 Lorenz power return map

For a symmetric real saddle with one unstable direction and a strongly contracting transverse direction, composing the power-law local passage with a smooth global return gives this leading Poincaré return map, with $A>0$ and saddle index $\delta$. It is undefined at the stable-manifold crossing $x=0$. For $\delta >1$, small fixed points on each branch are attracting for $\mu <0$; for $\mu >0$ an attracting alternating-sign period-two orbit satisfies $s=\mu -A{s}^{\delta }$, giving gluing of two one-lobe flow cycles into one two-lobe cycle. For $\delta <1$ and small $\mu >0$, on $I=[-c\mu ,c\mu ]$, $0<c<1$, the inverse branchesmap into $I$ and contract. Infinite branch choices define an invariant Cantor set with full two-symbol dynamics. In the two-dimensional return, transverse contraction turns this mechanism into a Smale horseshoe. An attracting Lorenz set additionally requires appropriate trapping and global-return properties.