Past exam of the flows, fluctuations and complexity course of the University of Oxford / 2016 / 3 / c / 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 isFor $r<1$, $\sinh y/y>1$ 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 $\pm y_{\ast }$ appears, where $r=\sinh y_{\ast }/y_{\ast }$. Each nonzero root is attracting, since $r-\cosh y_{\ast }<0$. Close to the threshold,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.