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The associated even map is simply $g(x)=\mu -x^2$. For nonzero $x$, the odd quadratic Lorenz map has the two branches $f_L(x)=x^2-\mu$ on the positive side and $f_L(x)=\mu -x^2$ on the negative side. Their limiting values at zero are $-\mu$ and $+\mu$, while the prescribed point value is $f_L(0)=-\mu$. Thus the branches join continuously only at $\mu =0$, where the origin is a fixed point and $f_L(x)=x|x|$.

The small fixed point of $g$ isFor small negative $\mu$, $r<0$ and the odd quadratic Lorenz map has two stable fixed points $r,-r$. Their periodic-orbit multiplier is $2|r|<1$. As $\mu \uparrow 0$ these fixed points collide at zero. For small positive $\mu$, $r>0$ andso a stable symmetric two-cycle is present, with periodic-orbit multiplier $4r^2<1$. Hence the global transition isThe large fixed points $\pm (1+r)$ remain unstable near this transition and are not created or destroyed there. The parabola $g$ itself is smooth as $\mu$ passes zero; its fixed point crosses the critical point and changes the sign of its periodic-orbit multiplier. The change of orbit type occurs in the sign-lifted discrete dynamical system, whose discontinuity switches orientation.

For a noncritical least-period-$n$ cycle of $g(x)=\mu -x^2$, the multiplier sign is $(-1{)}^n{\prod }_k\mathrm{sgn}(x_k)$. A positive multiplier gives two symmetry-related period-$n$ cycles of the odd quadratic Lorenz map; a negative multiplier gives one period-$2n$ cycle. Their multipliers are respectively $|(g^n{)}^{\prime }|$ and $|(g^n{)}^{\prime }{|}^2$. Distinct points of a quadratic-map cycle cannot have the same absolute value, since the map is even and acts bijectively on its cycle; this proves the asserted least periods.

// Target: dynamical-systems.bigb