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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: Lorenz system is dissapative. This means:
- Volume in phase space contracts with flow? - Volume in phase space contracts with flow?
This introduces some questions... How do volumes evolve? This introduces some questions... How do volumes evolve?
n
Suppose a surface $S(t)$ encloses volume $V(t)$, with normal vectors pointing away from the surface ($\vec{n}$). 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 A trajectory starts on S. let them evolve for $dt$. With a flux vector $\vec{f}$, we have
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- fixed points - fixed points
- limit cycles - limit cycles
- strong attractors - strong attractors
Proving which type something will end up on Proving which type something will end up on is much harder. But, repellers will always result in a positive $\dot V$.