Notes On Dynamical Systems

Author : Jurgen Moser,Jürgen Moser,Eduard Zehnder
Publisher : American Mathematical Soc.
Page : 266 pages
File Size : 50,7 Mb
Release : 2005
Category : Combinatorial dynamics
ISBN : 9780821835777

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Notes on Dynamical Systems by Jurgen Moser,Jürgen Moser,Eduard Zehnder Pdf

This book is an introduction to the field of dynamical systems, in particular, to the special class of Hamiltonian systems. The authors aimed at keeping the requirements of mathematical techniques minimal but giving detailed proofs and many examples and illustrations from physics and celestial mechanics. After all, the celestial $N$-body problem is the origin of dynamical systems and gave rise in the past to many mathematical developments. Jurgen Moser (1928-1999) was a professor atthe Courant Institute, New York, and then at ETH Zurich. He served as president of the International Mathematical Union and received many honors and prizes, among them the Wolf Prize in mathematics. Jurgen Moser is the author of several books, among them Stable and Random Motions in DynamicalSystems. Eduard Zehnder is a professor at ETH Zurich. He is coauthor with Helmut Hofer of the book Symplectic Invariants and Hamiltonian Dynamics. Information for our distributors: Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.

Author : Arjan J. van der Schaft,Hans Schumacher
Publisher : Springer
Page : 189 pages
File Size : 47,9 Mb
Release : 2007-10-03
Category : Technology & Engineering
ISBN : 9781846285424

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An Introduction to Hybrid Dynamical Systems by Arjan J. van der Schaft,Hans Schumacher Pdf

This book is about dynamical systems that are "hybrid" in the sense that they contain both continuous and discrete state variables. Recently there has been increased research interest in the study of the interaction between discrete and continuous dynamics. The present volume provides a first attempt in book form to bring together concepts and methods dealing with hybrid systems from various areas, and to look at these from a unified perspective. The authors have chosen a mode of exposition that is largely based on illustrative examples rather than on the abstract theorem-proof format because the systematic study of hybrid systems is still in its infancy. The examples are taken from many different application areas, ranging from power converters to communication protocols and from chaos to mathematical finance. Subjects covered include the following: definition of hybrid systems; description formats; existence and uniqueness of solutions; special subclasses (variable-structure systems, complementarity systems); reachability and verification; stability and stabilizability; control design methods. The book will be of interest to scientists from a wide range of disciplines including: computer science, control theory, dynamical system theory, systems modeling and simulation, and operations research.

Author : Antonio Giorgilli
Publisher : Cambridge University Press
Page : 474 pages
File Size : 46,9 Mb
Release : 2022-05-05
Category : Science
ISBN : 9781009174862

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Notes on Hamiltonian Dynamical Systems Notes on Hamiltonian Dynamical Systems by Antonio Giorgilli Pdf

Starting with the basics of Hamiltonian dynamics and canonical transformations, this text follows the historical development of the theory culminating in recent results: the Kolmogorov–Arnold–Moser theorem, Nekhoroshev's theorem and superexponential stability. Its analytic approach allows students to learn about perturbation methods leading to advanced results. Key topics covered include Liouville's theorem, the proof of Poincaré's non-integrability theorem and the nonlinear dynamics in the neighbourhood of equilibria. The theorem of Kolmogorov on persistence of invariant tori and the theory of exponential stability of Nekhoroshev are proved via constructive algorithms based on the Lie series method. A final chapter is devoted to the discovery of chaos by Poincaré and its relations with integrability, also including recent results on superexponential stability. Written in an accessible, self-contained way with few prerequisites, this book can serve as an introductory text for senior undergraduate and graduate students.

Author : George Osipenko
Publisher : Springer
Page : 288 pages
File Size : 51,8 Mb
Release : 2006-10-28
Category : Mathematics
ISBN : 9783540355953

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Dynamical Systems, Graphs, and Algorithms by George Osipenko Pdf

This book describes a family of algorithms for studying the global structure of systems. By a finite covering of the phase space we construct a directed graph with vertices corresponding to cells of the covering and edges corresponding to admissible transitions. The method is used, among other things, to locate the periodic orbits and the chain recurrent set, to construct the attractors and their basins, to estimate the entropy, and more.

