Dissertations / Theses on the topic 'Chaos Theory (Mathematics)'

Didactic proposals on modelling in mathematics education mostly give priority to models which describe, explain as well as partially forecast and provide mathematical solutions to real situations. A view of the modelling concept of informatics, which also initiates rapidly generalised deliberations of models, can also make a contribution to the spectrum of models, which are treated in a meaningful sense in mathematics lessons so as to expand some interesting aspects. In this paper, this is illustrated by means of conceptual design models – and, here, especially of process models – using the example of elevator organisation in a multi-storey construction.

This dissertation is about some of the most important philosophical aspects of chaos research, a famous recent mathematical area of research about deterministic yet unpredictable and irregular, or even random behaviour. It consists of three parts. First, as a basis for the dissertation, I examine notions of unpredictability in ergodic theory, and I ask what they tell us about the justification and formulation of mathematical definitions. The main account of the actual practice of justifying mathematical definitions is Lakatos's account on proof-generated definitions. By investigating notions of unpredictability in ergodic theory, I present two previously unidentified but common ways of justifying definitions. Furthermore, I criticise Lakatos's account as being limited: it does not acknowledge the interrelationships between the different kinds of justification, and it ignores the fact that various kinds of justification - not only proof-generation - are important. Second, unpredictability is a central theme in chaos research, and it is widely claimed that chaotic systems exhibit a kind of unpredictability which is specific to chaos. However, I argue that the existing answers to the question "What is the unpredictability specific to chaos?" are wrong. I then go on to propose a novel answer, viz. the unpredictability specific to chaos is that for predicting any event all sufficiently past events are approximately probabilistically irrelevant. Third, given that chaotic systems are strongly unpredictable, one is led to ask: are deterministic and indeterministic descriptions observationally equivalent, i.e., do they give the same predictions? I treat this question for measure-theoretic deterministic systems and stochastic processes, both of which are ubiquitous in science. I discuss and formalise the notion of observational equivalence. By proving results in ergodic theory, I first show that for many measure-preserving deterministic descriptions there is an observationally equivalent indeterministic description, and that for all indeterministic descriptions there is an observationally equivalent deterministic description. I go on to show that strongly chaotic systems are even observationally equivalent to some of the most random stochastic processes encountered in science. For instance, strongly chaotic systems give the same predictions at every observation level as Markov processes or semi-Markov processes. All this illustrates that even kinds of deterministic and indeterministic descriptions which, intuitively, seem to give very different predictions are observationally equivalent. Finally, I criticise the claims in the previous philosophical literature on observational equivalence.

This dissertation presents an effort to implement nonlinear dynamic tools adapted from chaos theory in financial applications. Chaos theory might be useful in explaining the dynamics of financial markets, since chaotic models are capable of exhibiting behaviour similar to that observed in empirical financial data. In this context, the scope of this research is to provide an insight into the role that nonlinearities and, in particular, chaos theory may play in explaining the dynamics of financial markets. From a theoretical point of view, the basic features of chaos theory, as well as, the rationales for bringing chaos theory to the attention of financial researchers are discussed. Empirically, the fundamental issue of determining whether chaos can be observed in financial time series is addressed. Regarding the latter, empirical literature has been controversial. A quite exhaustive analysis of the existing literature is provided, revealing the inadequacies in terms of methodology and the testing framework adopted, so far. A new "multiple testing" methodology is developed combining methods and techniques from the fields of both Natural Sciences and the Economics, most of which have not been applied to financial data before. A serious effort has been made to fill, as much as possible, the gap which results from the lack of a proper statistical framework for the chaotic methods. To achieve this the bootstrap methodology is adopted. The empirical part of this work focuses on the comparison of two markets with different levels of maturity; the Athens Stock Exchange (ASE), an emerging market, and London Stock Exchange (LSE). Our aim is to determine whether structural differences exist in these markets in terms of chaotic dynamics. In the empirical level we find nonlinearities in both markets by the use of the BDS test. R/S analysis reveals fractality and long term memory for the ASE series only. Chaotic methods, such as the correlation dimension (and related methods and techniques) and the largest Lyapunov exponent estimation, cannot rule out a chaotic explanation for the ASE market, but no such indication could be found for the LSE market. Noise filtering by the SVD method does not alter these findings. Alternative techniques based on nonlinear nearest neighbour forecasting methods, such as the "piecewise polynomial approximation" and the "simplex" methods, support our aforementioned conclusion concerning the ASE series. In all, our results suggest that, although nonlinearities are present, chaos is not a widespread phenomenon in financial markets and it is more likely to exist in less developed markets such as the ASE. Even then, chaos is strongly mixed with noise and the existence of low-dimensional chaos is highly unlikely. Finally, short-term forecasts trying to exploit the dependencies found in both markets seem to be of no economic importance after accounting for transaction costs, a result which supports further our conclusions about the limited scope and practical implications of chaos in Finance.

