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Author: Jan Bouwe van den Berg Publisher: American Mathematical Soc. ISBN: 1470428148 Category : Mathematics Languages : en Pages : 226
Book Description
This volume is based on lectures delivered at the 2016 AMS Short Course “Rigorous Numerics in Dynamics”, held January 4–5, 2016, in Seattle, Washington. Nonlinear dynamics shapes the world around us, from the harmonious movements of celestial bodies, via the swirling motions in fluid flows, to the complicated biochemistry in the living cell. Mathematically these phenomena are modeled by nonlinear dynamical systems, in the form of ODEs, PDEs and delay equations. The presence of nonlinearities complicates the analysis, and the difficulties are even greater for PDEs and delay equations, which are naturally defined on infinite dimensional function spaces. With the availability of powerful computers and sophisticated software, numerical simulations have quickly become the primary tool to study the models. However, while the pace of progress increases, one may ask: just how reliable are our computations? Even for finite dimensional ODEs, this question naturally arises if the system under study is chaotic, as small differences in initial conditions (such as those due to rounding errors in numerical computations) yield wildly diverging outcomes. These issues have motivated the development of the field of rigorous numerics in dynamics, which draws inspiration from ideas in scientific computing, numerical analysis and approximation theory. The articles included in this volume present novel techniques for the rigorous study of the dynamics of maps via the Conley-index theory; periodic orbits of delay differential equations via continuation methods; invariant manifolds and connecting orbits; the dynamics of models with unknown nonlinearities; and bifurcations diagrams.
Author: Jan Bouwe van den Berg Publisher: American Mathematical Soc. ISBN: 1470428148 Category : Mathematics Languages : en Pages : 226
Book Description
This volume is based on lectures delivered at the 2016 AMS Short Course “Rigorous Numerics in Dynamics”, held January 4–5, 2016, in Seattle, Washington. Nonlinear dynamics shapes the world around us, from the harmonious movements of celestial bodies, via the swirling motions in fluid flows, to the complicated biochemistry in the living cell. Mathematically these phenomena are modeled by nonlinear dynamical systems, in the form of ODEs, PDEs and delay equations. The presence of nonlinearities complicates the analysis, and the difficulties are even greater for PDEs and delay equations, which are naturally defined on infinite dimensional function spaces. With the availability of powerful computers and sophisticated software, numerical simulations have quickly become the primary tool to study the models. However, while the pace of progress increases, one may ask: just how reliable are our computations? Even for finite dimensional ODEs, this question naturally arises if the system under study is chaotic, as small differences in initial conditions (such as those due to rounding errors in numerical computations) yield wildly diverging outcomes. These issues have motivated the development of the field of rigorous numerics in dynamics, which draws inspiration from ideas in scientific computing, numerical analysis and approximation theory. The articles included in this volume present novel techniques for the rigorous study of the dynamics of maps via the Conley-index theory; periodic orbits of delay differential equations via continuation methods; invariant manifolds and connecting orbits; the dynamics of models with unknown nonlinearities; and bifurcations diagrams.
Author: Gerald Teschl Publisher: American Mathematical Society ISBN: 147047641X Category : Mathematics Languages : en Pages : 370
Book Description
This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students. The first part begins with some simple examples of explicitly solvable equations and a first glance at qualitative methods. Then the fundamental results concerning the initial value problem are proved: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore, linear equations are considered, including the Floquet theorem, and some perturbation results. As somewhat independent topics, the Frobenius method for linear equations in the complex domain is established and Sturm–Liouville boundary value problems, including oscillation theory, are investigated. The second part introduces the concept of a dynamical system. The Poincaré–Bendixson theorem is proved, and several examples of planar systems from classical mechanics, ecology, and electrical engineering are investigated. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed. Finally, stability is studied, including the stable manifold and the Hartman–Grobman theorem for both continuous and discrete systems. The third part introduces chaos, beginning with the basics for iterated interval maps and ending with the Smale–Birkhoff theorem and the Melnikov method for homoclinic orbits. The text contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations.
Author: Lawrence Perko Publisher: Springer Science & Business Media ISBN: 1468402498 Category : Mathematics Languages : en Pages : 530
Book Description
Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence bf interest in the modern as well as the clas sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mat!!ematics (TAM). The development of new courses is a natural consequence of a high level of excitement oil the research frontier as newer techniques, such as numerical and symbolic cotnputer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Math ematical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. Preface to the Second Edition This book covers those topics necessary for a clear understanding of the qualitative theory of ordinary differential equations and the concept of a dynamical system. It is written for advanced undergraduates and for beginning graduate students. It begins with a study of linear systems of ordinary differential equations, a topic already familiar to the student who has completed a first course in differential equations.
