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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.

A chaotic attractor of a Rössler system for suitable parameter values. Local expansion near an unstable spiral combines with excursions in the third coordinate and global return; the existence of a local expanding direction by itself does not prove a chaotic attractor.

An autonomous differential equation in three variables. Its linear planar rotation couples to a nonlinear vertical variable through $xz$. Depending on parameters, its solutions can approach an equilibrium point, a periodic orbit, or a more complicated attractor. The specialization $b=a$ is used in the 2011 Oxford combined paper.