An observer-based referential that is consistent with cosmological observations

In a universe of arbitrary spacetime geometry, we introduce a referential centered at the observer that constitutes a consistent representation of its environment and in which he performs measurements. We postulate some natural rules applying in this representation referential, that match our everyday experience of the physical world. We show that these assumptions turn out to be consistent with cosmological observations, provided we consider a particularly simple cosmological solution of the equations of general relativity: the static de Sitter space

On the generalized dimensions of chaotic attractors

We prove that if $\mu$ is the physical measure of a chaotic flow diffeomorphically conjugated to a suspension flow based on a Poincar\'{e} application $R$ with physical measure $\mu_{R}$, $D_{q}(\mu)=D_{q}(\mu _{R})+1$, where $D_{q}$ denotes the generalized dimension of order $q$ with $q \neq1$. We also prove that a similar result holds for the local dimensions so that, under the additional hypothesis of exact-dimensionality of $\mu_{R}$, our result extends to the case $q=1$. We then apply these results to estimate the $D_{q}$ spectrum of the R\"ossler attractor and prove the existence of the information dimension $D_{1}$ for the Lorenz '63 flow.Comment: 34 pages, 6 figure

Dynamical properties of chimera states for globally coupled map lattices

We study the stability properties and long-term dynamical behavior of chimera states in globally coupled map lattices. In particular, we give a formula for the transverse Lyapunov exponent associated with blocks of synchronized sites. We use these results to study clustered dynamics from a numerical perspective, and give numerical evidence of attracting chimeras having chaotic dynamics, as well as periodic behaviors. Finally, we obtain some results ruling out the existence of absolutely continuous invariant measures supported on chimera states in strong coupling regimes

Matching of observations of dynamical systems, with applications to sequence matching

International audienceWe study the statistical distribution of the closest encounter between generic smooth observations computed along dierent trajectories of a rapidly mixing dynamical system. At the limit of large trajectories, we obtain a distribution of Gumbel type that depends on both the length of the trajectories and on the Generalized Dimensions of the image measure. It is also modulated by an Extremal Index, that informs on the tendency of nearby observations to diverge along with the evolution of the dynamics. We give a formula for this quantity for a class of chaotic maps of the interval and regular observations. We present diverse numerical applications illustrating the theory and discuss the implications of these results for the study of physical systems. Finally, we discuss the connection between this problem and the problem of the longest matching block common to dierent sequences of symbols. In particular, we obtain a distributional result for strongly mixing processes

Théorie des valeurs extrêmes pour les systèmes dynamiques, avec applications au climat et aux 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 have full mathematical rigor, 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 abjects that are characteristic of the invariant measure of the system, namely local dimensions and generalized dimensions, w1 devote the following chapters to the use of EVT to compute such dimensional quantities. One of these method defines naturally a navel global indicator on the hyperbolic properties of the system. ln these chapters, we will present several numerical applications of the methods, bath 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 importanc 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 betweer 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 navel 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 d'intérêt et présenteronsr de nouveaux indicateurs utiles associés à la dynamique. Nos résultats auront une rigueur mathématique totale, 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 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 des 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 et bruyants. 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 de frappe, temps de retour) et les dimensions généralisées du système, 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 la distribution de ces quantités locales calculées 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 observable

TOPOLOGICAL SYNCHRONISATION OR A SIMPLE ATTRACTOR?

International audienceA few recent papers introduced the concept of topological synchronisation. We refer in particular to [11], where the theory was illustrated by means of a skew product system, coupling two logistic maps. In this case, we show that the topological synchronisation could be easily explained as the birth of an attractor for increasing values of the coupling strength and the mutual convergence of two marginal empirical measures. Numerical computations based on a careful analysis of the Lyapunov exponents suggest that the attractor supports an absolutely continuous physical measure (acpm). We finally show that for some unimodal maps such acpm exhibit a multifractal structure

A statistical physics and dynamical systems perspective on geophysical extreme events

Statistical physics and dynamical systems theory are key tools to study high-impact geophysical events such as temperature extremes, cyclones, thunderstorms, geomagnetic storms and many more. Despite the intrinsic differences between these events, they all originate as temporary deviations from the typical trajectories of a geophysical system, resulting in well-organised, coherent structures at characteristic spatial and temporal scales. While statistical extreme value analysis techniques are capable to provide return times and probabilities of occurrence of certain geophysical events, they are not apt to account for their underlying physics. Their focus is to compute the probability of occurrence of events that are large or small with respect to some specific observable (e.g. temperature, precipitation, solar wind), rather than to relate rare or extreme phenomena to the underlying anomalous geophysical regimes. This paper delineates this knowledge gap, presenting some related challenges and new formalisms which arise in the study of geophysical extreme events and may help better understand them