The mechanism by which periodic nonrandom forces lead to stochastic acceleration of particles is examined. Two examples considered are (i) the Fermi problem of a ball bouncing between a fixed and an oscillating wall, with various wall-oscillation functions, and (ii) cyclotron-resonance heating in a magnetic mirror. Numerical studies show that the phase plane consists of a complicated but regular structure of islands embedded in a stochastic sea. These islands may have the character of either adiabatic barriers or sinks for particles. The islands can be described analytically by expansions about elliptic singular points. A velocity below which no islands exist is observed computationally and is predicted from Floquet theory. Computations also demonstrate that in some cases an adiabatic wall forms an upper limit to particle diffusion in velocity space. A lower bound on this wall is predicted analytically. The Fermi problem is reduced to a Hamiltonian form, and the nonlinear stability and approximate location of the adiabatic wall is predicted from adiabatic invariance theory. Introduction of an external random-force component modifies, but does not destroy, the basic results. For velocities below which no islands exist, it is shown that the random-phase assumption holds, and that the particle motion can be described by a Fokker-Planck equation. The solution to the Fokker-Planck equation is found to agree with the numerical calculations. Above the stochastic transition velocity, strong phase correlations exist, and a Fokker-Planck description is inappropriate.