Dispersive to non-dispersive transition and phase velocity transient for linear waves in plane wake and channel flows

In this study we analyze the phase and group velocity of three-dimensional linear traveling waves in two sheared flows, the plane channel and the wake flows. This was carried out by varying the wave number over a large interval of values at a given Reynolds number inside the ranges 20-100, 1000-8000, for the wake and channel flow, respectively. Evidence is given about the possible presence of both dispersive and non-dispersive effects which are associated with the long and short ranges of wavelength. We solved the Orr-Sommerfeld and Squire eigenvalue problem and observed the least stable mode. It is evident that, at low wave numbers, the least stable eigenmodes in the left branch of the spectrum beave in a dispersive manner. By contrast, if the wavenumber is above a specific threshold, a sharp dispersive to non-dispersive transition can be observed. Beyond this transition, the dominant mode belongs to the right branch of the spectrum. The transient behavior of the phase velocity of small three-dimensional traveling waves was also considered. Having chosen the initial conditions, we then show that the shape of the transient highly depends on the transition wavelength threshold value. We show that the phase velocty can oscillate with a frequency which is equal to the frequency width of the eigenvalue spectrum. Furthermore, evidence of intermediate self-similarity is given for the perturbation field.Comment: 19 pages, 11 figures. Text and discussion improved with respect to the first version. Accepted for publication on Physical Review

Boolean Delay Equations: A simple way of looking at complex systems

Boolean Delay Equations (BDEs) are semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs can be classified into conservative or dissipative, in a manner that parallels the classification of ordinary or partial differential equations. Solutions to certain conservative BDEs exhibit growth of complexity in time. They represent therewith metaphors for biological evolution or human history. Dissipative BDEs are structurally stable and exhibit multiple equilibria and limit cycles, as well as more complex, fractal solution sets, such as Devil's staircases and ``fractal sunbursts``. All known solutions of dissipative BDEs have stationary variance. BDE systems of this type, both free and forced, have been used as highly idealized models of climate change on interannual, interdecadal and paleoclimatic time scales. BDEs are also being used as flexible, highly efficient models of colliding cascades in earthquake modeling and prediction, as well as in genetics. In this paper we review the theory of systems of BDEs and illustrate their applications to climatic and solid earth problems. The former have used small systems of BDEs, while the latter have used large networks of BDEs. We moreover introduce BDEs with an infinite number of variables distributed in space (``partial BDEs``) and discuss connections with other types of dynamical systems, including cellular automata and Boolean networks. This research-and-review paper concludes with a set of open questions.Comment: Latex, 67 pages with 15 eps figures. Revised version, in particular the discussion on partial BDEs is updated and enlarge

Analytic urns

This article describes a purely analytic approach to urn models of the generalized or extended P\'olya-Eggenberger type, in the case of two types of balls and constant ``balance,'' that is, constant row sum. The treatment starts from a quasilinear first-order partial differential equation associated with a combinatorial renormalization of the model and bases itself on elementary conformal mapping arguments coupled with singularity analysis techniques. Probabilistic consequences in the case of ``subtractive'' urns are new representations for the probability distribution of the urn's composition at any time n, structural information on the shape of moments of all orders, estimates of the speed of convergence to the Gaussian limit and an explicit determination of the associated large deviation function. In the general case, analytic solutions involve Abelian integrals over the Fermat curve x^h+y^h=1. Several urn models, including a classical one associated with balanced trees (2-3 trees and fringe-balanced search trees) and related to a previous study of Panholzer and Prodinger, as well as all urns of balance 1 or 2 and a sporadic urn of balance 3, are shown to admit of explicit representations in terms of Weierstra\ss elliptic functions: these elliptic models appear precisely to correspond to regular tessellations of the Euclidean plane.Comment: Published at http://dx.doi.org/10.1214/009117905000000026 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Théorie des valeurs extrêmes pour systèmes dynamiques, avec applications au climat et en neurosciences

