Romanenko E. The elements of Qualitative theory of continuous argument difference equations.

We suggest a method to analyzing the asymptotic dyna-mics of nondissipative systems on noncompact function spa-ces, which is applied to the dynamical systems generated by difference equations and boundary value problems for partial differential equations. We build the basics of qualitative theory of nonlinear continuous time difference equations x(t+1)=f(x(t), typical solutions are showed to be those tending to upper semicontinuous functions whose graphs are self-similar and, at times, fractal. We introduce the notion of self-stochasticity in deterministic systems (the attractor contains random functions). Substantiated is a scenario for a spatial-temporal chaos in systems with regular dynamics on attractor (the attractor consists of cycles only and the onset of chaos results from the complicated structure of attractor «points»). We study several classes of q-difference and differential-difference equations. We develop a method to research into boundary value problems, that combines the reduction to difference equations and going to dynamical systems. We put forward a formalisms for describing self-organization and dynamical chaos.