Past exam of the flows, fluctuations and complexity and symmetry and relativity paper of Part B of the physics course of the University of Oxford / 2011 / 3 / 1

An equilibrium point of a dynamical system is a state ${\mathbf{x}}_{\ast }$ with $\mathbf{f}({\mathbf{x}}_{\ast })=0$, so the trajectory stays there. In a planar phase portrait, an unstable spiral has a complex-conjugate pair of Jacobian matrix eigenvalues with positive real part: nearby trajectories rotate while moving outward. An attractor is an invariant set approached by trajectories from a neighborhood, its basin of attraction. It can be an equilibrium point, a limit cycle, or a more complicated set; outward spiralling near one equilibrium point does not preclude attraction to a larger set.