Quantifying Chaos in Models of the Solar Neighbourhood

{} {To quantify the amount of chaos that exists in the local phase space.} {A sample of orbits from four different models of the Solar neighbourhood phase space are analysed by a new chaos identification (and quantification) technique. While three of the used models bear the signature of the perturbation due to both the Galactic bar and the spiral pattern, the last of the models is a bar only one. We explore the models by inter-comparing the corresponding values of chaos strength that is induced at the various energy levels .}{(1) We find that of all the viable models that have been demonstrated to successfully reproduce the local phase space structure, i.e. those that include the bar as well as the spiral, bear strong chaoticity, though the model that implies the highest degree of chaos is the one in which overlap of the major resonances of the bar and the spiral occurs. The bar only model is found to display regularity. (2) We advance chaos to be primarily responsible for the splitting of the Hyades-Pleiades mode (the larger mode) of the local velocity distribution}{}Comment: 6 pages; 4 figures; accepted for publication in Astronomy & Astrophysic

Development of singularities for the compressible Euler equations with external force in several dimensions

We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singularity formation, "the best sufficient condition", in the sense that one can explicitly construct a global in time smooth solution for which this condition is not satisfied "arbitrary little". Also compactly supported perturbation of nontrivial constant state is considered. We generalize the known theorem by Sideris on initial data resulting in singularities. Finally, we investigate the influence of frictional damping and rotation on the singularity formation.Comment: 23 page

Chaotic mixing in noisy Hamiltonian systems

This paper summarises an investigation of the effects of low amplitude noise and periodic driving on phase space transport in 3-D Hamiltonian systems, a problem directly applicable to systems like galaxies, where such perturbations reflect internal irregularities and.or a surrounding environment. A new diagnsotic tool is exploited to quantify how, over long times, different segments of the same chaotic orbit can exhibit very different amounts of chaos. First passage time experiments are used to study how small perturbations of an individual orbit can dramatically accelerate phase space transport, allowing `sticky' chaotic orbits trapped near regular islands to become unstuck on suprisingly short time scales. Small perturbations are also studied in the context of orbit ensembles with the aim of understanding how such irregularities can increase the efficacy of chaotic mixing. For both noise and periodic driving, the effect of the perturbation scales roughly in amplitude. For white noise, the details are unimportant: additive and multiplicative noise tend to have similar effects and the presence or absence of a friction related to the noise by a Fluctuation- Dissipation Theorem is largely irrelevant. Allowing for coloured noise can significantly decrease the efficacy of the perturbation, but only when the autocorrelation time, which vanishes for white noise, becomes so large that t here is little power at frequencies comparable to the natural frequencies of the unperturbed orbit. This suggests strongly that noise-induced extrinsic diffusion, like modulational diffusion associated with periodic driving, is a resonance phenomenon. Potential implications for galaxies are discussed.Comment: 15 pages including 18 figures, uses MNRAS LaTeX macro

Interplay Between Chaotic and Regular Motion in a Time-Dependent Barred Galaxy Model

We study the distinction and quantification of chaotic and regular motion in a time-dependent Hamiltonian barred galaxy model. Recently, a strong correlation was found between the strength of the bar and the presence of chaotic motion in this system, as models with relatively strong bars were shown to exhibit stronger chaotic behavior compared to those having a weaker bar component. Here, we attempt to further explore this connection by studying the interplay between chaotic and regular behavior of star orbits when the parameters of the model evolve in time. This happens for example when one introduces linear time dependence in the mass parameters of the model to mimic, in some general sense, the effect of self-consistent interactions of the actual N-body problem. We thus observe, in this simple time-dependent model also, that the increase of the bar's mass leads to an increase of the system's chaoticity. We propose a new way of using the Generalized Alignment Index (GALI) method as a reliable criterion to estimate the relative fraction of chaotic vs. regular orbits in such time-dependent potentials, which proves to be much more efficient than the computation of Lyapunov exponents. In particular, GALI is able to capture subtle changes in the nature of an orbit (or ensemble of orbits) even for relatively small time intervals, which makes it ideal for detecting dynamical transitions in time-dependent systems.Comment: 21 pages, 9 figures (minor typos fixed) to appear in J. Phys. A: Math. Theo

Chaos and the continuum limit in the gravitational N-body problem II. Nonintegrable potentials