Author : Martine Queffélec
Publisher : Springer
Page : 252 pages
File Size : 50,6 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540480884

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Substitution Dynamical Systems - Spectral Analysis by Martine Queffélec Pdf

Author : Hieu Trinh,Tyrone Fernando
Publisher : Springer Science & Business Media
Page : 226 pages
File Size : 55,8 Mb
Release : 2011-09-24
Category : Technology & Engineering
ISBN : 9783642240638

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Functional Observers for Dynamical Systems by Hieu Trinh,Tyrone Fernando Pdf

The theory of linear functional observers, which is the subject of this book, is increasingly becoming a popular researched topic because of the many advantages it presents in state observation and control system design. This book presents recent information on the current state of the art research in this field. This book will serve as a useful reference to researchers in this area of research to understand the fundamental concepts relevant to the theory of functional observers and to gather most recent advancements in the field. This book is useful to academics and postgraduate students researching into the theory of linear functional observers. This book can also be useful for specialized final year undergraduate courses in control systems engineering and applied mathematics with a research focus.

Author : Hendrik W. Broer,George B. Huitema,Mikhail B. Sevryuk
Publisher : Springer
Page : 203 pages
File Size : 48,6 Mb
Release : 2009-01-25
Category : Mathematics
ISBN : 9783540496137

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Quasi-Periodic Motions in Families of Dynamical Systems by Hendrik W. Broer,George B. Huitema,Mikhail B. Sevryuk Pdf

This book is devoted to the phenomenon of quasi-periodic motion in dynamical systems. Such a motion in the phase space densely fills up an invariant torus. This phenomenon is most familiar from Hamiltonian dynamics. Hamiltonian systems are well known for their use in modelling the dynamics related to frictionless mechanics, including the planetary and lunar motions. In this context the general picture appears to be as follows. On the one hand, Hamiltonian systems occur that are in complete order: these are the integrable systems where all motion is confined to invariant tori. On the other hand, systems exist that are entirely chaotic on each energy level. In between we know systems that, being sufficiently small perturbations of integrable ones, exhibit coexistence of order (invariant tori carrying quasi-periodic dynamics) and chaos (the so called stochastic layers). The Kolmogorov-Arnol'd-Moser (KAM) theory on quasi-periodic motions tells us that the occurrence of such motions is open within the class of all Hamiltonian systems: in other words, it is a phenomenon persistent under small Hamiltonian perturbations. Moreover, generally, for any such system the union of quasi-periodic tori in the phase space is a nowhere dense set of positive Lebesgue measure, a so called Cantor family. This fact implies that open classes of Hamiltonian systems exist that are not ergodic. The main aim of the book is to study the changes in this picture when other classes of systems - or contexts - are considered.

Author : William Edward Fitzgibbon
Publisher : Pitman Advanced Publishing Program
Page : 388 pages
File Size : 41,5 Mb
Release : 1984
Category : Mathematics
ISBN : UOM:39015049315677

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Partial Differential Equations and Dynamical Systems by William Edward Fitzgibbon Pdf

There has recently been a great amount of activity and a rapid growth in the areas of partial differential equations and dynamical systems. This interest has been encouraged by the development of powerful new techniques in nonlinear analysis and a renewed scientific interest in applied mathematical analysis. This book has been designed to make the reader aware of progress and current problems in this exciting and useful area. The book consists of articles by internationally known mathematical scientists, based on lectures given during a year-long program at the University of Houston.

Author : Sergei Yu. Pilyugin
Publisher : Springer
Page : 284 pages
File Size : 54,5 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540484295

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Shadowing in Dynamical Systems by Sergei Yu. Pilyugin Pdf

This book is an introduction to the theory of shadowing of approximate trajectories in dynamical systems by exact ones. This is the first book completely devoted to the theory of shadowing. It shows the importance of shadowing theory for both the qualitative theory of dynamical systems and the theory of numerical methods. Shadowing Methods allow us to estimate differences between exact and approximate solutions on infinite time intervals and to understand the influence of error terms. The book is intended for specialists in dynamical systems, for researchers and graduate students in the theory of numerical methods.

Author : Eduard Zehnder
Publisher : European Mathematical Society
Page : 372 pages
File Size : 41,5 Mb
Release : 2010
Category : Dynamics
ISBN : 3037190817

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Lectures on Dynamical Systems by Eduard Zehnder Pdf

This book originated from an introductory lecture course on dynamical systems given by the author for advanced students in mathematics and physics at ETH Zurich. The first part centers around unstable and chaotic phenomena caused by the occurrence of homoclinic points. The existence of homoclinic points complicates the orbit structure considerably and gives rise to invariant hyperbolic sets nearby. The orbit structure in such sets is analyzed by means of the shadowing lemma, whose proof is based on the contraction principle. This lemma is also used to prove S. Smale's theorem about the embedding of Bernoulli systems near homoclinic orbits. The chaotic behavior is illustrated in the simple mechanical model of a periodically perturbed mathematical pendulum. The second part of the book is devoted to Hamiltonian systems. The Hamiltonian formalism is developed in the elegant language of the exterior calculus. The theorem of V. Arnold and R. Jost shows that the solutions of Hamiltonian systems which possess sufficiently many integrals of motion can be written down explicitly and for all times. The existence proofs of global periodic orbits of Hamiltonian systems on symplectic manifolds are based on a variational principle for the old action functional of classical mechanics. The necessary tools from variational calculus are developed. There is an intimate relation between the periodic orbits of Hamiltonian systems and a class of symplectic invariants called symplectic capacities. From these symplectic invariants one derives surprising symplectic rigidity phenomena. This allows a first glimpse of the fast developing new field of symplectic topology.