Systems whose time evolutions are entirely deterministic can nevertheless be studied probabilistically, i. e. in terms of the evolution of probability distributions rather than individual trajectories. This approach is central to the dynamics of ensembles (statistical mechanics) and systems with uncertainty in the initial conditions. It is also the basis of ergodic theory--the study of probabilistic invariants of dynamical systems--which provides one framework for understanding chaotic systems whose time evolutions are erratic and for practical purposes unpredictable. Delay differential equations (DDEs) are a particular class of deterministic systems, distinguished by an explicit dependence of the dynamics on past states. DDEs arise in diverse applications including mathematics, biology and economics. A probabilistic approach to DDEs is lacking. The main problems we consider in developing such an approach are (1) to characterize the evolution of probability distributions for DDEs, i. e. develop an analog of the Perron-Frobenius operator; (2) to characterize invariant probability distributions for DDEs; and (3) to develop a framework for the application of ergodic theory to delay equations, with a view to a probabilistic understanding of DDEs whose time evolutions are chaotic. We develop a variety of approaches to each of these problems, employing both analytical and numerical methods. In transient chaos, a system evolves erratically during a transient period that is followed by asymptotically regular behavior. Transient chaos in delay equations has not been reported or investigated before. We find numerical evidence of transient chaos (fractal basins of attraction and long chaotic transients) in some DDEs, including the Mackey-Glass equation. Transient chaos in DDEs can be analyzed numerically using a modification of the "stagger-and-step" algorithm applied to a discretized version of the DDE.

Ce travail met en jeu plusieurs systèmes dynamiques sur des tores en dimension finie, pour lesquels on sait établir des théorèmes limites, qui permettent de préciser leur comportement stochastique. On généralise d'abord le théorème limite local usuel sur un sous-shift de type fini, en ajoutant un terme de perturbation, en reprenant la preuve classique, par des techniques d'opérateurs. On en déduit un théorème limite local pour les sommes de « Riesz-Raïkov unitaires étendues », et des observables höldériennes. Pour cela, on reprend une méthode employée par Bernard Petit, en utilisant des codages symboliques, et le théorème limite local avec perturbation. Puis, on présente plusieurs situations de composées d'automorphismes hyperboliques du tore en dimension deux pour lesquelles on sait établir un théorème limite central quelque soit le choix de la composée. En particulier, on aborde le cas des matrices à coefficients entiers positifs.

This thesis advances the theory of network specialization by characterizing the effect of network specialization on the eigenvectors of a network. We prove and provide explicit formulas for the eigenvectors of specialized graphs based on the eigenvectors of their parent graphs. The second portion of this thesis applies network specialization to learning problems. Our work focuses on training reservoir computers to mimic the Lorentz equations. We experiment with random graph, preferential attachment and small world topologies and demonstrate that the random removal of directed edges increases predictive capability of a reservoir topology. We then create a new network model by growing networks via targeted application of the specialization model. This is accomplished iteratively by selecting top preforming nodes within the reservoir computer and specializing them. Our generated topology out-preforms all other topologies on average.

