Efficient uncertainty propagation schemes for dynamical systems with stochastic finite element analysis.

Efficient uncertainty propagation schemes for dynamical systems are investigated here within the framework of stochastic finite element analysis. Uncertainty in the mathematical models arises from the incomplete knowledge or inherent variability of the various parametric and geometric properties of the physical system. These input uncertainties necessitate the use of stochastic mathematical models to accurately capture their behavior. The resolution of such stochastic models is computationally quite expensive. This work is concerned with development of model order reduction techniques for obtaining the dynamical response statistics of stochastic finite element systems. Efficient numerical methods have been proposed to propagate the input uncertainty of dynamical systems to the response variables. Response statistics of randomly parametrized structural dynamic systems have been investigated with a reduced spectral function approach. The frequency domain response and the transient evolution of the response of randomly parametrized structural dynamic systems have been studied with this approach. An efficient discrete representation of the input random field in a finite dimensional stochastic space is proposed here which has been integrated into the generic framework of the stochastic finite element weak formulation. This framework has been utilized to study the problem of random perturbation of the boundary surface of physical domains. Truncated reduced order representation of the complex mathematical quantities which are associated with the stochastic isoparametric mapping of the random domain to a deterministic master domain within the stochastic Galerkin framework have been provided. Lastly, an a-priori model reduction scheme for the resolution of the response statistics of stochastic dynamical systems has also been studied here which is based on the concept of balanced truncation. The performance and numerical accuracy of the methods proposed in this work have been exemplified with numerical simulations of stochastic dynamical systems and the convergence behavior of various error indicators

An efficient Jeans modelling of axisymmetric galaxies with multiple stellar components

Dynamical models of stellar systems represent a powerful tool to study their internal structure and dynamics, to interpret the observed morphological and kinematical fields, and also to support numerical simulations of their evolution. We present a method especially designed to build axisymmetric Jeans models of galaxies, assumed as stationary and collisionless stellar systems. The aim is the development of a rigorous and flexible modelling procedure of multicomponent galaxies, composed of different stellar and dark matter distributions, and a central supermassive black hole. The stellar components, in particular, are intended to represent different galaxy structures, such as discs, bulges, halos, and can then have different structural (density profile, flattening, mass, scale-length), dynamical (rotation, velocity dispersion anisotropy), and population (age, metallicity, initial mass function, mass-to-light ratio) properties. The theoretical framework supporting the modelling procedure is presented, with the introduction of a suitable nomenclature, and its numerical implementation is discussed, with particular reference to the numerical code JASMINE2, developed for this purpose. We propose an approach for efficiently scaling the contributions in mass, luminosity, and rotational support, of the different matter components, allowing for fast and flexible explorations of the model parameter space. We also offer different methods of the computation of the gravitational potentials associated of the density components, especially convenient for their easier numerical tractability. A few galaxy models are studied, showing internal, and projected, structural and dynamical properties of multicomponent galaxies, with a focus on axisymmetric early-type galaxies with complex kinematical morphologies. The application of galaxy models to the study of initial conditions for hydro-dynamical and $N$-body simulations of galaxy evolution is also addressed, allowing in particular to investigate the large number of interesting combinations of the parameters which determine the structure and dynamics of complex multicomponent stellar systems

Information Theoretic Analysis of the Structure-Dynamics Relationships in Complex Biological Systems

