Odd quadratic Lorenz map ($f_{\mu }(x)=\mathrm{sgn}(x)(x^2-\mu )$)

Away from zero this discontinuous map is odd, and its branch derivatives are $2|x|$. With the convention $\mathrm{sgn}(0)=1$, its value at zero is $-\mu$, so it is not odd at zero unless $\mu =0$. A global transition at $\mu =0$ turns two small stable fixed points into one stable symmetric two-cycle. Its unsigned dynamics can be related to the even quadratic map $g_{\mu }(x)=\mu -x^2$, provided critical hits are treated separately.

// Target: dynamical-systems.bigb

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