Poincaré return map

A local transverse section to a flow is mapped back to itself by the next intersection of an orbit. Fixed points represent periodic orbits, and the derivative of this return map determines their transverse stability.

If a planar equilibrium with purely imaginary linear eigenvalues has radial return map $r\mapsto r+\kappa r^3+O(r^4)$, then $\kappa <0$ gives nonlinear attraction and $\kappa >0$ gives repulsion. Vanishing linear decay therefore does not imply a centre.