= Solution
Let a scalar <fixed point> have magnitude $r>0$. If $z=-r<0$, then $x$ and $y$ have the same sign; the two lifts are \b[the <fixed points> $(r,1)$ and $(-r,-1)$]. If $z=r>0$, their signs are opposite, and the lift is \b[the <period-two orbit> $(r,-1)\leftrightarrow(-r,1)$]. The scalar <fixed-point multiplier> is
$$
m=A\delta r^{\delta-1}>0.
$$
The two-return <fixed-point multiplier> of the period-two lift is $m^2$, so in either case <dynamical stability> requires $m<1$. The negative <fixed point> represents one of two mirror-related single-lobe <periodic orbits> in the flow, whereas the positive <fixed point> represents a double-lobe <periodic orbit>.
For $\delta>1$, the nearby <fixed point> is $z=\mu+o(\mu)$ and its <fixed-point multiplier> tends to zero. Thus \b[two stable single-lobe <limit cycles> for $\mu<0$ glue into one stable double-lobe <limit cycle> for $\mu>0$]. Their periods diverge as the <separatrix> is approached; $\mu=0,z=0$ is not a finite-period <limit cycle>.
For $0<\delta<1$, the small branch instead satisfies $z\sim-\operatorname{sgn}(\mu)(|\mu|/A)^{1/\delta}$ and its <fixed-point multiplier> tends to infinity. Hence \b[one unstable double-lobe <limit cycle> on $\mu<0$ splits into two unstable single-lobe <limit cycles> on $\mu>0$]. Additional finite-size stable branches depend on the surrounding map; near the resonance $\delta=1$ with $A<1$ they are resolved by the <saddle-node bifurcations> in part (c).
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