= Neimark–Sacker bifurcation
{c}
{title2=$\lambda_{1,2}=e^{\pm i\theta},\quad d\ne0$}
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 $r'=r(1+\eta+dr^2)+\cdots$ with $d\ne0$, 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.
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