Probing the local dynamics of periodic orbits by the generalized alignment index (GALI) method

As originally formulated, the Generalized Alignment Index (GALI) method of chaos detection has so far been applied to distinguish quasiperiodic from chaotic motion in conservative nonlinear dynamical systems. In this paper we extend its realm of applicability by using it to investigate the local dynamics of periodic orbits. We show theoretically and verify numerically that for stable periodic orbits the GALIs tend to zero following particular power laws for Hamiltonian flows, while they fluctuate around non-zero values for symplectic maps. By comparison, the GALIs of unstable periodic orbits tend exponentially to zero, both for flows and maps. We also apply the GALIs for investigating the dynamics in the neighborhood of periodic orbits, and show that for chaotic solutions influenced by the homoclinic tangle of unstable periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during which their amplitudes change by many orders of magnitude. Finally, we use the GALI method to elucidate further the connection between the dynamics of Hamiltonian flows and symplectic maps. In particular, we show that, using for the computation of GALIs the components of deviation vectors orthogonal to the direction of motion, the indices of stable periodic orbits behave for flows as they do for maps.Comment: 17 pages, 9 figures (accepted for publication in Int. J. of Bifurcation and Chaos

The Inevitability of Corruption in Greek Football

This is an Accepted Manuscript of an article published by Taylor & Francis in Soccer & Society on 14th March 2017, available online: http://www.tandfonline.com/10.1080/14660970.2017.1302936.From the late 1990s corrupt practices in Greek football have been considered to pose a serious threat to the integrity of the sport, with a number of schemes and measures being introduced as a response. The aim of this article is to show why corruption in Greek football is inevitable by offering a detailed account of three football-related corrupt practices and highlighting their contextual parameters, as well as juxtaposing them against the set of measures that have been implemented. By placing corruption in football in the wider landscape of the country and of global football, and examining the political, structural and economic factors that contribute to the overall managerial and financial implications of corruption, we present the reader with the new norm which, in reality, makes corruption the ‘only game in town’

On error estimates for Galerkin finite element methods for the Camassa-Holm equation

We consider the Camassa-Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order $L^{2}$-error estimates for the semidiscrete approximation. We also consider an initial-boundary-value problem on a finite interval for the system form of CH and analyze the convergence of its standard Galerkin semidiscretization. Using the fourth-order accurate, explicit, "classical" Runge-Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the `peakon' type

Case series and a systematic review concerning the level of the aortic bifurcation

Background: The aim of this study is to present the level of aortic bifurcation in a sample of Greek origin (case series) and to perform an up-to-date systematic review in the existing literature. Materials and methods: Seventy-six formalin-fixed adult cadavers were dissected and studied in order to research the level of aortic bifurcation. Additionally, PubMed and Google Scholar databases were searched for eligible articles concerning the level of aortic bifurcation for the period up to February 2020. Results: The mean level of aortic bifurcation according to our case series was the lower third of the L4 vertebral body (21/76, 27.6%). The level of aortic bifurcation ranged between the lower third of the L3 vertebral body and the lower third of the L5 body. No statistically significant correlation was found between the two sexes. The systematic review of the literature revealed 31 articles which were considered eligible and a total number of 3537 specimens were retracted. According to the recorded findings the most common mean level of aortic bifurcation was the body of L4 vertebra (1495/3537 cases, 42.2%), while the range of aortic bifurcation was described to occur from upper third of L3 vertebrae to the upper third of the S1 vertebrae in the 52.8% of the cases (1866/3537). Conclusions: The mean level of AA corresponds to the body of L4 and presents a great range (form L3U to S1U). Knowledge of the mean level of aortic bifurcation and its probable ranges is of great significance for interventional radiologists and especially vascular surgeons that deal with aneurism proximal to the aortic bifurcation

