= Lorenz power return map
{c}
{title2=$f(x)=\operatorname{sgn}(x)(-\mu+A|x|^\delta)$}
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 $s=\mu-As^\delta$, 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 branches
$$
g_+(y)=((\mu+y)/A)^{1/\delta},\qquad g_-(y)=-((\mu-y)/A)^{1/\delta}
$$
map 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.
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