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For a rational map of degree of a rational map of the Riemann sphere at least two, a parabolic cycle is a cycle of a holomorphic map whose periodic-orbit multiplier is a root of unity. If the exact period is $p$ and the multiplier has order $q$, the return iterate $f^{pq}$ has multiplier one. In a local coordinate $u$ centred at a point of the cycle it has the formThe nonzero nonlinear term gives attracting petals and repelling petals, with the former underlying a parabolic basin. A multiplier of modulus one with irrational argument is not parabolic.

For $f_c(z)=z^2+c$ at $c=1/4$, the fixed point $z=1/2$ has multiplier one. The translation $z=1/2+u$ gives the exact local map $u\mapsto u+u^2$, exposing the nonlinear term responsible for attraction from one direction and repulsion from another. The parabolic cycle belongs to the Julia set even though nearby points in its parabolic basin approach it.^[ref]