Index Theory for Dynamical Systems, Part 2: Poincaré-Hopf Index Theorem | You Can't Comb a Coconut

Index theory for compact manifolds like the sphere and torus puts a constraint on the type of vector fields allowed. For instance, the sum of the indices of fixed points on a sphere will always add up to +2, which means there must be at least one fixed point somewhere. This has been put as "You can't comb a coconut", "the hairy ball theorem", or the wind must vanish somewhere on the Earth. Some examples are given, such as Hamiltonian vector fields made by point vortices on the sphere, simulating the atmosphere.

By the way, you can comb a hairy donut, but do you want to?

► Next, limit cycles in dynamical systems

► Flows on the torus, with and without fixed points

► Introduction to index theory for 2D systems

► For background on 2D dynamical systems, see

► From 'Nonlinear Dynamics and Chaos' (online course).

► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)

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► Make your own phase portrait

► Course lecture notes (PDF)

Poincare Hopf compact continuously differentiable orientable manifold genus differential topology hairy doughnut Charles Conley index theory gradient system autonomous on the plane phase plane are introduced 2D ordinary differential equations 2d ODE vector field topology cylinder bifurcation robustness fragility cusp unfolding perturbations structural stability emergence critical point critical slowing down supercritical bifurcation subcritical bifurcations buckling beam model change of stability nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Point Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field Two-Dimensional 2-dimensional Functions Hamiltonian Hamilton streamlines weather vortex dynamics point vortices pendulum Newton's Second Law Conservation of Energy topology

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