= Solution
Restrict the complexified equation to the imaginary axis by writing its coordinate as $z=iy$ with real $y$. Transform the left-hand side as well: $i\dot y=iry-i\sinh y$. Thus the real equation along that axis is
$$
\boxed{\dot y=ry-\sinh y.}
$$
For $r<1$, $\sinh y/y>1$ away from zero, so the origin is the only <equilibrium point> and is attracting. At $r=1$ the leading term is $-y^3/6$, so it remains attracting with algebraic, rather than exponential, relaxation.
For $r>1$, the origin becomes repelling and a unique pair $\pm y_*$ appears, where $r=\sinh y_*/y_*$. Each nonzero root is attracting, since $r-\cosh y_*<0$. Close to the threshold,
$$
\boxed{y_*\simeq\sqrt{6(r-1)}.}
$$
This is a <supercritical pitchfork bifurcation>, opposite to the real-axis case. The hyperbolic sine confines large real $y$ rather than producing an infinite sequence of oscillatory lobes.
\Image[/past-exam-of-the-flows-fluctuations-and-complexity-course-of-the-university-of-oxford/2016-sine-and-sinh-bifurcations.png]
{title=Real-axis sinc equilibrium branches and the imaginary-axis supercritical pitchfork; solid branches attract and dashed branches repel, with a zoom on the small-r saddle-node sequence}
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