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

Lorenz system is dissapative. This means:

• Volume in phase space contracts with flow? 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}).

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.