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

@@ -13,3 +13,9 @@ $$V(t+dt) = V(t) + \int_S (\vec f \cdot \vec n dt)dA $$
 $$\dot{V} = \int_S (\vec f \cdot \vec n)dA $$
 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