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The paper introduces a one dimensional analogy of Poincare's "section" method. Its application to numerical study of sequences of fractional parts is considered. As a result, the possibility of Cantorian structure of time flow is deduced. A possible way of interrelating the time flow and time perception, is discussed.
By means of an updated renormalization method, we construct asymptotic expansions for unstable manifolds of hyperbolic fixed points in the double-well map and the dissipative Hénon map, both of which exhibit the strong homoclinic chaos. In terms of the asymptotic expansion, a simple formulation is presented to give the first homoclinic point in the double-well map. Even a truncated expansion of the unstable manifold is shown to reproduce the well-known many-leaved (fractal) structure of the strange attractor in the Hénon map.
After having closely re-examined the notion of a Lévy's stable vector, it is shown that the notion of a stable multivariate distribution is more general than previously defined. Indeed, a more intrinsic vector definition is obtained with the help of non isotropic dilations and a related notion of generalized scale. In this framework, the components of a stable vector may not only have distinct Levy's stability indices $α$'s, but the latter may depend on its norm. Indeed, we demonstrate that the Levy's stability index of a vector rather correspond to a linear application than to a scalar, and we show that the former should satisfy a simple spectral property.
We introduce a simple model for DNA evolution. Using the method of Peng et al.$^1$, we investigate the fractal properties of the system. For small chains and chains of intermediate size we find a fractal exponent that indicates the existence of long-range correlations, as in real DNA sequences. However, when very large chains are studied the fractal exponent asymptotically converge to the value of a random sequence. We verify that the mutations are responsible for the apparent existence of long-range correlations.
Nanometer scale electronics present a challenge for the computer architect. These quantum devices have small gain and are difficult to interconnect. I have analyzed current device capabilities and explored two general design requirements for the design of computers: error correction and long range connections. These two principles follow when Turing machines are implemented as integrated circuits. I consider the roles of electromigration through thin wires, circuit layout, and error rates for devices with small gain. The analysis brings into sharp focus the future of nanocomputers and suggests solutions to some of its difficulties. It gives a theoretical model for a nanocomputer, separating the roles of devices and algorithms. Within the model one can implement a stochastic computer, which operates despite quantum device limitations.
We propose an information-theoretic statistical model to describe the universal features of those chaotic scattering processes characterized by a prompt and an equilibrated component. The model, introduced in the past in nuclear physics, incorporates the average value of the scattering matrix to describe the prompt processes, and satisfies the requirements of flux conservation, causality, and ergodicity. We show that the model successfully describes electronic transport through chaotic quantum dots. The predicted distribution of the conductance may show a remarkable two-peak structure.
In this letter, I solve a model for the dynamics of vortices in a decaying two-dimensional turbulent fluid. The model describes their effective diffusion, and the merging of pairs of vortices of same vorticity sign, when they get too close. The merging process is characterized by the conservation of energy and of the quantity $Nr^n$, where $r$ is the mean vortex radius, and $N$ their number. $n=4$ corresponds to a constant peak vorticity, and $n=2$ to a constant kurtosis. I found the scaling laws for various physical quantities ($r$, enstrophy, kurtosis...), and for instance, it is shown that $N\sim (t/\ln(t))^{-\frac{2n}{3n-4}}$ for $n>2$, and $N\sim t^{-2}$ for $n=2$, in good agreement with extensive numerical simulations. I also discuss some recent experiments in view of these results.
We consider the quantum mechanical behavior of a driven particle in an infinite 1D potential well. We show that the time dependent perturbation series is induced by the delicate non-trivial properties of the momentum operator in this case, namely, its non-self-adjointness. Using this expansion, we calculate the first order contribution to the cross section and the energy gain, and discuss their classical limit. In this limit the one-period energy gain converges to its classical analog - the classical local (momentum space) diffusion coefficient. Both the classical and quantum mechanical results are compared with numerical simulations.
We generalize a method of control of chaos which uses delayed feedback at the period of an unstable orbit to stabilize that orbit. The generalization consists of substituting some portion of the nonlinear dynamical system with a delayed dynamics, rather than using a linear delay function for control. A further generalization, in which the control function retains memory of all previous periods, allows the region of the parameter space over which control can be achieved to be extended, but at the price of losing the ability to achieve superstability. Nonliner feedback results in a larger basin of attraction to the stabilized orbit than linear feedback. For a simple test mapping studied (the logistic map) the dimension of the system increases from one to two by introducing control. We show in the case involving memory, for a particular choice of the relationship between the control parameters, that the superstable orbit can be recovered without reducing the parameter space that can be controlled. This particular solution, in addition to having the largest basin of attraction of the methods considered, retains the dimension of the uncontrolled system.
