= Solution
A global <bifurcation> joins two mechanisms: a long passage near a <hyperbolic equilibrium point> and a finite excursion returning the <unstable manifold> to its neighborhood. <Eigenvalue> ratios control the local passage, while the geometry of the global return decides reinjection, folding and trapping. In the real symmetric case there are two returning branches and a power-law cusp. In the <saddle-focus> case rotation adds infinitely many oscillations to that cusp. The following maps display both mechanisms explicitly.
For the Lorenz configuration, enter on $z=h$ at $(x,y,h)$ and leave on $x=s h$, where $s=\operatorname{sgn}(x)$. Solving the linear <flow> gives a flight time
$$
T=\frac1{\lambda_+}\log\frac h{|x|}.
$$
Put $\gamma=-\widehat\lambda/\lambda_+>\delta=-\lambda_-/\lambda_+$. The outgoing coordinates are
$$
x_{\rm out}=s h,\qquad
y_{\rm out}=y(|x|/h)^\gamma,\qquad
z_{\rm out}=h(|x|/h)^\delta.
$$
The smooth global map from each outgoing section to the incoming section has, to leading order,
$$
x_{\rm new}=s[-\mu+a z_{\rm out}]+O(y_{\rm out},z_{\rm out}^2),\qquad
y_{\rm new}=s\nu+O(z_{\rm out},y_{\rm out}),
$$
up to the choice of which unstable branch carries the sign of $\nu$. Reflection <symmetry> makes the offsets opposite and the leading coefficients equal. Since the stable transverse coordinate is multiplied by the higher power $|x|^\gamma$, it contributes only subleading terms to the leading scalar map. Absorbing powers of $h$ into a positive constant gives the <Lorenz power return map>
$$
\boxed{x_{\rm new}=f_L(x)=\operatorname{sgn}(x)(-\mu+A|x|^\delta).}
$$
The line $x=0$ lies on the <stable manifold> and has no finite return; the discontinuity is therefore a genuine missing return, not an arbitrary value to assign at zero.
If $\delta>1$, the <derivative> $A\delta|x|^{\delta-1}$ tends to zero. For $\mu<0$, there are two attracting <fixed points>, $x=\pm s$ with $s=-\mu+As^\delta\simeq-\mu$. They describe two symmetry-related one-lobe attracting <flow> <periodic orbits>. For $\mu>0$, the orbit alternates between $s$ and $-s$, where $s=\mu-As^\delta\simeq\mu$; its multiplier is $(A\delta s^{\delta-1})^2<1$. This represents one attracting <flow> <periodic orbit> traversing both lobes. At $\mu=0$ the two homoclinic loops meet: this is the contracting gluing scenario, with diverging <flow> period at the connection.
If $\delta<1$, the local <derivative> is unbounded, and the small <periodic orbits> are repelling in the expanding return direction. For $\mu>0$, the small positive <fixed point> has $s\simeq(\mu/A)^{1/\delta}$, with a multiplier tending to infinity; there is a symmetry-related negative <fixed point> as well. More complex motion follows from the two branches, not merely from a large <derivative> at one <fixed point>. Choose $I=[-c\mu,c\mu]$ with $0<c<1$. The inverse branches are
$$
g_+(y)=((\mu+y)/A)^{1/\delta},\qquad
g_-(y)=-((\mu-y)/A)^{1/\delta}.
$$
For sufficiently small positive $\mu$, both lie inside $I$, their images are disjoint, and their <derivatives> are $O(\mu^{1/\delta-1})\to0$. Nested inverse images for every infinite sign sequence therefore define a Cantor invariant set. Iterating the forward map shifts the sequence. This constructs periodic itineraries of every symbolic period and nonperiodic itineraries; transverse <contraction> in the full return turns the expanding two-strip mechanism into a horseshoe.
This is the local chaotic Lorenz mechanism. A robust attracting Lorenz set additionally needs a global trapping region and the appropriate return properties. The scalar asymptote and $\delta<1$ alone do not prove that every nearby orbit is attracted to one chaotic set; periodic windows, escaping orbits and other global returns must be distinguished. <Symmetry> is crucial to obtaining the two homoclinic branches together in a single parameter variation.
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