Geometry of the ergodic quotient reveals coherent structures in flows

Dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach. However, by identifying coherent structures in their state spaces, i.e., regions of uniform and simpler behavior, we could hope to study each of the structures separately and then form the understanding of the system as a whole. The method we present in this paper uses trajectory averages of scalar functions on the state space to: (a) identify invariant sets in the state space, (b) form coherent structures by aggregating invariant sets that are similar across multiple spatial scales. First, we construct the ergodic quotient, the object obtained by mapping trajectories to the space of trajectory averages of a function basis on the state space. Second, we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space. Finally, we parametrize the ergodic quotient using intrinsic diffusion modes on it. By segmenting the ergodic quotient based on the diffusion modes, we extract coherent features in the state space of the dynamical system. The algorithm is validated by analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for alternative approaches: the Ulam's approximation of the transfer operator and the computation of Lagrangian Coherent Structures. Furthermore, we explain how the method extends the Poincar\'e map analysis for periodic flows. As a demonstration, we apply the method to a periodically-driven three-dimensional Hill's vortex flow, discovering unknown coherent structures in its state space. In the end, we discuss differences between the ergodic quotient and alternatives, propose a generalization to analysis of (quasi-)periodic structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen

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

Asymptotic forecast uncertainty and the unstable subspace in the presence of additive model error

It is well understood that dynamic instability is among the primary drivers of forecast uncertainty in chaotic, physical systems. Data assimilation techniques have been designed to exploit this phenomenon, reducing the effective dimension of the data assimilation problem to the directions of rapidly growing errors. Recent mathematical work has, moreover, provided formal proofs of the central hypothesis of the assimilation in the unstable subspace methodology of Anna Trevisan and her collaborators: for filters and smoothers in perfect, linear, Gaussian models, the distribution of forecast errors asymptotically conforms to the unstable-neutral subspace. Specifically, the column span of the forecast and posterior error covariances asymptotically align with the span of backward Lyapunov vectors with nonnegative exponents. Earlier mathematical studies have focused on perfect models, and this current work now explores the relationship between dynamical instability, the precision of observations, and the evolution of forecast error in linear models with additive model error. We prove bounds for the asymptotic uncertainty, explicitly relating the rate of dynamical expansion, model precision, and observational accuracy. Formalizing this relationship, we provide a novel, necessary criterion for the boundedness of forecast errors. Furthermore, we numerically explore the relationship between observational design, dynamical instability, and filter boundedness. Additionally, we include a detailed introduction to the multiplicative ergodic theorem and to the theory and construction of Lyapunov vectors. While forecast error in the stable subspace may not generically vanish, we show that even without filtering, uncertainty remains uniformly bounded due its dynamical dissipation. However, the continuous reinjection of uncertainty from model errors may be excited by transient instabilities in the stable modes of high variance, rendering forecast uncertainty impractically large. In the context of ensemble data assimilation, this requires rectifying the rank of the ensemble-based gain to account for the growth of uncertainty beyond the unstable and neutral subspace, additionally correcting stable modes with frequent occurrences of positive local Lyapunov exponents that excite model errors

Can quantum fractal fluctuations be observed in an atom-optics kicked rotor experiment?

We investigate the parametric fluctuations in the quantum survival probability of an open version of the delta-kicked rotor model in the deep quantum regime. Spectral arguments [Guarneri I and Terraneo M 2001 Phys. Rev. E vol. 65 015203(R)] predict the existence of parametric fractal fluctuations owing to the strong dynamical localisation of the eigenstates of the kicked rotor. We discuss the possibility of observing such dynamically-induced fractality in the quantum survival probability as a function of the kicking period for the atom-optics realisation of the kicked rotor. The influence of the atoms' initial momentum distribution is studied as well as the dependence of the expected fractal dimension on finite-size effects of the experiment, such as finite detection windows and short measurement times. Our results show that clear signatures of fractality could be observed in experiments with cold atoms subjected to periodically flashed optical lattices, which offer an excellent control on interaction times and the initial atomic ensemble.Comment: 18 pp, 7 figs., 1 tabl

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems

Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information useful for understanding dynamics. However, Koopman operators can have continuous spectra and infinite-dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data-driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we also compute smoothed approximations of spectral measures associated with measure-preserving dynamical systems. We prove explicit convergence theorems for our algorithms, which can achieve high-order convergence even for chaotic systems, when computing the density of the continuous spectrum and the discrete spectrum. We demonstrate our algorithms on the tent map, Gauss iterated map, nonlinear pendulum, double pendulum, Lorenz system, and an $11$-dimensional extended Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high-dimensional state-space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule that has a 20,046-dimensional state-space, and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number $>10^5$ that has a 295,122-dimensional state-space

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review