This paper continues a numerical investigation of orbits evolved in `frozen,' time-independent N-body realisations of smooth time-independent density distributions corresponding to both integrable and nonintegrable potentials, allowing for N as large as 300,000. The principal focus is on distinguishing between, and quantifying, the effects of graininess on initial conditions corresponding, in the continuum limit, to regular and chaotic orbits. Ordinary Lyapunov exponents X do not provide a useful diagnostic for distinguishing between regular and chaotic behaviour. Frozen-N orbits corresponding in the continuum limit to both regular and chaotic characteristics have large positive X even though, for large N, the `regular' frozen-N orbits closely resemble regular characteristics in the smooth potential. Viewed macroscopically both `regular' and `chaotic' frozen-N orbits diverge as a power law in time from smooth orbits with the same initial condition. There is, however, an important difference between `regular' and `chaotic' frozen-N orbits: For regular orbits, the time scale associated with this divergence t_G ~ N^{1/2}t_D, with t_D a characteristic dynamical time; for chaotic orbits t_G ~ (ln N) t_D. At least for N>1000 or so, clear distinctions exist between phase mixing of initially localised orbit ensembles which, in the continuum limit, exhibit regular versus chaotic behaviour. For both regular and chaotic ensembles, finite-N effects are well mimicked, both qualitatively and quantitatively, by energy-conserving white noise with amplitude ~ 1/N. This suggests strongly that earlier investigations of the effects of low amplitude noise on phase space transport in smooth potentials are directly relevant to real physical systems.Comment: 20 pages, including 21 FIGURES, uses RevTeX macro

Evaluating the educational environment of an international animal model-based wet lab course for undergraduate students

publisher: Elsevier articletitle: Evaluating the educational environment of an international animal model-based wet lab course for undergraduate students journaltitle: Annals of Medicine and Surgery articlelink: http://dx.doi.org/10.1016/j.amsu.2016.10.004 content_type: article copyright: © 2016 The Author(s). Published by Elsevier Ltd on behalf of IJS Publishing Group Ltd

Measurement of inclusive D*+- and associated dijet cross sections in photoproduction at HERA

Inclusive photoproduction of D*+- mesons has been measured for photon-proton centre-of-mass energies in the range 130 < W < 280 GeV and a photon virtuality Q^2 < 1 GeV^2. The data sample used corresponds to an integrated luminosity of 37 pb^-1. Total and differential cross sections as functions of the D* transverse momentum and pseudorapidity are presented in restricted kinematical regions and the data are compared with next-to-leading order (NLO) perturbative QCD calculations using the "massive charm" and "massless charm" schemes. The measured cross sections are generally above the NLO calculations, in particular in the forward (proton) direction. The large data sample also allows the study of dijet production associated with charm. A significant resolved as well as a direct photon component contribute to the cross section. Leading order QCD Monte Carlo calculations indicate that the resolved contribution arises from a significant charm component in the photon. A massive charm NLO parton level calculation yields lower cross sections compared to the measured results in a kinematic region where the resolved photon contribution is significant.Comment: 32 pages including 6 figure

Measurement of Jet Shapes in Photoproduction at HERA

The shape of jets produced in quasi-real photon-proton collisions at centre-of-mass energies in the range $134-277$ GeV has been measured using the hadronic energy flow. The measurement was done with the ZEUS detector at HERA. Jets are identified using a cone algorithm in the $\eta - \phi$ plane with a cone radius of one unit. Measured jet shapes both in inclusive jet and dijet production with transverse energies $E^{jet}_T>14$ GeV are presented. The jet shape broadens as the jet pseudorapidity ($\eta^{jet}$) increases and narrows as $E^{jet}_T$ increases. In dijet photoproduction, the jet shapes have been measured separately for samples dominated by resolved and by direct processes. Leading-logarithm parton-shower Monte Carlo calculations of resolved and direct processes describe well the measured jet shapes except for the inclusive production of jets with high $\eta^{jet}$ and low $E^{jet}_T$. The observed broadening of the jet shape as $\eta^{jet}$ increases is consistent with the predicted increase in the fraction of final state gluon jets.Comment: 29 pages including 9 figure

Theorems on existence and global dynamics for the Einstein equations

This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure or late-time asymptotics are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.Comment: Submitted to Living Reviews in Relativity, major update of Living Rev. Rel. 5 (2002)