 Neimark–Sacker bifurcation

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.