Strong resonance of a planar map (${\lambda }^k=1,k\le 4$)

At a strong resonance of a planar map the Floquet multiplier of a fixed point is a root of unity of low order. The usual nonresonant Neimark–Sacker bifurcation requires ${\lambda }^k\ne 1$ for $k=1,2,3,4$. At a 1:3 resonance a quadratic term ${\overline{z}}^2$ cannot be removed from the normal form; at a 1:4 resonance a cubic term ${\overline{z}}^3$ remains. Consequently one cannot infer a quasiperiodic invariant curve from the unit-modulus crossing alone.