Throughout the thesis, we will discuss, improve and provide a conceptual framework in which methods based on recurrence properties of chaotic dynamics can be understood. We will also provide new EVT-based methods to compute quantities of interest and introduce new useful indicators associated to the dynamics. Our results will be mathematically rigorous, although emphasis will be placed on physical applications and numerical computations, as the use of such methods is developing rapidly. We will start by an introductory chapter to the dynamical theory of extreme events, in which we will describe the principal results of the theory that will be used throughout the thesis. After a small chapter where we introduce some objects that are characteristic of the invariant measure of the system, namely local dimensions and generalized dimensions, we devote the following chapters to the use of EVT to compute such dimensional quantities. One of these methods defines naturally a novel global indicator on the hyperbolic properties of the system. In these chapters, we will present several numerical applications of the methods, both in real world and idealized systems, and study the influence of different kinds of noise on these indicators. We will then investigate a matter of physical importance related to EVT : the statistics of visits in some particular small target subsets of the phase-space, in particular for partly random, noisy systems. The results presented in this section are mostly numerical and conjectural, but reveal some universal behavior of the statistics of visits. The eighth chapter makes the connection between several local quantities associated to the dynamics and computed using a finite amount of data (local dimensions, hitting times, return times) and the generalized dimensions of the system, that are computable by EVT methods. These relations, stated in the language of large deviation theory (that we will briefly present), have profound physical implications, and constitute a conceptual framework in which the distribution of such computed local quantities can be understood. We then take advantage of these connections to design further methods to compute the generalized dimensions of a system. Finally, in the last part of the thesis, which is more experimental, we extend the dynamical theory of extreme events to more complex observables, which will allow us to study phenomena evolving over long temporal scales. We will consider the example of firing cascades in a model of neural network. Through this example, we will introduce a novel approach to study such complex systems.Tout au long de la thèse, nous discuterons, améliorerons et fournirons un cadre conceptuel dans lequel des méthodes basées sur les propriétés de récurrence de dynamiques chaotiques peuvent être comprises. Nous fournirons également de nouvelles méthodes basées sur l’EVT pour calculer les quantités importantes associées à la dynamique. Nos résultats sont rigoureux d’un point de vue mathématique, même si l’accent sera mis sur les applications physiques et les calculs numériques, car l’utilisation de telles méthodes se développe rapidement. Nous commencerons par un chapitre introductif à la théorie dynamique des événements extrêmes, dans lequel nous décrirons les principaux résultats de la théorie qui seront utilisés tout au long de la thèse. Après un petit chapitre dans lequel nous introduisons certains objets caractéristiques de la mesure invariante du système, à savoir les dimensions locales et les dimensions généralisées, nous consacrons les chapitres suivants à l’utilisation de l’EVT pour calculer de telles quantités dimensionnelles. L’une de ces méthodes définit naturellement un nouvel indicateur global sur les propriétés hyperboliques du système. Dans ces chapitres, nous présenterons plusieurs applications numériques de ces méthodes, à la fois dans des systèmes réels et idéalisés, et étudierons l’influence de différents types de bruit sur ces indicateurs. Nous examinerons ensuite une question d’importance physique liée à l’EVT : les statistiques de visites dans certains sous-ensembles cibles spécifiques de l’espace de phase, en particulier pour les systèmes partiellement aléatoires. Les résultats présentés dans cette section sont principalement numériques et hypothétiques, mais révèlent un comportement universel des statistiques de visites. Le huitième chapitre établit la connexion entre plusieurs quantités locales associées à la dynamique et calculées à l’aide d’une quantité finie de données (dimensions locales, temps d’entées, temps de retour) et les dimensions généralisées du système, qui calculables par les méthodes EVT. Ces relations, énoncées dans le langage de la théorie des grandes déviations (que nous exposerons brièvement), ont de profondes implications physiques et constituent un cadre conceptuel dans lequel le fait de calculer une distribution étalée de ces quantités locales peut être comprise. Nous tirons ensuite parti de ces connexions pour concevoir d’autres méthodes permettant de calculer les dimensions généralisées d’un système. Enfin, dans la dernière partie de la thèse, qui est plus expérimentale, nous étendons la théorie dynamique des événements extrêmes à des observables plus complexes, ce qui nous permettra d’étudier des phénomènes évoluant sur de longues échelles temporelles. Nous allons considérer l’exemple des cascades d’excitation dans un modèle de réseau de neurones. À travers cet exemple, nous allons introduire une nouvelle approche pour étudier de tels systèmes complexes

Generalizations of Bounds on the Index of Convergence to Weighted Digraphs

We study sequences of optimal walks of a growing length, in weighted digraphs, or equivalently, sequences of entries of max-algebraic matrix powers with growing exponents. It is known that these sequences are eventually periodic when the digraphs are strongly connected. The transient of such periodicity depends, in general, both on the size of digraph and on the magnitude of the weights. In this paper, we show that some bounds on the indices of periodicity of (unweighted) digraphs, such as the bounds of Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply to the weights of optimal walks when one of their ends is a critical node.Comment: 17 pages, 3 figure

Bifurcation and Chaos in Fractional-Order Systems

This book presents a collection of seven technical papers on fractional-order complex systems, especially chaotic systems with hidden attractors and symmetries, in the research front of the field, which will be beneficial for scientific researchers, graduate students, and technical professionals to study and apply. It is also suitable for teaching lectures and for seminars to use as a reference on related topics