The Definition and Measurement of the Topological Entropy per Unit Volume in Parabolic PDE's

We define the topological entropy per unit volume in parabolic PDE's such as the complex Ginzburg-Landau equation, and show that it exists, and is bounded by the upper Hausdorff dimension times the maximal expansion rate. We then give a constructive implementation of a bound on the inertial range of such equations. Using this bound, we are able to propose a finite sampling algorithm which allows (in principle) to measure this entropy from experimental data.Comment: 26 pages, 1 small figur

Complexity for extended dynamical systems

We consider dynamical systems for which the spatial extension plays an important role. For these systems, the notions of attractor, epsilon-entropy and topological entropy per unit time and volume have been introduced previously. In this paper we use the notion of Kolmogorov complexity to introduce, for extended dynamical systems, a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems. We introduce this notion as an almost sure limit on orbits of the system. Moreover we prove a kind of variational principle for this complexity.Comment: 29 page

Inelastic O+H collisions and the OI 777nm solar centre-to-limb variation

The OI 777 nm triplet is a key diagnostic of oxygen abundances in the atmospheres of FGK-type stars; however it is sensitive to departures from local thermodynamic equilibrium (LTE). The accuracy of non-LTE line formation calculations has hitherto been limited by errors in the inelastic O+H collisional rate coefficients: several recent studies have used the so-called Drawin recipe, albeit with a correction factor $\mathrm{S_{H}}$ that is calibrated to the solar centre-to-limb variation of the triplet. We present a new model oxygen atom that incorporates inelastic O+H collisional rate coefficients using an asymptotic two-electron model based on linear combinations of atomic orbitals, combined with a free electron model, based on the impulse approximation. Using a 3D hydrodynamic stagger model solar atmosphere and 3D non-LTE line formation calculations, we demonstrate that this physically-motivated approach is able to reproduce the solar centre-to-limb variation of the triplet to 0.02 dex, without any calibration of the inelastic collisional rate coefficients or other free parameters. We infer $\log\epsilon_{\mathrm{O}}=8.69\pm0.03$ from the triplet alone, strengthening the case for a low solar oxygen abundance.Comment: 13 pages, 8 figures; published in Astronomy & Astrophysic

Effective temperature determinations of late-type stars based on 3D non-LTE Balmer line formation

Hydrogen Balmer lines are commonly used as spectroscopic effective temperature diagnostics of late-type stars. However, the absolute accuracy of classical methods that are based on one-dimensional (1D) hydrostatic model atmospheres and local thermodynamic equilibrium (LTE) is still unclear. To investigate this, we carry out 3D non-LTE calculations for the Balmer lines, performed, for the first time, over an extensive grid of 3D hydrodynamic STAGGER model atmospheres. For H$\alpha$, H$\beta$, and H$\gamma$, we find significant 1D non-LTE versus 3D non-LTE differences (3D effects): the outer wings tend to be stronger in 3D models, particularly for H$\gamma$, while the inner wings can be weaker in 3D models, particularly for H$\alpha$. For H$\alpha$, we also find significant 3D LTE versus 3D non-LTE differences (non-LTE effects): in warmer stars ($T_{\text{eff}}\approx6500$K) the inner wings tend to be weaker in non-LTE models, while at lower effective temperatures ($T_{\text{eff}}\approx4500$K) the inner wings can be stronger in non-LTE models; the non-LTE effects are more severe at lower metallicities. We test our 3D non-LTE models against observations of well-studied benchmark stars. For the Sun, we infer concordant effective temperatures from H$\alpha$, H$\beta$, and H$\gamma$; however the value is too low by around 50K which could signal residual modelling shortcomings. For other benchmark stars, our 3D non-LTE models generally reproduce the effective temperatures to within $1\sigma$ uncertainties. For H$\alpha$, the absolute 3D effects and non-LTE effects can separately reach around 100K, in terms of inferred effective temperatures. For metal-poor turn-off stars, 1D LTE models of H$\alpha$ can underestimate effective temperatures by around 150K. Our 3D non-LTE model spectra are publicly available, and can be used for more reliable spectroscopic effective temperature determinations.Comment: 19 pages, 10 figures, abstract abridged; accepted for publication in Astronomy & Astrophysic

Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds, and demonstrate $L^1$ convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.Comment: 34 page

Dynamical estimates of chaotic systems from Poincar\'e recurrences

We show that the probability distribution function that best fits the distribution of return times between two consecutive visits of a chaotic trajectory to finite size regions in phase space deviates from the exponential statistics by a small power-law term, a term that represents the deterministic manifestation of the dynamics, which can be easily experimentally detected and theoretically estimated. We also provide simpler and faster ways to calculate the positive Lyapunov exponents and the short-term correlation function by either realizing observations of higher probable returns or by calculating the eigenvalues of only one very especial unstable periodic orbit of low-period. Finally, we discuss how our approaches can be used to treat data coming from complex systems.Comment: subm. for publication. Accepted fpr publication in Chao

Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems

In this paper, we prove an inequality, which we call "Devroye inequality", for a large class of non-uniformly hyperbolic dynamical systems (M,f). This class, introduced by L.-S. Young, includes families of piece-wise hyperbolic maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas), unimodal and H{\'e}non-like maps. Devroye inequality provides an upper bound for the variance of observables of the form K(x,f(x),...,f^{n-1}(x)), where K is any separately Holder continuous function of n variables. In particular, we can deal with observables which are not Birkhoff averages. We will show in \cite{CCS} some applications of Devroye inequality to statistical properties of this class of dynamical systems.Comment: Corrected version; To appear in Nonlinearit

Simultaneous multi-band detection of Low Surface Brightness galaxies with Markovian modelling

We present an algorithm for the detection of Low Surface Brightness (LSB) galaxies in images, called MARSIAA (MARkovian Software for Image Analysis in Astronomy), which is based on multi-scale Markovian modeling. MARSIAA can be applied simultaneously to different bands. It segments an image into a user-defined number of classes, according to their surface brightness and surroundings - typically, one or two classes contain the LSB structures. We have developed an algorithm, called DetectLSB, which allows the efficient identification of LSB galaxies from among the candidate sources selected by MARSIAA. To assess the robustness of our method, the method was applied to a set of 18 B and I band images (covering 1.3 square degrees in total) of the Virgo cluster. To further assess the completeness of the results of our method, both MARSIAA, SExtractor, and DetectLSB were applied to search for (i) mock Virgo LSB galaxies inserted into a set of deep Next Generation Virgo Survey (NGVS) gri-band subimages and (ii) Virgo LSB galaxies identified by eye in a full set of NGVS square degree gri images. MARSIAA/DetectLSB recovered ~20% more mock LSB galaxies and ~40% more LSB galaxies identified by eye than SExtractor/DetectLSB. With a 90% fraction of false positives from an entirely unsupervised pipeline, a completeness of 90% is reached for sources with r_e > 3" at a mean surface brightness level of mu_g=27.7 mag/arcsec^2 and a central surface brightness of mu^0 g=26.7 mag/arcsec^2. About 10% of the false positives are artifacts, the rest being background galaxies. We have found our method to be complementary to the application of matched filters and an optimized use of SExtractor, and to have the following advantages: it is scale-free, can be applied simultaneously to several bands, and is well adapted for crowded regions on the sky.Comment: 39 pages, 18 figures, accepted for publication in A

Viscoelastic impact between a cylindrical striker and a long cylindrical bar

International audienceAxial impact between a cylindrical striker of finite length and a long cylindrical bar, both of linearly viscoelastic materials, is considered under uni-axial conditions. General results are derived for the impact force, the particle velocity and the strain in the bar in terms of closed-contour integrals suitable for numerical evaluation. Such results are derived also for the transfer of momentum and energy from the striker to the bar. Numerical results for elastic and viscoelastic impact of a striker and a bar with different cross-sectional areas are compared. In viscoelastic impact, unlike elastic impact, the duration of impact may be finite but larger than two transit times for a wave front through the striker due to the for-mation of a tail after the main pulse. Furthermore, repeated contacts and separations of the striker and the bar may occur within a range of striker-to bar characteristic impedance ratios smaller than one. In viscoelastic impact, the duration of impact is at least as long and the momentum and energy transferred are at most as large as in elastic impact. Strains measured at three locations of a PMMA bar impacted by PMMA strikers of three different lengths agree well with the theoretical results