\vec f \cdot \vec n$ - normal, outward component of velocity
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In $dt$ time, $dA$ sweeps out a volume.
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Volume: $(\vec f \cdot \vec n dt)dA$
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$$V(t+dt) = V(t) + \int_S (\vec f \cdot \vec n dt)dA $$
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$$\dot{V} = \int_S (\vec f \cdot \vec n)dA $$
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Now we can apply the divergence theorem:
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$$\dot{V} = \int_V (\nabla \cdot \vec f )dV $$
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