Nous nous intéressons dans cette thèse à la théorie cinétique et aux systèmes de particules dans le cadre des équations de Boltzmann et Landau. Premièrement, nous étudions la dérivation des équations cinétiques comme des limites de champ moyen des systèmes de particules, en utilisant le concept de propagation du chaos. Plus précisément, nous étudions les probabilités chaotiques sur l'espace de phase de ces systèmes de particules : la sphère de Boltzmann, qui correspond à l'espace de phase d'un système de particules qui évolue conservant le moment et l'énergie ; et la sphère de Kac, correspondant à un système de particules qui conserve seulement l'énergie. Ensuite, nous nous intéressons à la propagation du chaos, avec des estimations quantitatives et uniforme en temps, pour les équations de Boltzmann et Landau. Deuxièmement, nous étudions le comportement asymptotique en temps grand des solutions de l'équation de Landau.

Hydraulisk modellering är ett viktigt verktyg vid utvärdering av lämpliga åtgärder för torrfåror. Modelleringen påverkas dock alltid av osäkerheter och om dessa är stora kan en modells simuleringsresultat bli opålitligt. Det kan därför vara viktigt att presentera dess simuleringsresultat tillsammans med osäkerheter. Denna studie utreder olika typer av osäkerheter som kan påverka hydrauliska modellers simuleringsresultat. Dessutom utförs känslighetsanalyser där en andel av osäkerheten i simuleringsresultatet tillskrivs de olika inmatningsvariablerna som beaktas. De parametrar som ingår i analysen är upplösningen i använd terrängmodell, upplösning i den hydrauliska modellens beräkningsnät, inflöde till modellen och råheten genom Mannings tal. Studieobjektet som behandlades i denna studie var en torrfåra som ligger nedströms Sandforsdammen i Skellefteälven och programvaran TELEMAC-MASCARET nyttjades för samtliga hydrauliska simuleringar i denna studie.  För att analysera osäkerheter kopplade till upplösning i en terrängmodell och ett beräkningsnät användes ett kvalitativt tillvägagångsätt. Ett antal simuleringar utfördes där alla parametrar förutom de kopplade till upplösning fixerades. Simuleringsresultaten illustrerades sedan genom profil, sektioner, enskilda raster och raster som visade differensen mellan olika simuleringar. Resultaten för analysen visade att en låg upplösning i terrängmodeller och beräkningsnät kan medföra osäkerheter lokalt där det är högre vattenhastigheter och där det finns stor variation i geometrin. Några signifikanta effekter kunde dock inte skönjas på större skala.  Separat gjordes kvantitativa osäkerhets- och känslighetsanalyser för vattendjup och vattenhastighet i torrfåran. Inmatningsparametrarna inflöde till modellen och råhet genom Mannings tal ansågs medföra störst påverkan och övriga parametrar fixerades således. Genom script skapade i programmeringsspråket Python tillsammans med biblioteket OpenTURNS upprättades ett stort urval av möjliga kombinationer för storlek på inflöde och Mannings tal. Alla kombinationer som skapades antogs till fullo täcka upp för den totala osäkerheten i inmatningsparametrarna. Genom att använda urvalet för simulering kunde osäkerheten i simuleringsresultaten också beskrivas. Osäkerhetsanalyser utfördes både genom klassisk beräkning av statistiska moment och genom Polynomial Chaos Expansion. En känslighetsanalys följde sedan där Polynomial Chaos Expansion användes för att beräkna Sobols känslighetsindex för inflödet och Mannings tal i varje kontrollpunkt. Den kvantitativa osäkerhetsanalysen visade att det fanns relativt stora osäkerheter för både vattendjupet och vattenhastighet vid behandlat studieobjekt. Flödet bidrog med störst påverkan på osäkerheten medan Mannings tals påverkan var insignifikant i jämförelse, bortsett från ett område i modellen där dess påverkan ökade markant.
Hydraulic modelling is an important tool when measures for dry river stretches are assessed. The modelling is however always affected by uncertainties and if these are big the simulation results from the models could become unreliable. It may therefore be important to present its simulation results together with the uncertainties. This study addresses various types of uncertainties that may affect the simulation results from hydraulic models. In addition, sensitivity analysis is conducted where a proportion of the uncertainty in the simulation result is attributed to the various input variables that are included. The parameters included in the analysis are terrain model resolution, hydraulic model mesh resolution, inflow to the model and Manning’s roughness coefficient. The object studied in this paper was a dry river stretch located downstream of Sandforsdammen in the river of Skellefteälven, Sweden. The software TELEMAC-MASCARET was used to perform all hydraulic simulations for this thesis.  To analyze the uncertainties related to the resolution for the terrain model and the mesh a qualitative approach was used. Several simulations were run where all parameters except those linked to the resolution were fixed. The simulation results were illustrated through individual rasters, profiles, sections and rasters that showed the differences between different simulations. The results of the analysis showed that a low resolution for terrain models and meshes can lead to uncertainties locally where there are higher water velocities and where there are big variations in the geometry. However, no significant effects could be discerned on a larger scale.  Separately, quantitative uncertainty and sensitivity analyzes were performed for the simulation results, water depth and water velocity for the dry river stretch. The input parameters that were assumed to have the biggest impact were the inflow to the model and Manning's roughness coefficient. Other model input parameters were fixed. Through scripts created in the programming language Python together with the library OpenTURNS, a large sample of possible combinations for the size of inflow and Manning's roughness coefficient was created. All combinations were assumed to fully cover the uncertainty of the input parameters. After using the sample for simulation, the uncertainty of the simulation results could also be described. Uncertainty analyses were conducted through both classical calculation of statistical moments and through Polynomial Chaos Expansion. A sensitivity analysis was then conducted through Polynomial Chaos Expansion where Sobol's sensitivity indices were calculated for the inflow and Manning's M at each control point. The analysis showed that there were relatively large uncertainties for both the water depth and the water velocity. The inflow had the greatest impact on the uncertainties while Manning's M was insignificant in comparison, apart from one area in the model where its impact increased.

