Attractor for lattice system of dissipative Zakharov equation

We consider the asymptotic behavior of solutions of an infinite lattice dynamical system of dissipative Zakharov equation. By introducing new weight inner product and norm in the space and establishing uniform estimate on ＂Tail End＂ of solutions, we overcome some difficulties caused by the lack of Sobolev compact embedding under infinite lattice system, and prove the existence of the global attractor; then by using element decomposition and the covering property of a polyhedron in the finite-dimensional space, we obtain an upper bound for the Kolmogorov ε-entropy of the global attractor; finally, we present the upper semicontinuity of the global attractor.     

One-dimensional global attractor for the damped and driven sine-Gordon equation

The damped and driven sine-Gordon equation with Neumann boundary conditions is studied. It is shown that it has a one-dimensional global attractor in a suitable functional space when the “damping” and the “diffusing” are not very small. Project supported by the National Natural Science Foundation of China.     

Kolmogorov＇s ε-Entropy of Bounded Sets in Discrete Spaces and Attractors of Dissipative Lattice Systems

We obtain an estimate of the upper bound for Kolmogorov＇s ε-entropy for the bounded sets with small ＂tail＂ in discrete spaces, then we present a sufficient condition for the existence of a global attractor for dissipative lattice systems in a reflexive Banach discrete space and establish an upper bound of Kolmogorov＇s ε-entropy of the global attractor for lattice systems.     

Micro-chaos in digital control

Summary In this paper we analyze a model for the effect of digital control on one-dimensional, linearly unstable dynamical systems. Our goal is to explain the existence of small, irregular oscillations that are frequently observed near the desired equilibrium. We derive a one-dimensional map that captures exactly the dynamics of the continuous system. Using thismicro-chaos map, we prove the existence of a hyperbolic strange attractor for a large set of parameter values. We also construct an “instability chart” on the parameter plane to describe how the size and structure of the chaotic attractor changes as the parameters are varied. The applications of our results include the stick-and-slip motion of machine tools and other mechanical problems with locally negative dissipation.     

A simple proof of the ergodic theorem using nonstandard analysis

A simple proof of the individual ergodic theorem is given. The essential tool is the nonstandard measure theory developed by P. Loeb. Any dynamical system on an abstract Lebesgue space can be represented as a factor of a “cyclic” system with a hyperfinite cycle. The ergodic theorem for such a “cyclic” system is almost trivial because of its simple structure. The general case follows from this special case.     

Tracing of pseudotrajectories of dynamical systems and stability of prolongations of orbits

We investigate properties of dynamical systems associated with the approximation of pseudotrajectories of a dynamical system by its trajectories. According to modern terminology, a property of this sort is called the “property of tracing pseudotrajectories” (also known in the English literature as the “shadowing property”). We prove that dynamical systems given by mappings of a compact set into itself and possessing this property are systems with stable prolongation of orbits. We construct examples of mappings of an interval into itself that prove that the inverse statement is not true, i.e., that dynamical systems with stable prolongation of orbits may not possess the property of tracing pseudotrajectories. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1016–1024, August, 1997.     

Rang des systèmes dynamiques gaussiens

Although some Gaussian dynamical systems, called “Gaussian-Kronecker”, share with rank one systems some unusual properties, we show in this work that a Gaussian dynamical system is never of finite rank. In fact, such a system can’t even be of local rank one.      

Attractors for a Three-Dimensional Thermo-Mechanical Model of Shape Memory Alloys

Abstract In this note, we consider a Frémond model of shape memory alloys. Let us imagine a piece of a shape memory alloy which is fixed on one part of its boundary, and assume that forcing terms, e.g., heat sources and external stress on the remaining part of its boundary, converge to some time-independent functions, in appropriate senses, as time goes to infinity. Under the above assumption, we shall discuss the asymptotic stability for the dynamical system from the viewpoint of the global attractor. More precisely, we generalize the paper [12] dealing with the one-dimensional case. First, we show the existence of the global attractor for the limiting autonomous dynamical system; then we characterize the asymptotic stability for the non-autonomous case by the limiting global attractor. * Project supported by the MIUR-COFIN 2004 research program on “Mathematical Modelling and Analysis of Free Boundary Problems”.     

Upper Semicontinuity and Kolmogorov ε-Entropy of Global Attractor for κ-Dimensional Lattice Dynamical System Corresponding to Klein-Gordon-SchrSdinger Equation

In this paper, we establish the existence of a global attractor for a coupled κ-dimensional lattice dynamical system governed by a discrete version of the Klein-Gordon-SchrSdinger Equation. An estimate of the upper bound of the Kohnogorov ε-entropy of the global attractor is made by a method of element decomposition and the covering property of a polyhedron by balls of radii ε in a finite dimensional space. Finally, a scheme to approximate the global attractor by the global attractors of finite-dimensional ordinary differential systems is presented .     