Author: Gary A. Sod Publisher: Cambridge University Press ISBN: 9780521259248 Category : Mathematics Languages : en Pages : 464
Book Description
Here is an introduction to numerical methods for partial differential equations with particular reference to those that are of importance in fluid dynamics. The author gives a thorough and rigorous treatment of the techniques, beginning with the classical methods and leading to a discussion of modern developments. For easier reading and use, many of the purely technical results and theorems are given separately from the main body of the text. The presentation is intended for graduate students in applied mathematics, engineering and physical sciences who have a basic knowledge of partial differential equations.
Author: Àlex Haro Publisher: Springer ISBN: 3319296620 Category : Mathematics Languages : en Pages : 280
Book Description
This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online. The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems.
Author: Xueming Yang Publisher: World Scientific ISBN: 9812385681 Category : Technology & Engineering Languages : en Pages : 653
Book Description
The field of chemical reaction dynamics has made huge progress during the last decade or so. The aim of these volumes is to provide graduate students and experts in the field with a picture of the current status of advanced experimental and theoretical research in chemical reaction dynamics.
Author: Erik M. Bollt Publisher: SIAM ISBN: 1611972639 Category : Mathematics Languages : en Pages : 376
Book Description
Until recently, measurable dynamics has been held as a highly theoretical mathematical topic with few generally known obvious links for practitioners in areas of applied mathematics. However, the advent of high-speed computers, rapidly developing algorithms, and new numerical methods has allowed for a tremendous amount of progress and sophistication in efforts to represent the notion of a transfer operator discretely but to high resolution. This book connects many concepts in dynamical systems with mathematical tools from areas such as graph theory and ergodic theory. The authors introduce practical tools for applications related to measurable dynamical systems, coherent structures, and transport problems. The new and fast-developing computational tools discussed throughout the book allow for detailed analysis of real-world problems that are simply beyond the reach of traditional methods.
Author: Dan B. Marghitu Publisher: Springer Science & Business Media ISBN: 1461434750 Category : Technology & Engineering Languages : en Pages : 616
Book Description
Advanced Dynamics: Analytical and Numerical Calculations with MATLAB provides a thorough, rigorous presentation of kinematics and dynamics while using MATLAB as an integrated tool to solve problems. Topics presented are explained thoroughly and directly,allowing fundamental principles to emerge through applications from areas such as multibody systems, robotics, spacecraft and design of complex mechanical devices. This book differs from others in that it uses symbolic MATLAB for both theory and applications. Special attention is given to solutions that are solved analytically and numerically using MATLAB. The illustrations and figures generated with MATLAB reinforce visual learning while an abundance of examples offer additional support.
Author: Kopin Liu Publisher: World Scientific ISBN: 9814485268 Category : Science Languages : en Pages : 653
Book Description
The field of chemical reaction dynamics has made tremendous progress during the last decade or so. This is due largely to the development of many new, state-of-the-art experimental and theoretical techniques during that period. It is beneficial to present these advances, both theoretical and experimental, in a review volume published in two parts (Parts I and II). The primary purpose of this review volume is to provide graduate students and experts in the field with a rather detailed picture of the current status of advanced experimental and theoretical research in chemical reaction dynamics. All chapters in these two parts have been written by world-renowned experts active in such research.
Author: F. Zanolin Publisher: Springer ISBN: 3709126800 Category : Technology & Engineering Languages : en Pages : 214
Book Description
The area covered by this volume represents a broad choice of some interesting research topics in the field of dynamical systems and applications of nonlinear analysis to ordinary and partial differential equations. The contributed papers, written by well known specialists, make this volume a useful tool both for the experts (who can find recent and new results) and for those who are interested in starting a research work in one of these topics (who can find some updated and carefully presented papers on the state of the art of the corresponding subject).
Author: Jan Bouwe van den Berg
Author: Jan Bouwe van den Berg
Author: Gerald Teschl
Author: Lawrence Perko
Author: Gary A. Sod
Author: Àlex Haro
Author: Xueming Yang
Author: Erik M. Bollt
Author: Dan B. Marghitu
Author: Kopin Liu
Author: F. Zanolin