In this work methods for performing time series prediction on complex real world time series are examined. In particular series exhibiting non-linear or chaotic behaviour are selected for analysis. A range of methodologies based on Takens' embedding theorem are considered and compared with more conventional methods. A novel combination of methods for determining the optimal embedding parameters are employed and tried out with multivariate financial time series data and with a complex series derived from an experiment in biotechnology. The results show that this combination of techniques provide accurate results while improving dramatically the time required to produce predictions and analyses, and eliminating a range of parameters that had hitherto been fixed empirically. The architecture and methodology of the prediction software developed is described along with design decisions and their justification. Sensitivity analyses are employed to justify the use of this combination of methods, and comparisons are made with more conventional predictive techniques and trivial predictors showing the superiority of the results generated by the work detailed in this thesis.

The thermally-driven rotating annulus is a laboratory experiment used to study the dynamics of planetary atmospheres under controlled and reproducible conditions. The predictability of this experiment is studied by applying the same principles used to predict the atmosphere. A forecasting system for the annulus is built using the analysis correction method for data assimilation and the breeding method for ensemble generation. The results show that a range of flow regimes with varying complexity can be accurately assimilated, predicted, and studied in this experiment. This framework is also intended to demonstrate a proof-of-concept: that the annulus could be used as a testbed for meteorological techniques under laboratory conditions. First, a regime diagram is created using numerical simulations in order to select points in parameter space to forecast, and a new chaotic flow regime is discovered within it. The two components of the framework are then used as standalone algorithms to measure predictability in the perfect model scenario and to demonstrate data assimilation. With a perfect model, regular flow regimes are found to be predictable until the end of the forecasts, and chaotic regimes are predictable over hundreds of seconds. There is a difference in the way predictability is lost between low-order chaotic regimes and high-order chaos. Analysis correction is shown to be accurate in both regular and chaotic regimes, with residual velocity errors about 3-8 times the observational error. Specific assimilation scenarios studied include information propagation from data-rich to data-poor areas, assimilation of vortex shedding observations, and assimilation over regime and rotation rate transitions. The full framework is used to predict regular and chaotic flow, verifying the forecasts against laboratory data. The steady wave forecasts perform well, and are predictable until the end of the available data. The amplitude and structural vacillation forecasts lose quality and skill by a combination of wave drift and wavenumber transition. Amplitude vacillation is predictable up to several hundred seconds ahead, and structural vacillation is predictable for a few hundred seconds.

