Neimark–Sacker bifurcation (${\lambda }_{1,2}=e^{\pm i\theta },d\ne 0$)

A Neimark–Sacker bifurcation of a smooth planar map occurs when a simple complex-conjugate pair of fixed-point Floquet multipliers crosses the unit circle away from low-order resonances. If its radial normal form is $r^{\prime }=r(1+\eta +d{r}^2)+\cdots$ with $d\ne 0$, a small invariant closed curve has $r^2=-\eta /d+\cdots$. For $d<0$ it appears on $\eta >0$, where the fixed point has become unstable, and its radial Floquet multiplier is $1-2\eta +\cdots$, so the bifurcation is supercritical. For $d>0$ the curve lies on $\eta <0$ and is unstable, giving a subcritical bifurcation. Area-preserving maps have a vanishing cubic radial drift at an elliptic point and do not meet this dissipative nondegeneracy condition.

In a complex coordinate write $z^{\prime }=\lambda z+f_{20}{z}^2+f_{11}z\overline{z}+f_{02}{\overline{z}}^2$, with $|\lambda |=1$. Away from strong resonances of a planar map, the quadratic change $z=w+h_{20}{w}^2+h_{11}w\overline{w}+h_{02}{\overline{w}}^2$ has $h_{20}=f_{20}/({\lambda }^2-\lambda )$, $h_{11}=f_{11}/(1-\lambda )$ and $h_{02}=f_{02}/({\overline{\lambda }}^2-\lambda )$. Substituting and collecting $w^2\overline{w}$ gives $g_{21}=2f_{20}{h}_{11}+f_{11}{h}_{20}+f_{11}{\overline{h}}_{11}+2f_{02}{\overline{h}}_{02}$. The cubic radial coefficient is $\ell =\mathrm{Re}(\overline{\lambda }{g}_{21})$; $\ell <0$ gives a supercritical crossing when the multiplier modulus increases through one.

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.