John N. Tsitsiklis, Michael Athans. / "February 1983." / Bibliography: p. 10-11. / grant NASA/NGL-22-009-124

Orientador: Ricardo Egydio de Carvalho / Banca: Ana Paula Mijolaro / Banca: Luiz Antônio Barreiro / Resumo: De maneira geral, o comportamento dinâmico de sistemas não lineares é caracterizado pela imprevisibilidade e extrema sensibilidade às condições iniciais e aos parâmetros do sistema. A sensibilidade dessas condições pode ser analisada a partir dos expoentes de Lyapunov, quando são consideradas órbitas infinitesimalmente próximas. O mapa escolhido para análise é o modelo denominado "Mapa padrão não - twist dissipativo labiríntico", que apresenta as chamadas curvas shearless. O estudo desenvolvido analisa esse sistema com a introdução de dissipação e com parâmetros de perturbação variáveis na presença de três curvas shearless. O objetivo é compreender a evolução da dinâmica destas curvas no espaço de fase e no diagrama de Lyapunov a fim de caracterizar qual shearless é mais robusta frente á variação dos parâmetros de dissipação e perturbação / Abstract: In general, the dynamical behavior of non-linear systems is characterized by unpredictability and extreme sensibility to the initial conditions and to the parameters of the system. The sensitivity of these conditions can be analyzed from the Lyapunov exponents, when infinitesimally close orbits are considered. The map we have chosen for analysis is the model denoted as "Labyrinthic non-twist standard map", which presents the so-called "shearless" curves. The present study analyzes this system with the introduction of dissipation and with changeable parameters of perturbation in the presence of three shearless curves. The objective is to understand the evolution of the dynamics of the curves in the phase space and in the diagram of Lyapunov in order to characterize which shearless is more robust under the variation of both parameters, dissipation and perturbation / Mestre

Made available in DSpace on 2014-06-11T19:26:15Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-09-11Bitstream added on 2014-06-13T20:47:15Z : No. of bitstreams: 1 lenarduzzi_fn_me_sjrp.pdf: 432607 bytes, checksum: 6e0e82d0a71ba0e530e2f097612c9be5 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / O objetivo principal deste trabalho e estudar os resultados apresentados por R. Zeimuller em Poisson Suspensions of Compactly Regenerative Transformations[Z0]. Neste artigo, partindo de um espaço de medida σ-finito (X;A;μ) com uma transformação ergódica T, o autor consideração de T em poeiras enumeráveis de pontos, o que define uma transformação T num espaço de probabilidade ~ X. Será mostrado que ~ T e invariante e ergódica para uma medida ~μ em ~ X, que est a relacionada com estes conjuntos enumer aveis de pontos. Apesar de não valer o teorema de Birkhoff para o espaço inicial (X;A;μ ) que tem medida infinita, vale a convergência das médias ergódicas neste novo espaço, o que permite recuperar a medida de um conjunto A em termos do número de visitas a A se forem consideradas órbitas de conjuntos enumeráveis ~ μ-típicos ao invés de olhar para a órbita de um só ponto. São estabelecidas ainda condições suficientes para obter um Teorema Central do Limite que acompanha o teorema ergódico de Birkhoff para ~Sn . Também em faremos um breve estudo sobre conservatividade de aplicações em espa ços σ-nito com medida total infinita, taxa de errância de conjuntos de medida positiva e medida aleatória de Poisson / The main purpose of this work is to understand the results presented by R. Zeimuller on his paper Poisson Suspensions of Compactly Regenerative Transformati-ons[Z0]. In this paper, considering σ- nite space (X;A;μ) and a ergodic transformation T, the author considers the action of T on a countable ensemble of points, which de nes a transformation ~ acting on another probability space ~ X. It will be proved that ~ T is invariant and ergodic for a measure ~μ on ~ X, which is related to this countable set of points. We know that Birkhoff's ergodic theorem is not valid on its classical formulation to a in nite measure space (X;A;μ), however we have the convergence of the ergodic means on this new space. This allows us to, somehow, recover the measure of a given set A just looking at the number of its visits considering the orbits of a ~ μ-typical coun-table set instead of looking at the orbit of one single point. It is also established some su cient conditions in order to get a Central Limit Theorem for ~ Sn . We'll also make a brief discussion on conservativity of maps on σ-finite spaces with full measure in nity, wandering rate of positive measure and Poisson random measure. We'll also make a brief discussion on conservativity of maps on σ-finite spaces with full measure in nity, wandering rate of positive measure and Poisson random measure

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.