Thermodynamical formalism is a relatively recent area of pure mathematics owing a lot to some classical notions of thermodynamics. On this thesis we state and prove some of the main results in the area of thermodynamical formalism. The first chapter is an introduction to ergodic theory. Some of the main theorems are proved and there is also a quite thorough study of the topology that arises in Borel probability measure spaces. In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. Finally the sixth chapter introduces the notion of conformal measures.

The principal’s leadership and curriculum development are considered the core elements for creating a high performing junior high school. In Taiwan, mathematics curriculum reform has been an ongoing topic since 1994. The pedagogy, classroom interactions, and the underlying philosophy of mathematics education have varied with different versions of guidelines. These changes inevitably increase the requirement for principals’ leadership in order to effectively implement the curriculum reform. Principals’ leadership is essential to the success of the implementation in their school. This study aimed to explore and identify the leadership of junior high school principals whose schools had been judged by the Taipei City Government as Grade A junior high schools. Principals’ implementations of the reformed mathematics curriculum were used as examples to generate insights of their leadership. This study drew upon a multiple-case study approach. Data were collected from interviews, observations, and documentations. Bass and Avolio’s (1997) full range leadership theory provided a structure for gaining insight into these principals’ leadership practices. Five Grade A Taipei junior high school principals participated and shared their leadership concepts and experiences. Findings revealed that the leadership preferences of the five principles varied considerably. Management by exception-active, contingent reward, individualised consideration, and idealised influence were Grade A Taipei junior high school principals’ preferred leadership practices. In addition, principals’ leadership strategies associated with these practices were identified. These principals had adopted a range of leadership strategies according to the staff and school needs. Results of this study have implications for both Taiwanese principals and education departments. Principals can enhance their leadership by gaining more understanding about the Grade A principals’ leadership practices and strategies. Taiwanese education departments can improve school leadership training programs by focusing on these practices and strategies, which may also lead to more effective strategies for implementing national curriculum reform.

Les systèmes dynamiques discrets, œuvrant en itérations chaotiques ou asynchrones, se sont avérés être des outils particulièrement intéressants à utiliser en sécurité informatique, grâce à leur comportement hautement imprévisible, obtenu sous certaines conditions. Ces itérations chaotiques satisfont les propriétés de chaos topologiques et peuvent être programmées de manière efficace. Dans l’état de l’art, elles ont montré tout leur intérêt au travers de schémas de tatouage numérique. Toutefois, malgré leurs multiples avantages, ces algorithmes existants ont révélé certaines limitations. Cette thèse a pour objectif de lever ces contraintes, en proposant de nouveaux processus susceptibles de s’appliquer à la fois au domaine du tatouage numérique et au domaine de la stéganographie. Nous avons donc étudié ces nouveaux schémas sur le double plan de la sécurité dans le cadre probabiliste. L’analyse de leur biveau de sécurité respectif a permis de dresser un comparatif avec les autres processus existants comme, par exemple, l’étalement de spectre. Des tests applicatifs ont été conduits pour stéganaliser des processus proposés et pour évaluer leur robustesse. Grâce aux résultats obtenus, nous avons pu juger de la meilleure adéquation de chaque algorithme avec des domaines d’applications ciblés comme, par exemple, l’anonymisation sur Internet, la contribution au développement d’un web sémantique, ou encore une utilisation pour la protection des documents et des donnés numériques. Parallèlement à ces travaux scientifiques fondamentaux, nous avons proposé plusieurs projets de valorisation avec pour objectif la création d’une entreprise de technologies innovantes
Discrete dynamical systems by chaotic or asynchronous iterations have proved to be highly interesting toolsin the field of computer security, thanks to their unpredictible behavior obtained under some conditions. Moreprecisely, these chaotic iterations possess the property of topological chaos and can be programmed in anefficient way. In the state of the art, they have turned out to be really interesting to use notably through digitalwatermarking schemes. However, despite their multiple advantages, these existing algorithms have revealedsome limitations. So, these PhD thesis aims at removing these constraints, proposing new processes whichcan be applied both in the field of digital watermarking and of steganography. We have studied these newschemes on two aspects: the topological security and the security based on a probabilistic approach. Theanalysis of their respective security level has allowed to achieve a comparison with the other existing processessuch as, for example, the spread spectrum. Application tests have also been conducted to steganalyse and toevaluate the robustness of the algorithms studied in this PhD thesis. Thanks to the obtained results, it has beenpossible to determine the best adequation of each processes with targeted application fields as, for example,the anonymity on the Internet, the contribution to the development of the semantic web, or their use for theprotection of digital documents. In parallel to these scientific research works, several valorization perspectiveshave been proposed, aiming at creating a company of innovative technology

