The purpose of this review article is to demonstrate via a few simple models the mechanism for a very general, universal instability-the
Arnold diffusion - which occurs in the oscillating systems having more than two degrees of freedom. A peculiar feature of this instability results in
an irregular, or stochastic, motion of the system as if the latter were influenced by a random perturbation even though, in fact, the motion is
governed by purely dynamical equations. The instability takes place generally for very special initial conditions (inside the so-called stochastic
layers) which are, however, everywhere dense in the phase space of the system.
The basic and simplest one of the models considered is that of a pendulum under an external periodic perturbation. This model represents the
behavior of nonlinear oscillations near a resonance, including the phenomenon of the stochastic instability within the stochastic layer of
resonance. All models are treated both analytically and numerically. Some general regulations concerning the stochastic instability are presented,
including a general, semi-quantitative method-the overlap criterion-to estimate the conditions for this stochastic instability as well as its main
characteristics.