Existence of localized periodic motions in systems of coupled integrable symplectic maps

We study a one-dimensional chain of discrete non-linear maps with a weak coupling. We consider solutions of the integrable anticontinuous limit, where one or more “central” oscillators move in resonant non-isolated periodic orbits while the other oscillators are at rest. We prove the continuation of these motions for weak non-zero coupling and determine their initial conditions and stability. We apply the above results and perform numerical investigation in the case where the uncoupled motion of each oscillator is described by the integrable Suris map.