In this thesis I hope to provide a clear and concise introduction to Feigenbaum scaling accessible to undergraduate students. This is accompanied by a description of how to obtain numerical results by various means. A more intricate approach drawing from renormalization theory as well as a short consideration of some of the topological properties will also be presented. I was furthermore trying to put great emphasis on diagrams throughout the text to make the contents more comprehensible and intuitive.

La résolution du problème à N-corps constitue aussi bien en mécanique classique qu'en mécanique quantique un des grands enjeux de la physique. En physique nucléaire, diverses méthodes ont été développées pour obtenir des solutions approchées permettant de décrire convenablement les propriétés des noyaux (spectres, transitions électromagnétiques...). Dans cette thèse, nous avons tout d'abord rappelé comment les symétries pouvaient être utilisées pour obtenir des solutions exactes. Nous avons notamment insisté sur le rôle occupé par l'algèbre unitaire en mécanique quantique et nous avons développé et implémenté une façon de construire les représentations irréductibles de cette algèbre à partir d'un état dit de poids maximal et dans lesquelles ont été calculés les spectres de systèmes bosoniques et fermioniques aussi bien avec des interactions réalistes qu'avec des interactions aléatoires. L'utilisation d'interactions aléatoires à 1- et 2-corps conservant le moment angulaire a révélé que certaines caractéristiques des spectres (état fondamental de moment angulaire nul, existence de bandes rotationnelles, vibrationnelles...) étaient robustes. Ainsi dans une seconde partie, nous avons montré que le choix de l'espace de valence conditionne fortement les spectres possibles d'un système quantique : en particulier, nous avons élaboré une méthode géométrique qui, dans certains cas, permet de prévoir les propriétés du fondamental. Nous avons également présenté des résultats numériques dans des situations où la méthode géométrique ne s'applique pas. Dans la dernière partie, nous nous sommes intéressés au lien entre le chaos et les spectres des noyaux obtenus avec des interactions réalistes.

