Long-time behavior of Navier-Stokes flow on a two-dimensional torus excited by an external sinusoidal force

In this paper we study the Navier-Stokes flow on the two-dimensional torusS ¹ ×S ¹ excited by the external force (k ² sinky, 0) and find the long-time behavior for the flow starting from some states, whereS ¹=[0,2](mod 2). Especially for the casek=2, it follows from an analysis and computation that the Navier-Stokes flow with the initial state cos(mx+ny) or sin(mx+ny) will likely evolve through at most one step bifurcation to either a steady-state solution or a time-dependent periodic solution for any Reynolds number and integersm andn.     

Noise and bifurcations

The influence of while noise on bifurcating dynamical systems is investigated using both Fokker-Planck and functional integral methods. Noise leads to fuzzy bifurcations where physically relevant quantities become smooth functions of the bifurcation parameters. We study dynamical and probabilistic quantities, such as invariant measures, Liapunov exponents, correlation functions, and exit times. The behavior of these quantities near the deterministic bifurcation point changes for distinct values of the control parameter. Therefore the very concept of bifurcation point becomes meaningless and must be replaced by the notion of bifurcation region.     

A dynamical system approach to SOC models of Zhang's type

We discuss Zhang's model of SOC in the framework of hyperbolic dynamical systems with singularities. The fractal structure of the invariant energy distribution, correlation decay-like phenomena, and symbolic coding are discussed.     

The stochastic bifurcation behaviour of speculative financial markets

This paper establishes a continuous-time stochastic asset pricing model in a speculative financial market with fundamentalists and chartists by introducing a noisy fundamental price. By application of stochastic bifurcation theory, the limiting market equilibrium distribution is examined numerically. It is shown that speculative behaviour of chartists can cause the market price to display different forms of equilibrium distributions. In particular, when chartists are less active, there is a unique equilibrium distribution which is stable. However, when the chartists become more active, a new equilibrium distribution will be generated and become stable. The corresponding stationary density will change from a single peak to a crater-like density. The change of stationary distribution is characterized by a bimodal logarithm price distribution and fat tails. The paper demonstrates that stochastic bifurcation theory is a useful tool in providing insight into various types of financial market behaviour in a stochastic environment.     

Nonequilibrium Phenomena: Outlines and bibliographies of a workshop

The following is a set of outlines and bibliographies for lectures presented at a Summer Workshop on Nonequilibrium Phenomena held from June 22 to July 3, 1981 at the Institute for Theoretical Physics in Santa Barbara. These outlines were distributed to the participants in lieu of formal proceedings, and they are being presented for publication in the same form, in the belief that the information they contain will be useful to a wider audience. It should be clearly stated, however, that the compilation is an informal one which does not claim to be a complete survey of the subject.     

Analysis of grazing bifurcations of quasiperiodic system attractors

This paper presents the first application of the discontinuity-mapping approach to the study of near-grazing bifurcations of originally quasiperiodic, co-dimension-two system attractors. The paper establishes an exact formulation for the discontinuity-mapping methodology under the assumption that a Poincaré section can be found that is everywhere transversal to the grazing attractor. In particular, it is shown that, while a reduced formulation may be employed successfully in the case of co-dimension-one attractors, it fails to capture dynamics in directions transversal to the original quasiperiodic attractor. This shortcoming necessitates the full machinery presented here. The generality of the proposed approach is illustrated through numerical analysis of two nonlinear dynamical systems of dimension three and four.     

Universal multifractal properties of circle maps from the point of view of critical phenomena I. Phenomenology

The strange attractor for maps of the circle at criticality has been shown to be characterized by a remarkable infinite set of exponents. This characterization by an infinite set of exponents has become known as the multifractal approach. The present paper reformulates the multifractal properties of the strange attractor in a way more akin to critical phenomena. This new approach allows one to study the universal properties of both the critical point and of its vicinity within the same framework, and it allows universal properties to be extracted from experimental data in a straightforward manner. Obtaining Feigenbaum's scaling function from the experimental data is, by contrast, much more difficult. In addition to the infinite set of exponents, universal amplitude ratios here appear naturally. To study the crossover region near criticality, a correlation time, which plays a role analogous to the correlation length in critical phenomena, is introduced. This new approach is based on the introduction of a joint probability distribution for the positive integer moments of the closest-return distances. This joint probability distribution is physically motivated by the large fluctuations of the multifractal moments with respect to the choice of origin. The joint probability distribution has scaling properties analogous to those of the free energy close to a critical point.     

A cell dynamical system model of chemical turbulence

A cellular-automaton-like caricature of chemical turbulence on an infinite one-dimensional lattice is studied. The model exhibits apparently turbulent space-time patterns. To make this statement precise, the following problems or points are discussed: (1) The infinite-system-size limit of such cell-dynamical systems and its observability is defined. (2) It is proved that the invariant state in the large-system-size limit of the turbulent phase exhibits spatial patterns governed by a Gibbs random field. (3) Potential characteristics of turbulent space-time patterns are critically surveyed and a working definition of (weak) turbulence is proposed. (4) It is proved that the invariant state of the turbulent phase is actually (weak) turbulent. Furthermore, we conjecture that the turbulent phase of our model is an example of a K system that is not Bernoulli.     

