 Hyperbolic attracting limit cycle

For an autonomous ordinary differential equation, the tangent to a periodic orbit gives a unit Floquet multiplier. A hyperbolic attracting cycle has that multiplier simple and all transverse multipliers strictly inside the unit disk. Thus the periodic variational operator has a one-dimensional null space spanned by the tangent. Nonlinear asymptotic attraction without hyperbolicity does not suffice: $\dot{r}=-(r-1{)}^3$, $\dot{\phi }=2\pi$ attracts to a unit-period circle but its radial variational equation is neutral.