Complex systems and networks is an emerging scientific field, with applications in every area of human enquiry, for which a solid theoretical, computational and experimental foundation is lacking. As our technological capability of generating and gathering vast amounts of data from such systems is increasing, precise methods are needed to describe, analyse and synthesize such systems. Systems biology is a prime example of an interdisciplinary field aiming at tackling the complexity of biological organisms and dedicated to understanding their organizing principles and to devising efficient intervention strategies for curing diseases.A very important topic in the study of complex systems and networks is to uncover the laws that govern their structure-dynamics relationships. A complete description of the system’s behaviour as a whole can only be achieved if the structure and the dynamics are investigated together, as well as the intricate ways in which they influence each other. The understanding of structure-dynamics relationships is a key step in the control of complex systems and networks. For example, in biology, understanding these relationships in organisms would enable us to find more precise drug targets and to design better drugs to cure diseases. In gene regulatory networks, it would help devise control strategies to change the network from faulty states that correspond to disease states, to normal states that give the healthy phenotype. When we observe a dynamical behaviour that is different from the normal, healthy one, the knowledge about the structure-dynamics relationships would help us identify which part of the structure gives rise to such behaviour. Then, we would know where and how to change the structure, to return the system to its normal dynamics, that is, to obtain a desired dynamical behaviour.A feasible way of investigating the structure-dynamics relationships is by measuring the amount of information that is communicated in the system and by analysing the patterns of information propagation within its elements. These objectives can be achieved by means of information theory. To this end, with concepts from Kolmogorov complexity and from Shannon’s information theory, we create novel analysis methods of the structure-dynamics relationships in two models of complex systems: an executable model of the human immune systems and the random Boolean network model of gene regulatory networks.In these endeavours, the information-theoretic means of identifying and measuring the information propagation in complex systems and networks needs to be improved and extended. Research is needed into the theoretical foundations of information theory, to refine existing equations and to introduce new ones that can give more accurate results in the investigation of the propagation of information and its applications to the structure-dynamics relationships. To this end, we bring analytical contributions to the generalization of Shannon’s information theory, named Rényi’s information theory. Thus, we continue the development of the theoretical foundations of information theory, for new and better applications in complex systems science and engineering.The goal of this thesis is to characterize various aspects of the structure-dynamics relationships in models of complex biological systems, by means of information theory. Moreover, our goal is to prove that information theory is a model independent analysis framework that can be applied to any class of models. We pursue our objective, by analysing two different classes of models: an executable model of the human immune system and the random Boolean network model of gene regulatory networks.In the executable model of the regulation of cytokines within the human immune system, our aim is to develop computationally feasible analysis methods that can extract meaningful biological information from the complex encoding of the dynamical behaviour of different perturbations of the wild type system. We aim at classifying several structural perturbations of the system, using only their dynamical information. We endeavour to create methods that can make predictions about the structural parameters that should be changed in order to obtain a desired dynamical behaviour. These conclusions have direct applications to the fine-tuning of the real-world biological experiments performed on the system, of whose computational model we analyse. The benefits of our predictions would be increased efficiency and increased reduction of the time required to optimize the parameters of the real-world biological experiments.In the random Boolean network model of gene regulatory networks, our goal is to develop an experimental order parameter that can characterize the dynamical regime of the network, from the dynamical behaviour that simulates that obtained from the measurements of real-world biological experiments. Moreover, we aim at proving that structural information is hidden in the dynamics of random Boolean networks and that it can be extracted with methods from information theory. We study ensembles of random Boolean networks from two distinct structural classes, which take into account the stochasticity present in real biological systems.Another goal of this study is to bring analytical contributions to the field of Rényi’s information theory, which is a generalization of Shannon’s information theory. Recently, it has found novel applications in the study of the structure-dynamics relationships in complex systems and networks

Deterministic Dynamics and Chaos: Epistemology and Interdisciplinary Methodology

We analyze, from a theoretical viewpoint, the bidirectional interdisciplinary relation between mathematics and psychology, focused on the mathematical theory of deterministic dynamical systems, and in particular, on the theory of chaos. On one hand, there is the direct classic relation: the application of mathematics to psychology. On the other hand, we propose the converse relation which consists in the formulation of new abstract mathematical problems appearing from processes and structures under research of psychology. The bidirectional multidisciplinary relation from-to pure mathematics, largely holds with the "hard" sciences, typically physics and astronomy. But it is rather new, from the social and human sciences, towards pure mathematics