Gibbsian Dynamics and Invariant Measures for Stochastic Dissipative PDEs

We present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative PDEs. It consists of two main steps. The first step is the reduction to a finite dimensional Gibbsian dynamics of the low modes. The second step is to prove the equivalence between measures induced by different past histories using Girsanov theorem. As applications, we prove ergodicity for Ginzburg–Landau, Kuramoto–Sivashinsky and Cahn–Hilliard equations with stochastic forcing.     

Accelerator driven systems from the radiological safety point of view

In the proposed accelerator driven systems (ADS) the possible use of several milliamperes of protons of about 1 GeV incident on high mass targets like the molten lead-bismuth eutectic is anticipated to pose radiological problems that have so far not been encountered by the radiation protection community. Spallation reaction products like high energy gammas, neutrons, muons, pions and several radiotoxic nuclides including Po-210 complicate the situation. In the present paper, we discuss radiation safety measures like bulk shielding, containment of radiation leakage through ducts and penetration and induced activity in the structure to protect radiation workers as well as estimation of sky-shine, soil and ground water activation, release of toxic gases to the environment to protect public as per the stipulations of the regulatory authorities. We recommend the application of the probabilistic safety analysis technique by assessing the probability and criticality of different hazard-initiating events using HAZOP and FMECA.      

An Alternative Approach to Obtain the Exact Wave Functions of Time-Dependent Hamiltonian Systems Involving Quadratic, Inverse Quadratic, and (1/x)p+p(1/x) Terms

By using the Lewis-Riesenfeld theory and algebraic method, we present an alternative approach to obtain the exact solution of time-dependent Hamiltonian systems involving quadratic, inverse quadratic and (1/x)p+p(1/x) terms. This solution is discussed and compared with that obtained by Choi, J. R. (2003). International Journal of Theoretical Physics 42, 853]. PACS: 03.65Ge; 03.65Fd; 03.65Bz     

Invariant curves for area-preserving twist maps far from integrable

One-parameter families of area-preserving twist maps of the formF (x, y)=(x +y +f(x),y +f(x)) are considered. Various invariant curves, for the maps corresponding tof(x)=sin andf(x)=sinx+(1/50) sin(5x), are rigorously constructed forlarge values of the nonlinearity parameter . For larger values of , close to critical, some numerical experiments are briefly discussed.     

Exact invariants and adiabatic invariants of dynamical system of relative motion

Based on the theory of symmetries and conserved quantities, the exact invariants and adiabatic invariants of a dynamical system of relative motion are studied. The perturbation to symmetries for the dynamical system of relative motion under small excitation is discussed. The concept of high-order adiabatic invariant is presented, and the form of exact invariants and adiabatic invariants as well as the conditions for their existence are given. Then the corresponding inverse problem is studied.     

Experimental observation of noise-induced sensitivity to small signals in a system with on-off intermittency

An experimental (electronic circuit) realization and analytic studies of overdamped Kramers oscillator with an exponential nonlinearity under combined effect of a large multiplicative noise and a small periodic signal were performed. Under certain conditions, when the system exhibits on-off intermittency, it becomes sensitive to very small periodic signals, amplifying them greatly. Received 21 May 1999 and Received in final form 28 December 1999     

To understand matter from a point of view of clusters

We present an optimum valence bond scheme to study important stable geometric structures of clusters, by combining the characteristics of frontier molecular orbitals and the first-principle molecular dynamics simulation. It is interesting to note that even for small size clusters it has already provided clues in some macroscopic properties of the second- and the third-row elements in the period table (e.g. melting and boiling point).     

BEC from a time-dependent variational point of view

We use the time-dependent variational principle of Balian and Vénéroni to derive a set of equations governing the dynamics of a trapped Bose gas at finite temperature. We show that this dynamics generalizes the Gross-Pitaevskii equations in that it introduces a consistent dynamical coupling between the evolution of the condensate density, the thermal cloud, and the “anomalous” density.     

Geometry from the spectral point of view

In this Letter, we develop geometry from a spectral point of view, the geometric data being encoded by a triple (A. H. D.) of an algebraA represented in a Hilbert spaceH with selfadjoint operatorD. This point of view is dictated by the general framework of noncommutative geometry and allows us to use geometric ideas in many situations beyond Riemannian geometry.

This paper is dedicated to the memory of J. Schwinger     

Bifurcations of a lattice gas flow under external forcing

We study the behavior of a Frisch-Hasslacher-Pomeau lattice gas automaton under the effect of a spatially periodic forcing. It is shown that the lattice gas dynamics reproduces the steady-state features of the bifurcation pattern predicted by a properly truncated model of the Navier-Stokes equations. In addition, we show that the dynamical evolution of the instabilities driving the bifurcation can be modeled by supplementing the truncated Navier-Stokes equation with a random force chosen on the basis of the automaton noise.