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A selfcontained proof of the KAM theorem in the Thirring model is discussed, completely relaxing the ``strong diophantine property'' hypothesis used in previous papers. Keywords: \it KAM, invariant tori, classical mechanics, perturbation theory, chaos
We discuss various experiments on the time decay of velocity autocorrelation functions in billiards. We perform new experiments and find results which are compatible with an exponential mixing hypothesis, first put forward by [FM]: they do not seem compatible with the stretched exponentials believed, in spite of [FM], to describe the mixing. The analysis led us to several byproducts: we obtain information about the normal diffusive nature of the motion and we consider the probability distribution of the number of collisions in time $t_m$ (as $t_m\to\io$) finding a strong dependence on some geometric characteristics of the locus of the billiards obstacles.
The Jupiter-Saturn 2:5 near-commensurability is analyzed in a fully analytic Hamiltonian planetary theory. Computations for the Sun-Jupiter-Saturn system, extending to the third order of the masses and to the 8th degree in the eccentricities and inclinations, reveal an unexpectedly sensitive dependence of the solution on initial data and its likely nonconvergence. The source of the sensitivity and apparent lack of convergence is this near-commensurability, the so-called great inequality. This indicates that simple averaging, still common in current semi-analytic planetary theories, may not be an adequate technique to obtain information on the long-term dynamics of the Solar System. Preliminary results suggest that these difficulties can be overcome by using resonant normal forms.
The quantum density of states of the Hénon-Heiles potential displays a pronounced beating pattern. This has been explained by the interference of three isolated classical periodic orbits with nearby actions and periods. A singular magnetic flux line, passing through the origin, drastically alters the beats even though the classical Lagrangean equations of motion remain unchanged. Some of the changes can be easily understood in terms of the Aharonov-Bohm effect. However, we find that the standard periodic orbit theory does not reproduce the diffraction-like quantum effects on those classical orbits which intersect the singular flux line, and argue that corrections of relative order $\hbar$ are necessary to describe these effects. We also discuss the changes in the distribution of nearest-neighbour spacings in the eigenvalue spectrum, brought about by the flux line. pacs{PACS numbers: 05.45.+b, 02.50.+s,03.65.-w,03.65.Ge }
We establish a formula relating global diffusion in a space periodic dynamical system to cycles in the elementary cell which tiles the space under translations.
It is demonstrated, by means of analogue electronic simulation and theoretically, that external noise can markedly change the character of the response of a nonlinear system to a low-frequency periodic field. In general, noise of sufficient intensity {\it linearises} the response. For certain parameter ranges in particular cases, however, an increase in the noise intensity can sometime have the opposite effect and is shown to {\it delinearise} the response. The physical origins of these contrary behaviours are discussed.
A remarkable new definition of a self-delimiting universal Turing machine is presented that is easy to program and runs very quickly. This provides a new foundation for algorithmic information theory. This new universal Turing machine is implemented via software written in Mathematica and C. Using this new software, it is now possible to give a self-contained ``hands on'' mini-course presenting very concretely the latest proofs of the fundamental information-theoretic incompleteness theorems.
In the present paper which is a sequel to [N.B. Volkov and A.M. Iskoldsky The dynamics of vortex structures and states of current: 1;[1]], the dynamics of non-equilibrium phase transitions and states of current in electrophysical systems providing an external circuit and a nonlinear element the model of which has been developed in [1] (in the development of the model local kinetic transport coefficients were assumed to be constant), has been analyzed and simulated. A non-equilibrium phase transition has been shown to be induced by large-scale hydrodynamic fluctuations (vortex structures). Critical exponents of the amplitude singular behavior (the order parameters) for three types of circuits have been determined. Non-equilibrium phase transitions induced by external harmonic noise with a random or a determined phase have been studied. The detuning between an external noise frequency and natural frequencies of a nonlinear element determined by large-scale hydrodynamic fluctuations have been shown to give rise to random oscillations, which are typical of a strange attractor. When the frequencies coincide, a limit cycle appears in the system, which is characterized by the fact that the phase trajectory does not fill the phase space completely.
A short historical overview is given on the development of our knowledge of complex dynamical systems with special emphasis on ergodicity and chaos, and on the semiclassical quantization of integrable and chaotic systems. The general trace formula is discussed as a sound mathematical basis for the semiclassical quantization of chaos. Two conjectures are presented on the basis of which it is argued that there are unique fluctuation properties in quantum mechanics which are universal and, in a well defined sense, maximally random if the corresponding classical system is strongly chaotic. These properties constitute the quantum mechanical analogue of the phenomenon of chaos in classical mechanics. Thus quantum chaos has been found.
