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We give a straightforward derivation of formula (2), argued in a recent preprint of Hasley et al.[2]

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 Green's functions (CTPGF).

In the diagonal subspace of the Liapounov representation introduced by the Brussels group, the dominant terms for any matrix element of the time evolution operator are given explicitly if the Hamiltonian satisfies some general conditions held in all models studied so far. All these expressions have semi-definite positive forms so that, when they are different from zero for a particular isolated system, the "microscopic entropy" will increase monotonically until the system approaches the well-known microcanonical equilibrium.

The width of mode-locking tongues shrinks to zero at well-defined points in a piecewise linear circle map. The dimension of the set of quasiperiodic points along its critical line appears to be zero as compared to the "universal" value . Many properties of this map can be described analytically by rational fractions.

The Lorenz model and the antisymmetric cubic map enjoy the same discrete symmetry. A careful study of a one-dimensional bifurcation diagram obtained numerically from the Lorenz model reveals that the systematics of periodic windows is closely related to that of the cubic map. In addition, we determined the drift of the fundamental frequency and thus proposed a nomenclature for the periodic windows by associating an absolute period to each window, which in turn agrees with the corresponding word made of three letters for most of the observed periods.

A theoretical scheme using closed time-path Green's 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.

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.

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.

We show numerical evidence of exact U-sequences in the periodically forced trimolecular model (the forced Brusselator). Interspersed among period-doubling bifurcation sequences star-ting with RLn type periods, there are chaotic regions bounded on one side by period-doubling bifurcation sequences and on the other side by intermittent transitions. Along certain directions in the parameter space the most clearly seen periods appear in the same order as that in the logistic map, but along other directions the U-sequences may fold and give rise to deviations from the standard patterns. Our results show the coexistence of different "routes to chaos" in one and the same mathematical model and the necessity to enlarge the parameter space in both real and computer experiments on chaotic transitions.

A delay-differential system related to or device is discussed in this paper. We obtained the instability conditions for the steady state and the critical relations among the parameters and for the modes by solving the characteristic equation of the system. Based on the numerical result the chaos and bifurcation "spectrum" in the plane was shown for a simple model equation when higher modes were neglected. We discovered the coexistence of two different kinds of attractors for this equation in a certain region of the parameter plane.

There exists a simple scaling function relating the "operational" Hausdorff Dimension D=2/3 at each bifurcation point μk, k finite, to the universal Hausdorff dimension D∞ =0.538 . . . at μ∞ for the limiting sets of 1-dimensional unimodal mappings

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 Green's 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.

In this paper we suggest a semi-microscopic model to describe the phenomenon of acoustical activity and to extend the point group symmetry consideration first given by Portigal and Burstein.

A new way of constructing the steady state solutions of the Fokker-Planck Equations(FPE's) is described in which the concept of a vortex field plays an important role. The steady state solutions can be classified according to the type of . In particular, one easily obtains almost all the exact global solutions known so far for the stationary FPE's, as well as the exact local solutions near the fixed points of the corresponding deterministic equations. In this way some new properties of FPE are explored.

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.