A dynamical system approach to data assimilation in chaotic models

The Assimilation in the Unstable Subspace (AUS) was introduced by Trevisan and Uboldi in 2004, and developed by Trevisan, Uboldi and Carrassi, to minimize the analysis and forecast errors by exploiting the flow-dependent instabilities of the forecast-analysis cycle system, which may be thought of as a system forced by observations. In the AUS scheme the assimilation is obtained by confining the analysis increment in the unstable subspace of the forecast-analysis cycle system so that it will have the same structure of the dominant instabilities of the system. The unstable subspace is estimated by Breeding on the Data Assimilation System (BDAS). AUS- BDAS has already been tested in realistic models and observational configurations, including a Quasi-Geostrophicmodel and a high dimensional, primitive equation ocean model; the experiments include both fixed and“adaptive”observations. In these contexts, the AUS-BDAS approach greatly reduces the analysis error, with reasonable computational costs for data assimilation with respect, for example, to a prohibitive full Extended Kalman Filter. This is a follow-up study in which we revisit the AUS-BDAS approach in the more basic, highly nonlinear Lorenz 1963 convective model. We run observation system simulation experiments in a perfect model setting, and with two types of model error as well: random and systematic. In the different configurations examined, and in a perfect model setting, AUS once again shows better efficiency than other advanced data assimilation schemes. In the present study, we develop an iterative scheme that leads to a significant improvement of the overall assimilation performance with respect also to standard AUS. In particular, it boosts the efficiency of regime’s changes tracking, with a low computational cost. Other data assimilation schemes need estimates of ad hoc parameters, which have to be tuned for the specific model at hand. In Numerical Weather Prediction models, tuning of parameters — and in particular an estimate of the model error covariance matrix — may turn out to be quite difficult. Our proposed approach, instead, may be easier to implement in operational models.

Coarse-grained peptide models: conformational sampling, peptide association and dynamical properties for peptides

In dieser Arbeit wird ein vergröbertes (engl. coarse-grained, CG) Simulationsmodell für Peptide in wässriger Lösung entwickelt. In einem CG Verfahren reduziert man die Anzahl der Freiheitsgrade des Systems, so dass manrngrössere Systeme auf längeren Zeitskalen untersuchen kann. Die Wechselwirkungspotentiale des CG Modells sind so aufgebaut, dass die Peptid Konformationen eines höher aufgelösten (atomistischen) Modells reproduziert werden.rnIn dieser Arbeit wird der Einfluss unterschiedlicher bindender Wechsel-rnwirkungspotentiale in der CG Simulation untersucht, insbesondere daraufhin,rnin wie weit das Konformationsgleichgewicht der atomistischen Simulation reproduziert werden kann. Im CG Verfahren verliert man per Konstruktionrnmikroskopische strukturelle Details des Peptids, zum Beispiel, Korrelationen zwischen Freiheitsgraden entlang der Peptidkette. In der Dissertationrnwird gezeigt, dass diese “verlorenen” Eigenschaften in einem Rückabbildungsverfahren wiederhergestellt werden können, in dem die atomistischen Freiheitsgrade wieder in die CG-Strukturen eingefügt werden. Dies gelingt, solange die Konformationen des CG Modells grundsätzlich gut mit der atomistischen Ebene übereinstimmen. Die erwähnten Korrelationen spielen einerngrosse Rolle bei der Bildung von Sekundärstrukturen und sind somit vonrnentscheidender Bedeutung für ein realistisches Ensemble von Peptidkonformationen. Es wird gezeigt, dass für eine gute Übereinstimmung zwischen CG und atomistischen Kettenkonformationen spezielle bindende Wechselwirkungen wie zum Beispiel 1-5 Bindungs- und 1,3,5-Winkelpotentiale erforderlich sind. Die intramolekularen Parameter (d.h. Bindungen, Winkel, Torsionen), die für kurze Oligopeptide parametrisiert wurden, sind übertragbarrnauf längere Peptidsequenzen. Allerdings können diese gebundenen Wechselwirkungen nur in Kombination mit solchen nichtbindenden Wechselwirkungspotentialen kombiniert werden, die bei der Parametrisierung verwendet werden, sind also zum Beispiel nicht ohne weiteres mit einem andere Wasser-Modell kombinierbar. Da die Energielandschaft in CG-Simulationen glatter ist als im atomistischen Modell, gibt es eine Beschleunigung in der Dynamik. Diese Beschleunigung ist unterschiedlich für verschiedene dynamische Prozesse, zum Beispiel für verschiedene Arten von Bewegungen (Rotation und Translation). Dies ist ein wichtiger Aspekt bei der Untersuchung der Kinetik von Strukturbildungsprozessen, zum Beispiel Peptid Aggregation.rn

A two-body study of dynamical expansion in the XXZ spin chain and equivalent interacting fermion models

