A. N. Sharkovsky

Oscillations in Singularly Perturbed Delay Equations

This paper presents some recent results on the scalar singularly perturbed differentila delay equation

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

[v\dot x(t) + x(t) = f(x(t - 1)).

Introduction to the Theory of Dynamical Systems

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From one-dimensional to infinite-dimensional dynamical systems: Ideal turbulence

By replacing the parallel LC “resonator” in Chua’s circuit by a lossless transmission line, terminated by a short circuit, we obtain a “time-delayed Chua’s circuit” whose time evolution is described by a pair of linear partial differential equations with a nonlinear boundary condition. If we neglect the capacitance across the Chua’s diode, described by a nonsymmetric piecewiselinear vR–iR characteristic, the resulting idealized time-delayed Chua’s circuit is described exactly by a scalar nonlinear difference equation with continuous time, which makes it possible to characterize its associated nonlinear dynamics and spatial chaotic phenomena. From a mathematical viewpoint, circuits described by ordinary differential equations can generate only temporal chaos, while the time-delayed Chua’s circuit can generate spatiotem poral chaos. Except for stepwise periodic oscillations, the typical solutions of the idealized time-delayed Chua’s circuit consist of either weak turbulence, or strong turbulence, which are examples of “ideal” (or “dry”) turbulence. In both cases, we can observe infinite processes of spatiotemporal coherent structure formations. Under weak turbulence, the graphs of the solution tend to limit sets which are fractals with a Hausdorff dimension between 1 and 3, and is therefore larger than the topological dimension (of sets). Under strong turbulence, the “limit” oscillations are oscillations whose amplitudes are random functions. This means that the attractor of the idealized time-delayed Chua’s circuit already contains random functions, and spatial self-stochasticity phenomenon can be observed.

SELF-STRUCTURING AND SELF-SIMILARITY IN BOUNDARY VALUE PROBLEMS

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.

Trajectories of intervals in one-dimensional dynamical systems

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.

Recovering trajectories of chaotic piecewise linear dynamical systems

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