= Odd quadratic Lorenz map
{title2=$f_\mu(x)=\operatorname{sgn}(x)(x^2-\mu)$}
Away from zero this discontinuous map is odd, and its branch derivatives are $2|x|$. With the convention $\operatorname{sgn}(0)=1$, 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
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