Lo scopo di questa tesi è studiare l'espansione dinamica di due fermioni interagenti in una catena unidimensionale cercando di definire il ruolo degli stati legati durante l'evoluzione temporale del sistema. Lo studio di questo modello viene effettuato a livello analitico tramite la tecnica del Bethe ansatz, che ci fornisce autovalori ed autovettori dell'hamiltoniana, e se ne valutano le proprietà statiche. Particolare attenzione è stata dedicata alle caratteristiche dello spettro al variare dell'interazione tra le due particelle e alle caratteristiche degli autostati. Dalla risoluzione dell'equazione di Bethe vengono ricercate le soluzioni che danno luogo a stati legati delle due particelle e se ne valuta lo spettro energetico in funzione del momento del centro di massa. Si è studiato inoltre l'andamento del numero delle soluzioni, in particolare delle soluzioni che danno luogo ad uno stato legato, al variare della lunghezza della catena e del parametro di interazione. La valutazione delle proprietà dinamiche del modello è stata effettuata tramite l'utilizzo dell'algoritmo t-DMRG (time dependent - Density Matrix Renormalization Group). Questo metodo numerico, che si basa sulla decimazione dello spazio di Hilbert, ci permette di avere accesso a quantità che caratterizzano la dinamica quali la densità e la velocità di espansione. Da queste sono stati estratti i proli dinamici della densità e della velocità di espansione al variare del valore del parametro di interazione.

Dynamical local models for segmentation and prediction of financial time series

In the analysis and prediction of many real-world time series, the assumption of stationarity is not valid. A special form of non-stationarity, where the underlying generator switches between (approximately) stationary regimes, seems particularly appropriate for financial markets. We introduce a new model which combines a dynamic switching (controlled by a hidden Markov model) and a non-linear dynamical system. We show how to train this hybrid model in a maximum likelihood approach and evaluate its performance on both synthetic and financial data.

Border figure detection using a phase oscillator network with dynamical coupling

Oscillator networks have been developed in order to perform specific tasks related to image processing. Here we analytically investigate the existence of synchronism in a pair of phase oscillators that are short-range dynamically coupled. Then, we use these analytical results to design a network able of detecting border of black-and-white figures. Each unit composing this network is a pair of such phase oscillators and is assigned to a pixel in the image. The couplings among the units forming the network are also dynamical. Border detection emerges from the network activity.

The young, tight, and low-mass binary TWA22AB: a new calibrator for evolutionary models? Orbit, spectral types, and temperature

Context. Tight binaries discovered in young, nearby associations are ideal targets for providing dynamical mass measurements to test the physics of evolutionary models at young ages and very low masses. Aims. We report the binarity of TWA22 for the first time. We aim at monitoring the orbit of this young and tight system to determine its total dynamical mass using an accurate distance determination. We also intend to characterize the physical properties (luminosity, effective temperature, and surface gravity) of each component based on near-infrared photometric and spectroscopic observations. Methods. We used the adaptive-optics assisted imager NACO to resolve the components, to monitor the complete orbit and to obtain the relative near-infrared photometry of TWA22 AB. The adaptive-optics assisted integral field spectrometer SINFONI was also used to obtain medium-resolution (R(lambda) = 1500-2000) spectra in JHK bands. Comparison with empirical and synthetic librairies were necessary for deriving the spectral type, the effective temperature, and the surface gravity for each component of the system. Results. Based on an accurate trigonometric distance (17.5 +/- 0.2 pc) determination, we infer a total dynamical mass of 220 +/- 21 M(Jup) for the system. From the complete set of spectra, we find an effective temperature T(eff) = 2900(-200)(+200) K for TWA22A and T(eff) = 2900(-100)(+200) for TWA22 B and surface gravities between 4.0 and 5.5 dex. From our photometry and an M6 +/- 1 spectral type for both components, we find luminosities of log(L/L(circle dot)) = -2.11 +/- 0.13 dex and log(L/L(circle dot)) = -2.30 +/- 0.16 dex for TWA22 A and B, respectively. By comparing these parameters with evolutionary models, we question the age and the multiplicity of this system. We also discuss a possible underestimation of the mass predicted by evolutionary models for young stars close to the substellar boundary.

Bayesian online algorithms for learning in discrete Hidden Markov Models

We propose and analyze two different Bayesian online algorithms for learning in discrete Hidden Markov Models and compare their performance with the already known Baldi-Chauvin Algorithm. Using the Kullback-Leibler divergence as a measure of generalization we draw learning curves in simplified situations for these algorithms and compare their performances.

Dynamical mutation of dark energy

We discuss the intriguing possibility that dark energy may change its equation of state in situations where large dark energy fluctuations are present. We show indications of this dynamical mutation in some generic models of dark energy.

Periodic-orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system

Chaotic dynamical systems with two or more attractors lying on invariant subspaces may, provided certain mathematical conditions are fulfilled, exhibit intermingled basins of attraction: Each basin is riddled with holes belonging to basins of the other attractors. In order to investigate the occurrence of such phenomenon in dynamical systems of ecological interest (two-species competition with extinction) we have characterized quantitatively the intermingled basins using periodic-orbit theory and scaling laws. The latter results agree with a theoretical prediction from a stochastic model, and also with an exact result for the scaling exponent we derived for the specific class of models investigated. We discuss the consequences of the scaling laws in terms of the predictability of a final state (extinction of either species) in an ecological experiment.

Dynamical generation of mass in the D = (2+1) Wess-Zumino model

In this work we study the dynamical generation of mass in the massless N = 1 Wess-Zumino model in a three-dimensional spacetime. Using the tadpole method to compute the effective potential, we observe that supersymmetry is dynamically broken together with the discrete symmetry A(x) -> A(x). We show that this model, different from nonsupersymmetric scalar models, exhibits a consistent perturbative dynamical generation of mass after two-loop corrections to the effective potential.

Dynamical quantum noise in trapped Bose-Einstein condensates

We introduce the study of dynamical quantum noise in Bose-Einstein condensates through numerical simulation of stochastic partial differential equations obtained using phase-space representations. We derive evolution equations for a single trapped condensate in both the positive-P and Wigner representations and perform simulations to compare the predictions of the two methods. The positive-P approach is found to be highly susceptible to the stability problems that have been observed in other strongly nonlinear, weakly damped systems. Using the Wigner representation, we examine the evolution of several quantities of interest using from a variety of choices of initial stare for the condensate and compare results to those for single-mode models. [S1050-2947(98)06612-8].

Evolution of stress deficit and changing rates of seismicity in cellular automaton models of earthquake faults

We investigate the internal dynamics of two cellular automaton models with heterogeneous strength fields and differing nearest neighbour laws. One model is a crack-like automaton, transferring ail stress from a rupture zone to the surroundings. The other automaton is a partial stress drop automaton, transferring only a fraction of the stress within a rupture zone to the surroundings. To study evolution of stress, the mean spectral density. f(k(r)) of a stress deficit held is: examined prior to, and immediately following ruptures in both models. Both models display a power-law relationship between f(k(r)) and spatial wavenumber (k(r)) of the form f(k(r)) similar tok(r)(-beta). In the crack model, the evolution of stress deficit is consistent with cyclic approach to, and retreat from a critical state in which large events occur. The approach to criticality is driven by tectonic loading. Short-range stress transfer in the model does not affect the approach to criticality of broad regions in the model. The evolution of stress deficit in the partial stress drop model is consistent with small fluctuations about a mean state of high stress, behaviour indicative of a self-organised critical system. Despite statistics similar to natural earthquakes these simplified models lack a physical basis. physically motivated models of earthquakes also display dynamical complexity similar to that of a critical point system. Studies of dynamical complexity in physical models of earthquakes may lead to advancement towards a physical theory for earthquakes.

An Extension of Gompertzian Growth Dynamics Weibull and Frechet Models

In this work a new probabilistic and dynamical approach to an extension of the Gompertz law is proposed. A generalized family of probability density functions, designated by Beta* (p, q), which is proportional to the right hand side of the Tsoularis-Wallace model, is studied. In particular, for p = 2, the investigation is extended to the extreme value models of Weibull and Frechet type. These models, described by differential equations, are proportional to the hyper-Gompertz growth model. It is proved that the Beta* (2, q) densities are a power of betas mixture, and that its dynamics are determined by a non-linear coupling of probabilities. The dynamical analysis is performed using techniques of symbolic dynamics and the system complexity is measured using topological entropy. Generally, the natural history of a malignant tumour is reflected through bifurcation diagrams, in which are identified regions of regression, stability, bifurcation, chaos and terminus.

Synchronization in Von Bertalanffy’s models

Many data have been useful to describe the growth of marine mammals, invertebrates and reptiles, seabirds, sea turtles and fishes, using the logistic, the Gom-pertz and von Bertalanffy's growth models. A generalized family of von Bertalanffy's maps, which is proportional to the right hand side of von Bertalanffy's growth equation, is studied and its dynamical approach is proposed. The system complexity is measured using Lyapunov exponents, which depend on two biological parameters: von Bertalanffy's growth rate constant and the asymptotic weight. Applications of synchronization in real world is of current interest. The behavior of birds ocks, schools of fish and other animals is an important phenomenon characterized by synchronized motion of individuals. In this work, we consider networks having in each node a von Bertalanffy's model and we study the synchronization interval of these networks, as a function of those two biological parameters. Numerical simulation are also presented to support our approaches.

From weak Allee effect to no Allee effect in Richards’ growth models

Population dynamics have been attracting interest since many years. Among the considered models, the Richards’ equations remain one of the most popular to describe biological growth processes. On the other hand, Allee effect is currently a major focus of ecological research, which occurs when positive density dependence dominates at low densities. In this chapter, we propose the dynamical study of classes of functions based on Richards’ models describing the existence or not of Allee effect. We investigate bifurcation structures in generalized Richards’ functions and we look for the conditions in the (β, r) parameter plane for the existence of a weak Allee effect region. We show that the existence of this region is related with the existence of a dovetail structure. When the Allee limit varies, the weak Allee effect region disappears when the dovetail structure also disappears. Consequently, we deduce the transition from the weak Allee effect to no Allee effect to this family of functions. To support our analysis, we present fold and flip bifurcation curves and numerical simulations of several bifurcation diagrams.