We suggest a mathematical concept of self-stochasticity that may underlie a possible scenario of spacetime chaos in ideal medium (e.g. ideal turbulence). The self-stochasticity implies that the deterministic system which describes evolution of deterministic vector fields has trajectories whose ω-limit sets lie in the space of random vector fields. We consider a class of systems where almost all trajectories exhibit this property and, hence, the attractor consists of random fields.

Dedicated to the Memory of Professor Bernd AulbachThe necessity of considering trajectories just of sets rather than of points occurs both in dynamical systems theory by itself and in many evolutionary problems reducible to dynamical systems. In the case of one-dimensional dynamical systems, we present a number of conditions for the trajectory of an interval to be asymptotically periodic. The obtained results find significant applications in the theory of continuous time difference equations and some classes of boundary value problems for partial differential equations.In dynamical systems theory, one of the main subjects for study is the asymptotic dynamics of the trajectories. If a dynamical system possesses many unstable trajectories (for example, there is a strange attractor in a system), then in order to appreciate the dynamics of trajectories one needs to investigate trajectories of neighborhood's of unstable points. The necessity of considering trajectories just of sets rather than of points also occurs naturally in a variety of evolutionary problems reducible to dynamical systems. Below we give a survey of our recent works concerned with the asymptotic dynamics of trajectories of intervals.

A b s t r a c tIdeal turbulence is a mathematical phenomenon which occurs in certain infinite-dimensional deterministic dynamical systems, and implies that the attractor of a system lies off the phase space and among the attractor points there are fractal or evcn random funct,ions.Ideal turbulence is observed in various idealizcd models of real distributed systems, addresscd by electrodynamics, acoustics,' radiophysics, etc. Unlike real systems, in ideal systems (without internal resistancc), cascade processes are capable of giving birth to structures of arbitrarily small scale and even causing stochastization of the systems. Just these phenomena are readily described in the contecst of ideal turhulence, and allows to understand the mathematical scenarios for many features of real turbulence.A rna,thetnatically rigorous definition of ideal turbulence is based on standwd notions of dynamical systems theory and chaos theory. I n t r o d u c t i o nThe term turbulence, originally been in use only at flows of liquids and gases, in the broad sense is implied jn the progression of chaos in dcterministic distributcd parameter systems in any one of a number of characters. Many effects inherent in the phenomenon of turbulence can be observed in infinitelydimensional dynamical systems, induced, in particular, boundary value problems for partial differential equations.The distinguishing features,of turbulence are ca5cade processes of emergence of structures of decreasing scales and chaotic mixing. Which are mechanisms (scenarios) for these phenomena? In real distributed systems, processes of reducing structures to smaller and smaller size cannot go indefinitely because of their internal resistance. In ideal systems (without internal resistance), cascade processes are capable of giving birth to structures of arbitrarily small scale and even of causing stochastization of the systems (in which case their long-term behavior can be described in terms of probabilistic theory). Mceting just latter effects and an understanding of the mattiematical mechanisms responsible for these can be realizable through certain idealizations of mathematical models of real distributed systems and the use of the standard notions of dynaniical systems theory and chaos theory.Based upon the above ideas, we presents an original approach to modelling turbulent processes. This approach arised from the author's research into chadic dynamics in infinite-dimensional dynamical systems 0-7803-7939-X/03/$17.00 0 2003 IEEE 23 PhysCon 2003, St. Petersburg, Russia

We propose an approach to the analysis of turbulent oscillations described by nonlinear boundary-value problems for partial differential equations. This approach is based on passing to a dynamical system of shifts along solutions and uses the notion of ideal turbulence (a mathematical phenomenon in which an attractor of an infinite-dimensional dynamical system is contained not in the phase space of the system but in a wider functional space and there are fractal or random functions among the attractor "points"). A scenario for ideal turbulence in systems with regular dynamics on an attractor is described; in this case, the space-time chaotization of a system (in particular, intermixing, self-stochasticity, and the cascade process of formation of structures) is due to the very complicated internal organization of attractor "points" (elements of a certain wider functional space). Such a scenario is realized in some idealized models of distributed systems of electrodynamics, acoustics, and radiophysics.

Among evolutionary boundary value problems for partial differential equations, there is a wide class of problems reducible to difference, differential-difference and other relevant equations. Of especial promise for investigation are problems that reduce to difference equations with continuous argument. Such problems, even in their simplest form, may exhibit solutions with extremely complicated long-time behavior to the extent of possessing evolutions that are indistinguishable from random ones when time is large. It is owing to the reduction to a difference equation followed by the employment of the properties of the one-dimensional map associated with the difference equation, that, it is in many cases possible to establish mathematical mechanisms for one or other type of dynamical behavior of solutions. The paper presents the overall picture in the study of boundary value problems reducible to difference equations (on which the authors have a direct bearing over the last ten years) and demonstrates with several simplest examples the potentialities that such a reduction opens up.