In the course Discrete Deterministic Chaos, Dr. Mark Daniels introduces students to Chaos Theory and explores many topics within the field. Students prove many of the key results that are discussed in class and work through examples of each topic. Connections to the secondary mathematics curriculum are made throughout the course, and students discuss how the topics in the course could be implemented in the classroom. This paper will provide an overview of the topics covered in the course, Discrete Deterministic Chaos, and provide additional discussion on various related topics.
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Complex nonlinear and chaotic responses have been recently observed in various compliant ocean systems. These systems are characterized by a nonlinear mooring restoring force and a coupled fluid-structure interaction exciting force. A general class of ocean mooring system models is formulated by incorporating a variable mooring configuration and the exact form of the hydrodynamic excitation. The multi-degree of freedom system, subjected to combined parametric and external excitation, is shown to be complex, coupled and strongly nonlinear. Stability analysis by a Liapunov function approach reveals global system attraction which ensures that solutions remain bounded for small excitation. Construction of the system's Poincare map and stability analysis of the map's fixed points correspond to system stability of near resonance periodic orbits. Investigation of nonresonant solutions is done by a local variational approach. Tangent and period doubling bifurcations are identified by both local stability analysis techniques and are further investigated to reveal global bifurcations. Application of Melnikov's method to the perturbed averaged system provides an approximate criterion for the existence of transverse homoclinic orbits resulting in chaotic system dynamics. Further stability analysis of the subharmonic and ultraharmonic solutions reveals a cascade of period doubling which is shown to evolve to a strange attractor. Investigation of the bifurcation criteria obtained reveals a steady state superstructure in the bifurcation set. This superstructure identifies a similar bifurcation pattern of coexisting solutions in the sub, ultra and ultrasubharmonic domains. Within this structure strange attractors appear when a period doubling sequence is infinite and when abrupt changes in the size of an attractor occur near tangent bifurcations. Parametric analysis of system instabilities reveals the influence of the convective inertial force which can not be neglected for large response and the bias induced by the quadratic viscous drag is found to be a controlling mechanism even for moderate sea states. Thus, stability analyses of a nonlinear ocean mooring system by semi-analytical methods reveal the existence of bifurcations identifying complex periodic and aperiodic nonlinear phenomena. The results obtained apply to a variety of nonlinear ocean mooring and towing system configurations. Extensions and applications of this research are discussed.
Graduation date: 1992

A matemática do risco, enquanto o ramo da matemática que apoia a ciência do risco na investigação do seu objecto, o risco, enquanto tal, não possui um corpo conceptual formal unificado. A presente tese contribui para um tal corpo, investigando a matemática do risco a partir dos seus fundamentos, assumindo o risco, enquanto tal, como o objecto de investigação. É reconhecido que a teoria das categorias permite introduzir um substrato matemático comum, tanto para a matemática da medição do risco como para a modelação de sistemas em situações de risco. Assim, os fundamentos da matemática do risco são trabalhados a partir do formalismo da lógica da teoria das categorias, no seio da linguagem formal LCat, subjacente ao cálculo categorial, permitindo desenvolver a conexão entre o objecto da ciência do risco, o formalismo e a ciência dos sistemas, a qual constitui um substrato conceptual científico para a ciência do risco. Investigando a conexão entre a teoria das categorias, a topologia categorial, e as noções de acaso, aleatório, estocástico, caos e aleatoriedade algorítmica, o formalismo da teoria das categorias é directamente ligado à matemática do risco e, assim, demonstra-se que o mesmo fornece um enquadramento unificado, no qual, o risco, enquanto tal, assim como a modelação dos sistemas em situações de risco podem ser apreendidos e trabalhados, a partir de uma perspectiva matemática. Apresenta-se, também, uma aplicação do formalismo à captura de turbulência multifractal, risco de sincronização e efeitos de Malcolm, com implicações para as finanças e para a gestão do risco.
Risk mathematics, as the branch of mathematics that supports risk science in the research of its object, the risk, as such, lacks a unified formal conceptual body. The present thesis makes a contribution towards such a body, by researching risk mathematics from its foundations, assuming the risk, as such, as the object of research. It is recognized that category theory allows the introduction of a common mathematical substratum, both for the risk measurement mathematics as well as for the modelling of systems in situations of risk. Thus, the foundations of risk mathematics are worked upon from the formalism of category theory’s logic, within the formal language LCat, underlying categorial calculus, which allows the development of the connection between the object of risk science, the formalism itself and systems science, which constitutes a scientific conceptual substratum for risk science. By researching the connection between category theory, categorial topology, and the notions of acaso1, aleatorial, stochastic, chaos and algorithmic randomness, category theory’s formalism is directly linked to risk mathematics and, thus, it is shown to provide for a unified framework upon which risk, as such, as well as the modelling of systems in situations of risk can be apprehended and worked upon, from a mathematical perspective. It is also shown an application of the formalism to the capturing of multifractal turbulence, synchronization risk and Malcolm effects, with implications for finance and risk management.