21 lines
1.0 KiB
Markdown
21 lines
1.0 KiB
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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n
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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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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.
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Limiting set will consist of
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- fixed points
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- limit cycles
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- strong attractors
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Proving which type something will end up on is much harder. But, repellers will always result in a positive $\dot V$. |