Gluing of cycles in an iterated Lorenz map ($\mu =1:2\times \text{period }2⟷1\times \text{period }4$)

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