Control of complex networks requires both structure and dynamics

The study of network structure has uncovered signatures of the organization of complex systems. However, there is also a need to understand how to control them; for example, identifying strategies to revert a diseased cell to a healthy state, or a mature cell to a pluripotent state. Two recent methodologies suggest that the controllability of complex systems can be predicted solely from the graph of interactions between variables, without considering their dynamics: structural controllability and minimum dominating sets. We demonstrate that such structure-only methods fail to characterize controllability when dynamics are introduced. We study Boolean network ensembles of network motifs as well as three models of biochemical regulation: the segment polarity network in Drosophila melanogaster, the cell cycle of budding yeast Saccharomyces cerevisiae, and the floral organ arrangement in Arabidopsis thaliana. We demonstrate that structure-only methods both undershoot and overshoot the number and which sets of critical variables best control the dynamics of these models, highlighting the importance of the actual system dynamics in determining control. Our analysis further shows that the logic of automata transition functions, namely how canalizing they are, plays an important role in the extent to which structure predicts dynamics.Comment: 15 pages, 6 figure

Relative Stability of Network States in Boolean Network Models of Gene Regulation in Development

Progress in cell type reprogramming has revived the interest in Waddington's concept of the epigenetic landscape. Recently researchers developed the quasi-potential theory to represent the Waddington's landscape. The Quasi-potential U(x), derived from interactions in the gene regulatory network (GRN) of a cell, quantifies the relative stability of network states, which determine the effort required for state transitions in a multi-stable dynamical system. However, quasi-potential landscapes, originally developed for continuous systems, are not suitable for discrete-valued networks which are important tools to study complex systems. In this paper, we provide a framework to quantify the landscape for discrete Boolean networks (BNs). We apply our framework to study pancreas cell differentiation where an ensemble of BN models is considered based on the structure of a minimal GRN for pancreas development. We impose biologically motivated structural constraints (corresponding to specific type of Boolean functions) and dynamical constraints (corresponding to stable attractor states) to limit the space of BN models for pancreas development. In addition, we enforce a novel functional constraint corresponding to the relative ordering of attractor states in BN models to restrict the space of BN models to the biological relevant class. We find that BNs with canalyzing/sign-compatible Boolean functions best capture the dynamics of pancreas cell differentiation. This framework can also determine the genes' influence on cell state transitions, and thus can facilitate the rational design of cell reprogramming protocols.Comment: 24 pages, 6 figures, 1 tabl

Dynamical Systems, Stability, and Chaos

In this expository and resources chapter we review selected aspects of the mathematics of dynamical systems, stability, and chaos, within a historical framework that draws together two threads of its early development: celestial mechanics and control theory, and focussing on qualitative theory. From this perspective we show how concepts of stability enable us to classify dynamical equations and their solutions and connect the key issues of nonlinearity, bifurcation, control, and uncertainty that are common to time-dependent problems in natural and engineered systems. We discuss stability and bifurcations in three simple model problems, and conclude with a survey of recent extensions of stability theory to complex networks.Comment: 28 pages, 10 figures. 26/04/2007: The book title was changed at the last minute. No other changes have been made. Chapter 1 in: J.P. Denier and J.S. Frederiksen (editors), Frontiers in Turbulence and Coherent Structures. World Scientific Singapore 2007 (in press

Symptoms of complexity in a tourism system

Tourism destinations behave as dynamic evolving complex systems, encompassing numerous factors and activities which are interdependent and whose relationships might be highly nonlinear. Traditional research in this field has looked after a linear approach: variables and relationships are monitored in order to forecast future outcomes with simplified models and to derive implications for management organisations. The limitations of this approach have become apparent in many cases, and several authors claim for a new and different attitude. While complex systems ideas are amongst the most promising interdisciplinary research themes emerged in the last few decades, very little has been done so far in the field of tourism. This paper presents a brief overview of the complexity framework as a means to understand structures, characteristics, relationships, and explores the implications and contributions of the complexity literature on tourism systems. The objective is to allow the reader to gain a deeper appreciation of this point of view.Comment: 32 pages, 3 figures, 1 table; accepted in Tourism Analysi