Adomian decomposition method, nonlinear equations and spectral solutions of burgers equation

Tese de doutoramento. Ciências da Engenharia. 2006. Faculdade de Engenharia. Universidade do Porto, Instituto Superior Técnico. Universidade Técnica de Lisbo

Periodic time dependent Hamiltonian systems and applications

[eng] A dynamical system is one that evolves with time. This definition is so diffuse that seems to be completely useless, however, gives a good insight of the vast range of applicability of this field of Mathematics has. It is hard to track back in the history of science to find the origins of this discipline. The works by Fibonacci, in the twelfth century, concerning the population growth rate of rabbits can be already considered to belong to the above mentioned field. Newton's legacy changed the prism through the humankind watched the universe and established the starting shot of several areas of knowledge including the study of difierential equations. Newton's second law relates the acceleration, the second derivative of the position of a body with the net force acting upon it. The formulation of the law of universal gravitation settled the many body problem, the fundamental question around the field of celestial mechanics has grown. Newton itself solved the two body problem, providing an analytical proof of Kepler's laws. In the subsequent years a number of authors, among of them Euler and Lagrange, exhausted Newton's powerful ideas but none of them was able to find a closed solution of the many body problem. By the end of the nineteenth century, Poincaré changed again the point of view: The French mathematician realized that the many body problem could not be solved in the sense his predecessors expected, however, many other fundamental questions could be addressed by studying the solutions of not quantitatively but by means of their geometrical and topological properties. The ideas that bloomed in Poincaré's mind are nowadays a source of inspiration for modern scientist facing problems located along all the spectrum of human knowledge. Poincaré understood that invariant structures organize the long term behaviour of the solutions of the system. Invariant objects are, therefore, the skeleton of the dynamics. These invariant structures and their linear normal behaviour are to be analyzed carefully and this shall lead to a good insight on global aspects of the phase space. For nonintegrable systems the task of studying invariant objects and their stability is, in general, a problem which is hard to be handled rigorously. Usually, the hypotheses needed to prove specific statements on the solutions of the systems reduce the applicability of the results. This is especially relevant in physical problems: Indeed, we cannot, for instance, choose the mass of Sun to be suficiently small. The advent of the computers changed the way to undertake studies of dynamical systems. The task of writing programs for solving, numerically, problems related to specific examples is, at the present time, as important as theoretical studies. This has two main consequences: On the first hand, more involved models can be chosen to study real problems and this allow us to understand better the relation between abstract concepts and physical phenomena. Secondly, even when facing fundamental questions on dynamics, the numerical studies give us data from which build our theoretical developments. Nowadays, a large number of commercial (or public) software packages helps scientist to study simple problems avoiding the tedious work to master numerical algorithms and programming languages. These programs are coded to work in the largest possible number of different situations, therefore, they do not have the eficiency that programs written specifically for a certain purpose have. Some of the computations presented in this dissertation cannot be performed by using commercial software or, at least, not in a reasonable amount of time. For this reason, a large part of the work presented here has to do with coding and debugging programs to perform numerical computations. These programs are written to be highly eficient and adapted to each problem. At the same time, the design is done so that specific blocks of the code can be used for other computations, that is, there exist a commitment between eficiency and reusability which is hard to achieve without having full control on the code. Under these guiding principles we undertake the study of applied dynamical systems according to the following stages: From a particular problem we get a simple model, then perform a number of numerical experiments that permits us to understand the invariant objects of the system, with that information, we can isolate the relevant phenomena and identify the key elements playing a role on it. Next, we try to find an even simpler model in which we can develop theoretical arguments and produce theorems that, with more effort, can be generalized or related to other problems which, in principle, seem to be difierent to the original one. Paraphrasing Carles Simó, from a physical problem we can take the lift to the abstract world, use theoretical arguments, come out with conclusions and, finally, lift down to the real world and apply these conclusions to specific problems (maybe not only the original one). This methodology has been developed in the last decades over the world when it turned out to outstand among the most powerful approaches to cope with problems in applied mathematics. The group of Dynamical Systems from Barcelona has been one of the bulwarks of this development from the late seventies to the present days. Following the guidelines presented in the previous section, we concern with several problems, mostly from the field of celestial mechanics but we also deal with a phenomenon coming from high energy physics. All these situations can be modeled by means of periodically time dependent Hamiltonian systems. To cope with those investigations, we develop software which can be used to perform computations in any periodically perturbed Hamiltonian system. We split the contents of this dissertation in two parts. The first one is devoted to general tolos to handle periodically time dependent Hamiltonians, even though we fill this first part with a number of illustrating examples, the goal is to keep the exposition in the abstract setting. Most of the contents of Part I deal with the development of software used to be applied in the second part. Some of the software has not been applied to the specific contents of Part II, this is left for future work. We also devote a whole chapter to some theoretical issues that, while are motivated by physical problems, they fall out of the category of periodic time dependent Hamiltonians. This splitting of contents has the intention of reecting, somehow, the basic methodological principles presented in the previous paragraph, keeping separated the abstract and the physical world but keeping in mind the lift

