Short Essay on Difference Equations with Continuous Argument

The aim of the present paper is to draw attention to difference equations with continuous argument, which are paradigmatic for the simulation of complexity and chaos. Even a very simple equation x(t + 1) = f (x(t)); t > 0; has typical smooth solutions approaching discontinuous functions with, as a rule, infinitely many discontinuities on an interval of unit length. These solutions are excellent tools for the simulation of strongly nonlinear phenomena, such as deterministic chaos and cascade processes of the formation of structures, fractals, etc. One more argument in favor of difference equations with continuous argument is their use in the theory of differential-difference equations and in the theory of partial differential equations.