Hao Bai-lin

Elementary Symbolic Dynamics

In a sense the method of symbolic dynamics(1) is the only mathematically rigorous approach to the study of chaotic behavior in dynamical systems. However, having its roots in the topological theory of dynamical systems, symbolic dynamics remains a rather abstract chapter of mathematics and still seems to be useless for a practical physicist. The aim of this chapter is to show that there exists an elementary way to use at least a small “subset” of the method of symbolic dynamics to help the practical physicist in dealing with his data from computer or laboratory experiments. The adjective “elementary” means we shall use elementary mathematics only.

Scaling Property of Period-n-Tupling Sequences in One-Dimensional Mappings

We calculated the universal scaling function and the scaling factor as well as the convergence rate for periodtripling, -quadrapling and–quintupling sequences of and types. The superstable periods are closely connected to a set of polynomial defined recursively by the original mapping. Some notable properties of these polynomials are studied. Several approaches to solving the renormalization group equation and estimating the scaling factors are suggested.

Dimensions of the limiting sets of period-n-tupling sequences

We calculated the q-th order information dimension Dq and the spectrum f(α) of singularities in the measure for the limiting sets of period-n-tupling sequences in the case of the logistic map.

Analytical study of a bimodal mapping related to a hybrid optical bistable device using liquid crystal

We suggest a method to determine analytically the parameter values of all superstable periods and the equations of dark lines seen in various bifurcation diagrams, plotted for a bimodal mapping related to a hybrid optical bistable device in which liquid crystal was used as the nonlinear medium.

Hierarchy of Chaotic Bands and Periodicities Embedded in Them in a Forced Nonlinear Oscillator

Secondary direct and inverse bifurcation sequences exist in the chaotic bands of the primary inverse sequence. In secondary chaotic bands periodicities of the tertiary appear. This has been shown numerically in the forced Brusselator.

Analytical Study of a Bimodal Map Related to Optical Bistability

Using an analytical method proposed originally for the logistic map, we studied in detail the bifurcation diagram of a bimodal map related to a hybrid optical bistability device with liquid crystal as the nonlinear medium. In particular, we gave the equations of all the dark lines going through the chaotic region and all boundaries of the chaotic bands. We showed how to determine the parameter value for a given superstable period from corresponding word in the symbolic dynamics made of four letters.

On Theory of the Statistical Generating Functional for the Order Parameter(I) — General Formalism

A theoretical scheme using closed time-path Greens functions is proposed to describe the quantum statistical properties of the order parameter in terms of a generating functional. The dynamic evolution, is generated by a driving source, while the statistical correlation by a fluctuation source. The statistical causality is shown to hold explicitly and to give rise to a number of important consequences. The problem of determining the quantum statistical properties for the order parameter is reduced to finding a solution of the functional equation for it.

On Theory of the Statistical Generating Functional For the Order Parameter (II) —Density Matrix and the Field-Theoretical Structure of the Generating Functional

Several equivalent expressions for the generating functional of the order parameter are derived in this paper to elucidate how the statistical properties are incorporated in its field-theoretical structure. It is shown that in the process of determining the generating functional the dynamic evolution and the statistical information can be separated to a certain extent which facilitates solving the problem. The whole procedure is greatly simplified if the statistical correlation is Gaussian or a generalized fluctuation-dissipation theorem (FDT) holds. As an example, the Gaussian character of the thermal-equilibrium distribution is justified and the Matsubara technique along the imaginary time axis is extended to the real axis to provide a complete perturbation scheme for the closed time-path Greens functions (CTPGF).

On Theory of the Statistical Generating Functional for the Order Parameter (III) –Effective Action Formalism for the Order Parameter

A practical scheme is proposed in this paper to determine the quantum statistical properties of the order parameter in the frame work of the closed time-path Greens functions (CTPGF). As a microscopic quantum theory, this formalism is applicable in principle to both equilibrium and nonequilibrium phenomena and is capable of dealing with statistical systems both above and below their phase transition point. This formalism can be used to discuss the statistical properties of simple as well as composite order parameters in either uniform or nonuniform systems. As simple illustrations and check for the theory, the proposed prescription is applied to the contact-interaction model for superconductivity and a unimode laser system coupled with two-energy-level bound electrons.

THE KNOWLEDGE-BASED SYSTEM GRAPE AND ITS APPLICATION TO LANDAU THEORY ANALYSIS FOR MAGNETIC SPACE-GROUPS

Abstract A knowldge-based project, the GRAPE system(Group Representation and Application in Physics Environment), is described in this paper. The GRAPE system is designed to provide physicists with a group theoretical environment to help them solve problems in group theory and representation. The user can communicate with GRAPE in plain English. At the present stage, it contains the knowledge of crystallography point groups, space groups as well as magnetic space groups both in group structure and group representations. The GRAPE system consists of five modules besides the knowledge base and the data base: a natural language interface, a computation module, a tutprial module, a bibliography module, and a program library. Group theoretical analysis for the Landau theory of continuous phase transitions has been the first application of the GRAPE system. The calculation for determining directions of phase transition at the Γ point for 230 space groups, 230 grey space groups and 674 black and white magnetic space groups has been performed.