Book excerpt: The soil plays one of the most important roles in the hydrological cycle. It is a three-phase porous medium, where the phases are particles of ground rock orclay, water and a water vapour air combination. The most interesting nonlinear behaviour is observed in the region where the soil is not fully saturated. As far back as 1930s, it was shown experimentally that there is a hysteretic effect in the relationship of the water content and the capillary pressure in the unsaturated zone. This effect can be quite strong in certain types of soils, and therefore it is desirable to incorporate hysteresis in the dynamical models describing flows of water through the soil. In this thesis a model for the discontinuous flow of water through porous media such as soils is studied. This model uses ordinary differential equations with the Preisach operator under the derivative to describe the wetting and dry- ing process. The right hand side has discontinuities that occur when rain suddenly starts or stops. Numerical simulations are used to investigate the dynamics of soil-water content in the presence of hysteresis over a large period of time. The equations with the Preisach operator under the derivative are a very recent mathematical object. In this thesis the mathematical results obtained by P. Krejci et al are extended to the equations with a right-hand side that has discontinuities in time. In particular, existence, uniqueness, extendability and differentiability of solutions of such equations are established. A numerical algorithm for solving these equations is also proposed. In the final chapter equations of this type are applied to an electromagnetic model of a van der Pol type circuit with a ferromagnetic element in the inductance. The effect of hysteresis on periodic oscillations that are observed in thiscircuit is demonstrated.