Bifurcations in a system of coupled maps are investigated. Using symbolic dynamics it is proven that for coupled shift maps the well known space--time--mixing attractor becomes unstable at a critical coupling strength in favour of a synchronized state. For coupled non--hyperbolic maps analytical and numerical evidence is given that arbitrary small coupling changes the dynamical behaviour. The anomalous dependence of fluctuations on the system size is attributed to these bifurcations.
Voronin's theorem on the `Universality'' of Riemann zeta function is shown to imply that Riemann zeta function is a fractal (in the sense that Mandelbrot set is a fractal) and a concrete ``representation'' of the ``giant book of theorems'' that Paul Halmos referred to.
In order to study the stability of mode-I fracture, we consider a crack moving along the centerline of a very wide strip and compute its steady-state response to a small, spatially periodic shear stress. We find that, in the presence of this perturbation, the crack remains very nearly straight up to a critical speed $v_c$ of about $0.8\,v_R$, where $v_R$ is the Rayleigh speed. Deviations from straight-line propagation are suppressed by a factor proportional to $W^{-1/2}$, where $W$ is the width of the strip. At $v_c$, however, this suppression disappears and the steady-state crack follows the wavy curve along which the shear stress vanishes in the unbroken strip. We interpret this behavior as a loss of stability and discuss its implications for a more complete dynamical theory of fracture propagation.
Discrete arrays of Josephson junction elements differ from their continuum counterparts in two essential ways: i) localized dynamic states in discrete arrays, which are not present in the corresponding continuum system, can interact with other excitations that are present; ii) fluxoid quantization for the non-superconducting `holes' provides a constraint for discrete arrays that is not present in the corresponding continuum system. The consequences of these effects in one-dimensional systems are now beginning to be understood; in two-dimesional systems, on the other hand, the picture is not yet altogether clear. Progress in fabrication technology and potential applications in practical electronic devices -- as well as intrinsic interest in nonlinear dynamics -- have contributed significantly to the growing interest in these systems.
Velocity increments over a distance r and turbulent energy dissipation on a box of size r are well described by the multifractal models of fully developed turbulence. These quantities and models however, do not involve time-correlations and therefore are not a detailed test of the dynamics of the turbulent cascade. If the time development of the turbulent cascade, in the inertial range, is related to the lifetime of the eddies at different length scales, the time correlations may be described by a stochastic process on a tree with jumping kernels which are a function of the ultrametric (tree) distance. We obtain the solutions of the Chapman-Kolmogorov equation for such a stochastic process, with jumping kernels depending on the ultrametric distance, but with an arbitrarily specified invariant probability measure. We then show how to use these solutions to compute the time correlations in the turbulent cascade.
It is shown that statistical properties of developed hydrodynamic turbulence are characterized by an infinite set of independent anomalous exponents which describes the scaling behavior of hydrodynamic fields constructed from the second and larger powers of the velocity derivatives. A physical mechanism responsible for anomalous scaling, ``telescopic multi-step eddy interaction", is discovered and investigated. The essence of this mechanism is the existence of a very large number $(R/η)^{Δ_j}\gg 1$ of channels of interaction of large eddies of scale $R$ in the inertial interval with eddies of viscous scale $η$ via a set of eddies of all intermediate scales between $R$ and $η$. The description of this mechanism based on the NS equation in the quasi Lagrangian representation is presented. In the diagrammatic expansion of the correlation function of the energy dissipation field $K_ {\varepsilon \varepsilon}(R)$, we have found an infinite series of logarithmically diverging diagrams. Their summation leads to a renormalization of the normal Kolmogorov-41 dimensions. For a description of the scaling of various hydrodynamic fields an infinite set of primary fields $O_n$ with independent scaling exponents $Δ_n$ was introduced. We have proposed a symmetry classification of the fields $O_n$ enabling one to predict relations between scaling the behavior of different correlation functions.
The problem considered is the computation of an infinite product (composition) of Lie transformations generated by homogeneous polynomials of increasing order from a given convergent power series. Bounds are computed for the infinitesimal form of Lie transformations. The results obtained do not guarantee convergence of the product. Instead, the optimum truncation is determined by minimizing the terms of order n+1 that remain after the first n Lie transformations have been applied.
The (forced) periodic orbit spectrum is calculated for a chaotic time series (BZ reaction).
Some misleading statements in a recent paper in PRE are corrected.