Fixed points of nonlinear systems

Question

For nonlinear systems, I know the phase portrait at a fixed point is a spiral when the eigenvalues are complex conjugates with real parts, and centre when they have no real parts. But how should I determine if it's "left-handed" or "right-handed" spiral, or which way the centre is turning?

Answer 1

Find the direction of the velocity vector $d{\bf x}/dt$ at some point away from the origin in the linearized system. See e.g. these notes and these.