Modeling diversity by strange attractors with application to temporal pattern recognition

This thesis belongs to the general discipline of establishing black-box models from real-word data, more precisely, from measured time-series. This is an old subject and a large amount of papers and books has been written about it. The main difficulty is to express the diversity of data that has essentially the same origin without creating confusion with data that has a different origin. Normally, the diversity of time-series is modeled by a stochastic process, such as filtered white noise. Often, it is reasonable to assume that the time series is generated by a deterministic dynamical system rather than a stochastic process. In this case, the diversity of the data is expressed by the variability of the parameters of the dynamical system. The parameter variability itself is then, once again, modeled by a stochastic process. In both cases the diversity is generated by some form of exogenous noise. In this thesis a further step has been taken. A single chaotic dynamical system is used to model the data and their diversity. Indeed, a chaotic system produces a whole family of trajectories that are different but nonetheless very similar. It is believed that chaotic dynamics not only are a convenient means to represent diversity but that in many cases the origin of diversity stems actually from chaotic dynamic. Since the approach of this thesis explores completely new grounds the most suitable kind of data is considered, namely approximately periodic signals. In nature such time-series are rather common, in particular the physiological signal of living beings, such as the electrocardiograms (ECG), parts of speech signals, electroencephalograms (EEG), etc. Since there are strong arguments in favor of the chaotic nature of these signals, they appear to be the best candidates for modeling diversity by chaos. It should be stressed however, that the modeling approach pursued in this thesis is thought to be quite general and not limited to signals produced by chaotic dynamics in nature. The intended application of the modeling effort in this thesis is temporal signal classification. The reason for this is twofold. Firstly, classification is one of the basic building block of any cognitive system. Secondly, the recently studied phenomenon of synchronization of chaotic systems suggests a way to test a signal against its chaotic model. The essential content of this work can now be formulated as follows. Thesis: The diversity of approximately periodic signals found in nature can be modeled by means of chaotic dynamics. This kind of modeling technique, together with selective properties of the synchronization of chaotic systems, can be exploited for pattern recognition purposes. This Thesis is advocated by means of the following five points. Models of randomness (Chapter 2) It is argued that the randomness observed in nature is not necessarily the result of exogenous noise, but it could be endogenally generated by deterministic chaotic dynamics. The diversity of real signals is compared with signals produced by the most common chaotic systems. Qualitative resonance (Chapter 3) The behavior of chaotic systems forced by periodic or approximately periodic input signals is studied theoretically and by numerical simulation. It is observed that the chaotic system "locks" approximately to an input signal that is related to its internal chaotic dynamic. In contrast to this, its chaotic behavior is reinforced when the input signal has nothing to do with its internal dynamics. This new phenomenon is called "qualitative resonance". Modeling and recognizing (Chapter 4) In this chapter qualitative resonance is used for pattern recognition. The core of the method is a chaotic dynamical system that is able to reproduce the class of time-series that is to be recognized. This model is excited in a suitable way by an input signal such that qualitative resonance is realized. This means that if the input signal belongs to the modeled class of time-series, the system approximately "locks" into it. If not, the trajectory of the system and the input signal remain unrelated. Automated design of the recognizer (Chapters 5 and 6) For the kind of signals considered in this thesis a systematic design method of the recognizer is presented. The model used is a system of Lur'e type, i.e. a model where the linear dynamic and nonlinear static part are separated. The identification of the model parameters from the given data proceed iteratively, adapting in turn the linear and the nonlinear part. Thus, the difficult nonlinear dynamical system identification task is decomposed into the easier problems of linear dynamical and nonlinear static system identification. The way to apply the approximately periodic input signal in order to realize qualitative resonance is chosen with the help of periodic control theory. Validation (Chapter 7) The pattern recognition method has been validated on the following examples — A synthetic example — Laboratory measurement from Colpitts oscillator — ECG — EEG — Vowels of a speech signals In the first four cases a binary classification and in the last example a classification with five classes was performed. To the best of the knowledge of the author the recognition method is original. Chaotic systems have been already used to produce pseudo-noise and to model signal diversity. Also, parameter identification of chaotic systems has been already carried out. However, the direct establishment of the model from the given data and its subsequent use for classification based on the phenomenon of qualitative resonance is entirely new

Vibration, Control and Stability of Dynamical Systems

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”

15th Conference on Dynamical Systems Theory and Applications DSTA 2019 ABSTRACTS

From Preface: This is the fifteen time when the conference „Dynamical Systems – Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and the Ministry of Science and Higher Education. It is a great pleasure that our invitation has been accepted by so many people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcome nearly 255 persons from 47 countries all over the world. They decided to share the results of their research and many years experiences in the discipline of dynamical systems by submitting many very interesting papers. This booklet contains a collection of 338 abstracts, which have gained the acceptance of referees and have been qualified for publication in the conference edited books.Technical editor and cover design: Kaźmierczak, MarekCover design: Ogińska, Ewelina; Kaźmierczak, Mare

Mathematical foundations of elasticity

[Preface] This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is intended for mathematicians, engineers, and physicists who wish to see this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute