Figure 1

(a) Representative trajectories of the $\mathcal{PT}$-symmetric coupler on the surface of a cylinder. A periodic trajectory—the closed solid curve in the middle of the cylinder, plotted in purple (gray)—is confined within the separatrix loop (plotted in dashed red). The other two solid curves—dark blue (black) and green (light gray)—wind up to $Z\to \infty$ and represent blow-up regimes of the coupler. Here $\gamma =0.5$ and $\rho =3$. (b) A representative trajectory of the AC coupler. From the point shown by the red blob, the trajectory zaps onto the vertical axis, starts moving up, but gradually deviates from the vertical and leaves the imaginary cylinder of small radius. The trajectory ends up approaching a limit cycle. The blow-up is arrested. Here $\gamma =1.9$, $a=2$, and the initial conditions $(X,Y,Z)=(0,5,0)$.Reuse & Permissions

Figure 2

Top: The potential $V\left(q\right)=\sin (q+\chi )-{\kappa }^2{q}^2$, with the vertical lines indicating the positions of the extrema. Here $\kappa =0.2$ and $\chi =\pi$. The negative $q$ and the potential values $V>-1$ are inaccessible to the imaginary particle (shaded in yellow). Bottom: The corresponding phase portrait with fixed points and separatrices shown in dashed red, calculated via Eq. (16). Shaded is the region where $q<0$ or $E>-1$. Trajectories in this region do not correspond to any motions of the dimer.Reuse & Permissions