In the paper the simplest reversible mapping is considered. In this mapping islands of stability are realized as well as an attractor. New phenomena arising due to their interaction are revealed. The chaotic regimes are studied and the unpredictability of limit states (attractor or ± ∞) is demonstrated, the latter effect being caused by the fractal character of initial conditions corresponding to these states. An unusual class of models is described which admit a description of stochastic trajectories at large time scales.