Chapter 1 - Examples, basic problems, peculiar features of solutions

Abstract

This chapter discusses several dynamic systems described by differential equations of different types. The chapter also elaborates the features in the behaviors of solutions to these equations under random disturbances of parameters. A particle under random velocity field is described by the system of ordinary differential equations of the first order. The behavior of a system of particles essentially depends on whether the random field of velocities is divergence-free or divergent. The evolution of the two-dimensional system of particles uniformly distributed within the circle for a particular realization of the divergence-free steady field is analyzed. The system of equations also describes the behavior of a particle under the field of random external forces. The simplest nontrivial system with multiplicative impact can be illustrated using the stochastic parametric resonance as an example. The chapter also elaborates on a model that provides an insight into the difference between the diffusion processes in divergent and divergence-free velocity fields is also elaborated.