= Gluing of cycles in an iterated Lorenz map
{title2=$\mu=1:\quad 2\times\text{period }2\ \longleftrightarrow\ 1\times\text{period }4$}
The two-cycle of $g(x)=\mu-x^2$ has points $(1\pm\sqrt{4\mu-3})/2$ and multiplier $4(1-\mu)$. Crossing $\mu=1$ changes its sign and passes through the critical cycle $\{0,1\}$. The sign lift therefore changes from two stable two-cycles to one stable four-cycle. The second iterate is continuous at zero at the transition, but retains jumps at the nonzero preimages of zero. Thus the return-map interpretation concerns a neighborhood of zero, not continuity of the whole second iterate on the real line.
// Target: dynamical-systems.bigb
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