Orientador: Ali Messaoudi / Coorientador: Patricia Romano Cirilo / Banca: Carlos Gustavo Tamm de Araujo Moreira / Banca: Claudio Aguinaldo Buzzi / Resumo: O objetivo principal deste trabalho e estudar os resultados apresentados por R. Zeimuller em Poisson Suspensions of Compactly Regenerative Transformations[Z0]. Neste artigo, partindo de um espaço de medida σ-finito (X;A;μ) com uma transformação ergódica T, o autor consideração de T em "poeiras" enumeráveis de pontos, o que define uma transformação T num espaço de probabilidade ~ X. Será mostrado que ~ T e invariante e ergódica para uma medida ~μ em ~ X, que est a relacionada com estes conjuntos enumer aveis de pontos. Apesar de não valer o teorema de Birkhoff para o espaço inicial (X;A;μ ) que tem medida infinita, vale a convergência das médias ergódicas neste novo espaço, o que permite recuperar a medida de um conjunto A em termos do número de visitas a A se forem consideradas órbitas de conjuntos enumeráveis ~ μ-típicos ao invés de olhar para a órbita de um só ponto. São estabelecidas ainda condições suficientes para obter um Teorema Central do Limite que acompanha o teorema ergódico de Birkhoff para ~Sn . Também em faremos um breve estudo sobre conservatividade de aplicações em espa ços σ-nito com medida total infinita, taxa de errância de conjuntos de medida positiva e medida aleatória de Poisson / Abstract: The main purpose of this work is to understand the results presented by R. Zeimuller on his paper Poisson Suspensions of Compactly Regenerative Transformati-ons[Z0]. In this paper, considering σ- nite space (X;A;μ) and a ergodic transformation T, the author considers the action of T on a countable "ensemble" of points, which de nes a transformation ~ acting on another probability space ~ X. It will be proved that ~ T is invariant and ergodic for a measure ~μ on ~ X, which is related to this countable set of points. We know that Birkhoff's ergodic theorem is not valid on its classical formulation to a in nite measure space (X;A;μ), however we have the convergence of the ergodic means on this new space. This allows us to, somehow, recover the measure of a given set A just looking at the number of its visits considering the orbits of a ~ μ-typical coun-table set instead of looking at the orbit of one single point. It is also established some su cient conditions in order to get a Central Limit Theorem for ~ Sn . We'll also make a brief discussion on conservativity of maps on σ-finite spaces with full measure in nity, wandering rate of positive measure and Poisson random measure. We'll also make a brief discussion on conservativity of maps on σ-finite spaces with full measure in nity, wandering rate of positive measure and Poisson random measure / Mestre

We propose to interpret Cryogenic Electron Microscopy (CryoEM) data as a supervision for learning parameters of CryoEM microscopes. Following this formulation, we present a differentiable version of Transmission Electron Microscopy (TEM) Simulator that provides differentiability of all continuous inputs in a simulation. We demonstrate the learning capability of our simulator with two examples, detector parameter estimation and denoising. With our differentiable simulator, detector parameters can be learned from real data without time-consuming handcrafting. Besides, our simulator enables new way to denoising micrographs. We develop this simulator with the combination of Taichi and PyTorch, exploiting kernel-based and operator-based parallel differentiable programming, which results in good speed, low memory footprint and expressive code. We call our work as Differentiable TEM Detector as there are still challenges to implement a fully differentiable transmission electron microscope simulator that can further differentiate with respect to particle positions. This work presents first steps towards a fully differentiable TEM simulator. Finally, as a subsequence of our work, we abstract out the fuser that connects Taichi and PyTorch as an open-source library, Stannum, facilitating neural rendering and differentiable rendering in a broader context. We publish our code on GitHub.

This report describes the simulation of an inverted pendulum control system. The purpose is to provide an interesting learning process through high resolution color graphics animations in the control of dynamic systems. The software uses the graphic capabilities extensively to make it very user-friendly and highly interactive. A numerical analysis method is used to solve the systems of equations. The animation driven by the results is then displayed on the video terminal. Facilities range from selection of controllers, changing of system parameters, plotting graphs, and hardcopy outputs.

