= Weak focus
= Weak foci
{synonym}
A weak focus is an isolated planar <equilibrium point> with nonzero imaginary <eigenvalues> and nearby trajectories that spiral toward or away from it because of nonlinear terms. Its <linearization> is a <center equilibrium>, so linear analysis alone does not settle stability. The first nonzero radial drift in its <Poincaré return map> decides the direction.
For example, if the radial return map is $P(r)=r+\kappa r^{2m+1}+O(r^{2m+2})$ with $\kappa\ne0$, sufficiently small positive $r$ decreases under successive returns when $\kappa<0$ and increases when $\kappa>0$. The first case gives <asymptotic stability>; the second gives instability. Equivalently, if a positive local function $W=r^2+O(r^3)$ satisfies $\dot W=c r^{2m+2}+O(r^{2m+3})$ and the angular velocity stays positive, then the sign of $c$ settles the same question. A quadratic planar system can have its cubic radial drift vanish and still have a nonzero fifth-order radial drift.
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