A note on perturbed fixed slope iterations

AbstractAn approximation to the exact derivative leads to perturbed fixed slope iterations in the context of Inexact Newton methods. We prove an a posteriori convergence theorem for such an algorithm, and show an application to nonlinear differential boundary value problems. The abstract setting is a complex Banach space

Noble internal transport barriers and radial subdiffusion of toroidal magnetic lines

Single trajectories of magnetic line motion indicate the persistence of a central protected plasma core, surrounded by a chaotic shell enclosed in a double-sided transport barrier : the latter is identified as being composed of two Cantori located on two successive "most-noble" numbers values of the perturbed safety factor, and forming an internal transport barrier (ITB). Magnetic lines which succeed to escape across this barrier begin to wander in a wide chaotic sea extending up to a very robust barrier (as long as L<1) which is identified mathematically as a robust KAM surface at the plasma edge. In this case the motion is shown to be intermittent, with long stages of pseudo-trapping in the chaotic shell, or of sticking around island remnants, as expected for a continuous time random walk.Comment: TEX file, 84 pages including 32 color figures. Higher quality figures can be seen on the PDF file at http://membres.lycos.fr/fusionbfr/JHM/Tokamap/JSP.pd

The Edge of Quantum Chaos

We identify a border between regular and chaotic quantum dynamics. The border is characterized by a power law decrease in the overlap between a state evolved under chaotic dynamics and the same state evolved under a slightly perturbed dynamics. For example, the overlap decay for the quantum kicked top is well fitted with $[1+(q-1) (t/\tau)^2]^{1/(1-q)}$ (with the nonextensive entropic index $q$ and $\tau$ depending on perturbation strength) in the region preceding the emergence of quantum interference effects. This region corresponds to the edge of chaos for the classical map from which the quantum chaotic dynamics is derived.Comment: 4 pages, 4 figures, revised version in press PR

Evaluating the impact of binary parameter uncertainty on stellar population properties

Binary stars have been shown to have a substantial impact on the integrated light of stellar populations, particularly at low metallicity and early ages - conditions prevalent in the distant Universe. But the fraction of stars in stellar multiples as a function of mass, their likely initial periods and distribution of mass ratios are all known empirically from observations only in the local Universe. Each has associated uncertainties. We explore the impact of these uncertainties in binary parameters on the properties of integrated stellar populations, considering which properties and timescales are most susceptible to uncertainty introduced by binary fractions and whether observations of the integrated light might be sufficient to determine binary parameters. We conclude that the effects of uncertainty in the empirical binary parameter distributions are likely smaller than those introduced by metallicity and stellar population age uncertainties for observational data. We identify emission in the He II 1640Å emission line and continuum colour in the ultraviolet-optical as potential indicators of a high mass binary presence, although poorly constrained metallicity, dust extinction and degeneracies in plausible star formation history are likely to swamp any measurable signal

Stochastic perturbations in open chaotic systems: random versus noisy maps

We investigate the effects of random perturbations on fully chaotic open systems. Perturbations can be applied to each trajectory independently (white noise) or simultaneously to all trajectories (random map). We compare these two scenarios by generalizing the theory of open chaotic systems and introducing a time-dependent conditionally-map-invariant measure. For the same perturbation strength we show that the escape rate of the random map is always larger than that of the noisy map. In random maps we show that the escape rate $\kappa$ and dimensions $D$ of the relevant fractal sets often depend nonmonotonically on the intensity of the random perturbation. We discuss the accuracy (bias) and precision (variance) of finite-size estimators of $\kappa$ and $D$, and show that the improvement of the precision of the estimations with the number of trajectories $N$ is extremely slow ($\propto 1/\ln N$). We also argue that the finite-size $D$ estimators are typically biased. General theoretical results are combined with analytical calculations and numerical simulations in area-preserving baker maps.Comment: 12 pages, 3 figures, 1 table, manuscript submitted to Physical Review