We study the electronic band structure for a model one-dimensional periodic potential in the presence of a spacially homogeneous laser field. The statistical properties of the energy bands depend on the coupling between the crystal and the laser field, going from Poisson to Wigner-Dyson (GOE) and back to Poisson as the coupling increases. We argue that the classical dynamics of this system resembles that of a periodically driven pendulum. We find that the chaotic regime is not restricted solely to high-lying bands and should thus be of easier access to optical experiments.
We show that the condition for the appearance of quantum chaos (Wigner-Dyson distribution of energy eigenvalues, gaussian-random energy eigenfunctions) in a dilute gas of many hard spheres is $λ\ll \ell$, where $λ$ is the wavelength of a typical particle and $\ell$ is the mean free path between collisions. For fermions with Fermi wavelength $λ_F \ll \ell$, this implies that all energy eigenstates, including the ground state, are chaotic. Physical implications are discussed.
Lecture given at the International Meeting ``Boltzmann's Legacy - 150 Years after his Birth'', organized by the Accademia Nazionale dei Lincei, 25 - 28 May 1994, in Rome, to be published in: ``Atti dell"Accademia Nazionale dei Lincei'', 1997.
Closed expressions are derived for the resonance widths and Coulomb blockade conductance peak heights in quantum dots for the crossover regime between conserved and broken time-reversal symmetry. The results hold for leads with any number of possibly correlated and inequivalent channels. Our analytic predictions are in good agreement with simulations of both random matrices and a chaotic billiard with a magnetic flux line.
In this paper we will give a short rewiew about the conductance of a mesoscopic waveguide strip with a few impurities. Green-function method is used allowing to treat systems with low number of impurities, not only the completly clean case. Investigating a wire containing Dirac-delta potentials (modelling the impurities) we found that increasing number of impurities can cause a transition of the structure of conductance. The staircase like structure of the clean system vanishes and the conductance of a system containing finite number of impurities located periodically will be determined by the band structure of
Scaling properties of the BUX index are similar to those observed in other parts of the world. The main difference is that the traditional quantities like volatility, growth and autocorrelation of returns follows more closely the assumptions of the traditional stock market theory developed by Bachelier and by Black and Scholes.
This is an extended version of the talk given by the author at the NATO Advanced Study Institute "Supersymmetry and Trace Formulae" (Isaac Newton Institute, Cambridge, UK, 8-19 Septemebr 1997). Contains the following sections: 1. Eigenfunction correlations in the weak localization regime. 2. Strong correlations of eigenfunctions and level repulsion at the Anderson localization transition. 3. Fluctuations in the addition spectra of quantum dots.
An analytical expression for the maximal Lyapunov exponent $λ_1$ in generalized Fermi-Pasta-Ulam oscillator chains is obtained. The derivation is based on the calculation of modulational instability growth rates for some unstable periodic orbits. The result is compared with numerical simulations and the agreement is good over a wide range of energy densities $ε$. At very high energy density the power law scaling of $λ_1$ with $ε$ can be also obtained by simple dimensional arguments, assuming that the system is ruled by a single time scale. Finally, we argue that for repulsive and hard core potentials in one dimension $λ_1 \sim \sqrtε$ at large $ε$.
We give a microscopic representation of the stock-market in which the microscopic agents are the individual traders and their capital. Their basic dynamics consists in the auto-catalysis of the individual capital and in the global competition/cooperation between the agents mediated by the total wealth invested in the stock (which we identify with the stock-index). We show that such systems lead generically to (truncated) Pareto power-law distribution of the individual wealth. This, in turn, leads to intermittent market (short time) returns parametrized by a (truncated) Levy distribution. We relate the truncation in the Levy distribution to the (truncation in the Pareto Power Law i.e. to the) fact that at each moment no trader can own more than the current total wealth invested in the stock. In the cases where the system is dominated by the largest traders, the dynamics looks similar to noisy low-dimensional chaos. By introducing traders memory and/or feedback between individual and collective wealth fluctuations (the later identified with the stock returns), one obtains clustered "volatility" as well as market booms and crashes. The basic feedback loop consists in: - computing the market price of the stock as the sum of the individual wealths invested in the stock by the traders and - determining the time variation of the individual trader wealth as his/her previous wealth multiplied by the stock return (i.e. the variation of the stock price).