Fuzzy logic systems are widely used for control, system identification, and pattern recognition problems. In order to maximize their performance, it is often necessary to undertake a design optimization process in which the adjustable parameters defining a particular fuzzy system are tuned to maximize a given performance criterion. Some data to approximate are commonly available and yield what is called the supervised learning problem. In this problem we typically wish to minimize the sum of the squares of errors in approximating the data. We first introduce fuzzy logic systems and the supervised learning problem that, in effect, is a nonlinear optimization problem that at times can be non-differentiable. We review the existing approaches and discuss their weaknesses and the issues involved. We then focus on one of these problems, i.e., non-differentiability of the objective function, and show how current approaches that do not account for non-differentiability can diverge. Moreover, we also show that non-differentiability may also have an adverse practical impact on algorithmic performances. We reformulate both the supervised learning problem and piecewise linear membership functions in order to obtain a polynomial or factorable optimization problem. We propose the application of a global nonconvex optimization approach, namely, a reformulation and linearization technique. The expanded problem dimensionality does not make this approach feasible at this time, even though this reformulation along with the proposed technique still bears a theoretical interest. Moreover, some future research directions are identified. We propose a novel approach to step-size selection in batch training. This approach uses a limited memory quadratic fit on past convergence data. Thus, it is similar to response surface methodologies, but it differs from them in the type of data that are used to fit the model, that is, already available data from the history of the algorithm are used instead of data obtained according to an experimental design. The step-size along the update direction (e.g., negative gradient or deflected negative gradient) is chosen according to a criterion of minimum distance from the vertex of the quadratic model. This approach rescales the complexity in the step-size selection from the order of the (large) number of training data, as in the case of exact line searches, to the order of the number of parameters (generally lower than the number of training data). The quadratic fit approach and a reduced variant are tested on some function approximation examples yielding distributions of the final mean square errors that are improved (i.e., skewed toward lower errors) with respect to the ones in the commonly used pattern-by-pattern approach. Moreover, the quadratic fit is also competitive and sometimes better than the batch training with optimal step-sizes, thus showing an improved performance of this approach. The quadratic fit approach is also tested in conjunction with gradient deflection strategies and memoryless variable metric methods, showing errors smaller by 1 to 7 orders of magnitude. Moreover, the convergence speed by using either the negative gradient direction or a deflected direction is higher than that of the pattern-by-pattern approach, although the computational cost of the algorithm per iteration is moderately higher than the one of the pattern-by-pattern method. Finally, some directions for future research are identified. / Ph. D.

We develop a systematic approach to the problem of finding the perturbative expansion for the topological pressure for an analytic expanding dynamics (/, M) on a Riemannian manifold M. The method is based on the spectral analysis of the transfer operator C. We show that in typical cases, when / depends real-analytically on a set of perturbing parameters ,", the related operators C~ form an analytic family. This gives rise to the rigorous construction of the power series expansion for the pressure via the analytic perturbation theory for eigenvalues, [Kato]. Consequently, the pressure and related dynamical indices, such as dimension spectra, Lyapunov exponents, escape rates and Renyi entropies inherit the real-analyticity in ~ from (I,M). / Ph. D.

Thesis (MSc) -- University of Stellenbosch, 2000. / ENGLISH ABSTRACT: A discrete dynamical system is a pair (X, cf;) comprising a non-empty set X and a map cf; : X ---+ X. A study is made of the effect of repeated application of cf; on X, whereby points and subsets of X are classified according to their behaviour under iteration. These subsets include the JULIA and FATOU sets of the map and the sets of periodic and preperiodic points, and many interesting questions arise in the study of their properties. Such questions have been extensively studied in the case of complex dynamics, but much recent work has focussed on non-archimedean dynamical systems, when X is projective space over some field equipped with a non-archimedean metric. This work has uncovered many parallels to complex dynamics alongside more striking differences. In this thesis, various aspects of the theory of non-archimedean dynamics are presented, with particular reference to JULIA and FATOU sets and the relationship between good reduction of a map and the empty JULIA set. We also discuss questions of the finiteness of the sets of periodic points in special contexts. / AFRIKAANSE OPSOMMING: 'n Paar (X, <jJ) bestaande uit 'n nie-leë versameling X tesame met 'n afbeelding <jJ: X -+ X vorm 'n diskrete dinamiese sisteem. In die bestudering van so 'n sisteem lê die klem op die uitwerking op elemente van X van herhaalde toepassing van <jJ op die versameling. Elemente en subversamelings van X word geklasifiseer volgens dinamiese kriteria en op hierdie wyse ontstaan die JULIA en FATOU versamelings van die afbeelding en die versamelings van periodiese en preperiodiese punte. Interessante vrae oor die eienskappe van hierdie versamelings kom na vore. In die geval van komplekse dinamika is sulke vrae reeds deeglik bestudeer, maar onlangse werk is op nie-archimediese dinamiese sisteme gedoen, waar X 'n projektiewe ruimte is oor 'n liggaam wat met 'n nie-archimediese norm toegerus is. Hierdie werk het baie ooreenkomste maar ook treffende verskille met die komplekse dinamika uitgewys. In hierdie tesis word daar ondersoek oor verskeie aspekte van die teorie van nie-archimediese dinamika ingestel, in besonder met betrekking tot die JULIA en FATOU versamelings en die verband tussen goeie reduksie van 'n afbeelding en die leë JULIA versameling. Vrae oor die eindigheid van versamelings van periodiese punte in spesiale kontekste word ook aangebied.