Unfolding the torus: Oscillator geometry from time delays

Summary We present a simple method of plotting the trajectories of systems of weakly coupled oscillators. Our algorithm uses the time delays between the “firings” of the oscillators. For any system ofn weakly coupled oscillators there is an attracting invariantn-dimensional torus, and the attractor is a subset of this invariant torus. The invariant torus intersects a suitable codimension-1 surface of section at an (n−1)-dimensional torus. The dynamics ofn coupled oscillators can thus be reduced,in principle, to the study of Poincaré maps of the (n−1)-dimensional torus. This paper gives apractical algorithm for measuring then−1 angles on the torus. Since visualization of 3 (or higher) dimensional data is difficult we concentrate onn=3 oscillators. For three oscillators, a standard projection of the Poincaré map onto the plane yields a projection of the 2-torus which is 4-to-1 over most of the torus, making it difficult to observe the structure of the attractor. Our algorithm allows a direct measurement of the 2 angles on the torus, so we can plot a 1-to-1 map from the invariant torus to the “unfolded torus” where opposite edges of a square are identified. In the cases where the attractor is a torus knot, the knot type of the attractor is obvious in our projection.     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

Orthogonal and bounded orthogonal spectral representations

The concept of an orthogonal spectral representation (OTSR) of a Hilbert spaceH relative to a spectral measureE(.) is introduced and it is shown that every Hilbert space admits an OTSR relative to a given spectral measure. Apart from the various results obtained about OTSRs, the principal result of Allan Brown (1974) is deduced as an easy consequence of this study. A new complete system of unitary invariants called the “equivalence of OTSRs”, is given for spectral measures. Two special types of OTSRs called “BOTSR” and “COBOTSR” are introduced and characterized respectively in terms of the “GCGS-property” and “CGS-property” of the associated spectral measure. Various complete systems of unitary invariants are given for spectral measures with the GCGS-property. Finally, the Wecken-Plesner-Rohlin theorem on hermitian operators with simple spectra is generalized to arbitrary spectral measures.     

Generating series of classes of Hilbert schemes of points on orbifolds

The notion of power structure over the Grothendieck ring of complex quasi-projective varieties is used for describing generating series of classes of Hilbert schemes of zero-dimensional subschemes (“fat points”) on complex orbifolds.     

Complete observability of differential-algebraic systems with delays

We study the statement and solvability of complete observability problems for linear stationary differential-algebraic dynamical systems with delays (DAD systems. Since in the general case, the state space of such systems is infinite-dimensional and is not necessarily “minimal,” we consider various statements of problems depending on what states are observed. Our attention is focused on the simplest DAD system in symmetric form. We obtain efficient parametric criteria and analyze relationships between various notions of complete observability for DAD systems. In the case of DAD systems with scalar coefficients, we obtain a complete classification of notions of complete observability in the class of continuous initial functions with the continuous matching condition. We analyze the problem of computing the minimum number of outputs of a spectrally observable DAD system.     

An anyon model

We construct an infinite-dimensional dynamical Hamiltonian system that can be interpreted as a localized structure (“quasiparticle”) on the plane E ₂. The model is based on the theory of an infinite string in the Minkowski space E _1,3 formulated in terms of the second fundamental forms of the worldsheet. The model phase space H is parameterized by the coordinates, which are interpreted as “internal” (E(2)-invariant) and “external” (elements of T*E ₂) degrees of freedom. The construction is nontrivial because H contains a finite number of constraints entangling these two groups of coordinates. We obtain the expressions for the energy and for the effective mass of the constructed system and the formula relating the proper angular momentum and the energy. We consider a possible interpretation of the proposed construction as an anyon model.     

Projecting precipitousness

This paper is about “strong” ideals on small cardinals. It is shown that a typical property of large cardinal measures does not transfer to these ideals. More specifically, that precipitous ideals onP _ω1λ spaces may not project down to precipitous ideals on “smaller”P _ω1λ′ spaces. Also, that the existence of a presaturated ideal on the bigger space does not imply the existence of a presaturated ideal on the smaller space.     

Collapse of attractors for ODEs under small random perturbations

This paper concerns comparisons between attractors for random dynamical systems and their corresponding noiseless systems. It is shown that if a random dynamical system has negative time trajectories that are transient or explode with probability one, then the random attractor cannot contain any open set. The result applies to any Polish space and when applied to autonomous stochastic differential equations with additive noise requires only a mild dissipation of the drift. Additionally, following observations from numerical simulations in a previous paper, analytical results are presented proving that the random global attractors for a class of gradient-like stochastic differential equations consist of a single random point. Comparison with the noiseless system reveals that arbitrarily small non-degenerate additive white noise causes the deterministic global attractor, which may have non-zero dimension, to ‘collapse’. Unlike existing results of this type, no order preserving property is necessary.      

Attractors of Discrete Controlled Systems in Metric Spaces

A new scenario is described for the creation of a strange attractor in discrete dynamical systems acting in metric spaces. We investigate attractors for ensembles of dynamical systems and attractors for controlled systems with programmed piecewise-constant controls taking finitely many values.     

Deterministic chaos of a spherical pendulum under limited excitation

We investigate the appearance, development, and vanishing of deterministic chaos in a “spherical pendulum-electric motor of limited power” dynamical system. Chaotic attractors discovered in the system are described in detail. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 534–548, April, 2007.     

Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities” argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the “acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.