E. Yu. Romanenko

IDEAL TURBULENCE: ATTRACTORS OF DETERMINISTIC SYSTEMS MAY LIE IN THE SPACE OF RANDOM FIELDS

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.

FROM BOUNDARY VALUE PROBLEMS TO DIFFERENCE EQUATIONS: A METHOD OF INVESTIGATION OF CHAOTIC VIBRATIONS

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.

Introduction to the Theory of Dynamical Systems

The study of equations which describe the dynamics of real systems, and, in particular, the study of difference equations of the form

Stable periodic solutions of difference equations

x\left( {t + 1} \right) = f\left( {x\left( t \right)} \right)

Differential-difference equations reducible to difference and q-difference equations

(1.1) with t changing discretely or continuously, is usually connected with the following questions: How do individual solutions or sets of solutions (specified by the conditions of problems under consideration) behave as t grows ? How do they depend on variations of initial conditions and changes of the right-hand side (the function f)? What underlies and what governs the behavior of the solutions ? It is these very questions which must be answered by the theory of dynamical systems — the theory of groups or semi-groups of maps generated by solutions of equations in the space of states.

Trajectories of intervals in one-dimensional dynamical systems

There is a very short chain that joins dynamical systems with the simplest phase space (real line) and dynamical systems with the “most complicated” phase space containing random functions, as well. This statement is justified in this paper. By using “simple” examples of dynamical systems (one-dimensional and two-dimensional boundary-value problems), we consider notions that generally characterize the phenomenon of turbulence—first of all, the emergence of structures (including the cascade process of emergence of coherent structures of decreasing scales) and self-stochasticity.

Simulation of spatial-temporal chaos: The simplest mathematical patterns and computer graphics

Abstract Assuming certain conditions on an unperturbed continuous difference equation, we shall establish for a perturbed difference equation the existence of 1. (i) n asymptotically stable piecewise continuous n -periodic solutions, and 2. (ii) stable asymptotically n -periodic solutions.

Structural turbulence in boundary value problems

This paper aims at understanding the mathematical features of self-structuring in distributive media. We suggest a mathematical formalism for the description of structures and their dynamics in time and in space; in particular, the notions of coherent and self-similar structures are covered. A specific class of nonlinear boundary value problems presenting the evolution of some ideal medium is studied. Some regularities of structures appearing in its solutions with time increasing are established, The mathematical formulation is accompanied by computer pictures which visualize various properties of structures, in particular, the cascade process of appearance of coherent spatial-temporal structures and the process of forming spatially self-similar structures which become fractal as t → ∞.