Chaotic Dynamics of N-degree of Freedom Hamiltonian Systems

We investigate the connection between local and global dynamics of two N-degree of freedom Hamiltonian systems with different origins describing one-dimensional nonlinear lattices: The Fermi-Pasta-Ulam (FPU) model and a discretized version of the nonlinear Schrodinger equation related to Bose-Einstein Condensation (BEC). We study solutions starting in the vicinity of simple periodic orbits (SPOs) representing in-phase (IPM) and out-of-phase motion (OPM), which are known in closed form and whose linear stability can be analyzed exactly. Our results verify that as the energy E increases for fixed N, beyond the destabilization threshold of these orbits, all positive Lyapunov exponents exhibit a transition between two power laws, occurring at the same value of E. The destabilization energy E_c per particle goes to zero as N goes to infinity following a simple power-law. However, using SALI, a very efficient indicator we have recently introduced for distinguishing order from chaos, we find that the two Hamiltonians have very different dynamics near their stable SPOs: For example, in the case of the FPU system, as the energy increases for fixed N, the islands of stability around the OPM decrease in size, the orbit destabilizes through period-doubling bifurcation and its eigenvalues move steadily away from -1, while for the BEC model the OPM has islands around it which grow in size before it bifurcates through symmetry breaking, while its real eigenvalues return to +1 at very high energies. Still, when calculating Lyapunov spectra, we find for the OPMs of both Hamiltonians that the Lyapunov exponents decrease following an exponential law and yield extensive Kolmogorov--Sinai entropies per particle, in the thermodynamic limit of fixed energy density E/N with E and N arbitrarily large.Comment: 29 pages, 10 figures, published at International Journal of Bifurcation and Chaos (IJBC

Capsizing of ships : static and dynamic analysis of wind effect and cost implications

Thesis (Nav. E. and S.M. in Ocean Systems Management)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2006.Includes bibliographical references (p. 71).Capsizing of small vessels, such as commercial fishing vessels, is a frequent event. This phenomenon is generally associated with the combined action of storm seas, inadequate design parameter regulations, and dangerous operational procedures. In contrast, the capsizing of large ships is rare, but does occur. For these large vessels, more strict regulations exist to ensure safe operational procedures. While the storminess of the sea cannot be controlled, the navigation procedure can. Large offshore ships tend to navigate in a path to avoid forecasted severe weather, and in cases of stormy seas they temporarily operate at safe speeds and in the direction parallel to the waves. The work presented in this thesis investigates the effect of the wind in rolling and finally capsizing a ship. For the purposes of mechanical analysis, realistic hull forms are used and fundamental issues associated with moments and forces imposed by the wind, are applied. The platforms are examined for several wind speeds that strike the ship at different angles. Both static and dynamic cases were examined. Under the assumption of general conditions, the angles of heeling in each case and the wind speeds that caused the ship to capsize are calculated.(cont.) Furthermore, a cost analysis associated with the total loss of the ship due to capsize is also reviewed. An existing worldwide database of vessel total losses, dating from 1960 to present, is used to calculate the costs per ship capsize. Some simplifications are inevitably used, because the cost implications of total ship losses have both direct and indirect portions that are difficult to quantify. In addition, the actual numbers that result from such a catastrophe are not generally available to the public and are not found in the open literature. Given these limitations, a preliminary analysis of the capsize-associated costs is performed for several types of commercial vessels.by Angelos Antonopoulos.Nav.E.and S.M.in Ocean Systems Managemen

Anatomical study of the common iliac arteries

Background: The common iliac arteries (CIA) are the two terminal branches of the abdominal aorta which supply the pelvis and the lower extremities. The present study aims to examine the morphometric features of the CIA in a cadaveric sample and possible correlations between lengths.Materials and methods: Seventy-six formalin fixed cadavers of Greek origin were dissected in the Department of Anatomy, School of Medicine, National and Kapodistrian University of Athens. In each cadaver dissected, the abdominal aorta and the CIA were identified and their lengths were measured. Also the torso length was measured and the height of each cadaver. All the statistical analysis was done by SPSS 15.0.Results: The mean length of the left CIA was 6.12 cm (SD: ± 1.791, SE: 0.205) and that of the right one was 6.03 cm (SD: ± 1.607, SE: 0.184). The lengths of the CIA differed between the sexes, but no statistically significant difference was observed. Statistically significant differences regarding the torso lengths and body heights were found between the sexes, as well as a statistically strong correlation between the lengths of the left and right CIA in the cadavers dissected.Conclusions: The knowledge of the anatomy and morphology of the CIA is ofgreat clinical significance, given that abnormal course, length or branching pattern of these vessels are not uncommon and their clinical impact may be great. Mostly interventional radiologists and vascular surgeons should be aware of this knowledge

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