Quasi-energies, parametric resonances, and stability limits in ac-driven $\mathcal{PT}$-symmetric systems

We introduce a simple model for implementing the concepts of quasi-energy and parametric resonances (PRs) in systems with the $\mathcal{PT}$ symmetry, i.e., a pair of coupled and mutually balanced gain and loss elements. The parametric (ac) forcing is applied through periodic modulation of the coefficient accounting for the coupling of the two degrees of freedom. The system may be realized in optics as a dual-core waveguide with the gain and loss applied to different cores, and the thickness of the gap between them subject to a periodic modulation. The onset and development of the parametric instability for a small forcing amplitude ($V_{1}$) is studied in an analytical form. The full dynamical chart of the system is generated by systematic simulations. At sufficiently large values of the forcing frequency, $\omega$, tongues of the parametric instability originate, with the increase of $V_{1}$, as predicted by the analysis. However, the tongues following further increase of $V_{1}$ feature a pattern drastically different from that in usual (non-$\mathcal{PT}$)\ parametrically driven systems: instead of bending down to larger values of the dc coupling constant, $V_{0}$, they maintain a direction parallel to the $V_{1}$ axis. The system of the parallel tongues gets dense with the decrease of $\omega$, merging into a complex small-scale structure of alternating regions of stability and instability. The cases of $\omega \rightarrow 0$ and $\omega \rightarrow \infty$ are studied analytically by means of the adiabatic and averaging approximation, respectively. The cubic nonlinearity, if added to the system, alters the picture, destabilizing many originally robust dynamical regimes, and stabilizing some which were unstable.