Long-time stability of breathers in Hamiltonian $\cal PT$-symmetric lattices

We consider the Hamiltonian version of a $\cal PT$-symmetric lattice that describes dynamics of coupled pendula under a resonant periodic force. Using the asymptotic limit of a weak coupling between the pendula, we prove the nonlinear long-time stability of breathers (time-periodic solutions localized in the lattice) by using the Lyapunov method. Breathers are saddle points of the extended energy function, which are located between the continuous bands of positive and negative energy. Nevertheless, we construct an approximate Lyapunov function and estimate its evolution on a long but finite time interval. The nonlinear stability analysis becomes possible for the $\cal PT$-symmetric lattice only because of the existence of a Hamiltonian structure.