Author : D. A. Rand,L.-S. Young
Publisher : Springer
Page : 400 pages
File Size : 51,8 Mb
Release : 2006-11-14
Category : Science
ISBN : 9783540389453

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Dynamical Systems and Turbulence, Warwick 1980 by D. A. Rand,L.-S. Young Pdf

Author : Rabi Bhattacharya,Mukul Majumdar
Publisher : Cambridge University Press
Page : 5 pages
File Size : 51,7 Mb
Release : 2007-01-08
Category : Mathematics
ISBN : 9781139461627

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Random Dynamical Systems by Rabi Bhattacharya,Mukul Majumdar Pdf

This treatment provides an exposition of discrete time dynamic processes evolving over an infinite horizon. Chapter 1 reviews some mathematical results from the theory of deterministic dynamical systems, with particular emphasis on applications to economics. The theory of irreducible Markov processes, especially Markov chains, is surveyed in Chapter 2. Equilibrium and long run stability of a dynamical system in which the law of motion is subject to random perturbations is the central theme of Chapters 3-5. A unified account of relatively recent results, exploiting splitting and contractions, that have found applications in many contexts is presented in detail. Chapter 6 explains how a random dynamical system may emerge from a class of dynamic programming problems. With examples and exercises, readers are guided from basic theory to the frontier of applied mathematical research.

Author : Kenneth R. Meyer,Daniel C. Offin
Publisher : Springer
Page : 384 pages
File Size : 51,6 Mb
Release : 2017-05-04
Category : Mathematics
ISBN : 9783319536910

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Introduction to Hamiltonian Dynamical Systems and the N-Body Problem by Kenneth R. Meyer,Daniel C. Offin Pdf

This third edition text provides expanded material on the restricted three body problem and celestial mechanics. With each chapter containing new content, readers are provided with new material on reduction, orbifolds, and the regularization of the Kepler problem, all of which are provided with applications. The previous editions grew out of graduate level courses in mathematics, engineering, and physics given at several different universities. The courses took students who had some background in differential equations and lead them through a systematic grounding in the theory of Hamiltonian mechanics from a dynamical systems point of view. This text provides a mathematical structure of celestial mechanics ideal for beginners, and will be useful to graduate students and researchers alike. Reviews of the second edition: "The primary subject here is the basic theory of Hamiltonian differential equations studied from the perspective of differential dynamical systems. The N-body problem is used as the primary example of a Hamiltonian system, a touchstone for the theory as the authors develop it. This book is intended to support a first course at the graduate level for mathematics and engineering students. ... It is a well-organized and accessible introduction to the subject ... . This is an attractive book ... ." (William J. Satzer, The Mathematical Association of America, March, 2009) “The second edition of this text infuses new mathematical substance and relevance into an already modern classic ... and is sure to excite future generations of readers. ... This outstanding book can be used not only as an introductory course at the graduate level in mathematics, but also as course material for engineering graduate students. ... it is an elegant and invaluable reference for mathematicians and scientists with an interest in classical and celestial mechanics, astrodynamics, physics, biology, and related fields.” (Marian Gidea, Mathematical Reviews, Issue 2010 d)

Author : Nessim Sibony,Dierk Schleicher,Dinh Tien Cuong,Marco Brunella,Eric Bedford,Marco Abate
Publisher : Springer Science & Business Media
Page : 357 pages
File Size : 47,5 Mb
Release : 2010-07-31
Category : Mathematics
ISBN : 9783642131707

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Holomorphic Dynamical Systems by Nessim Sibony,Dierk Schleicher,Dinh Tien Cuong,Marco Brunella,Eric Bedford,Marco Abate Pdf

The theory of holomorphic dynamical systems is a subject of increasing interest in mathematics, both for its challenging problems and for its connections with other branches of pure and applied mathematics. A holomorphic dynamical system is the datum of a complex variety and a holomorphic object (such as a self-map or a vector ?eld) acting on it. The study of a holomorphic dynamical system consists in describing the asymptotic behavior of the system, associating it with some invariant objects (easy to compute) which describe the dynamics and classify the possible holomorphic dynamical systems supported by a given manifold. The behavior of a holomorphic dynamical system is pretty much related to the geometry of the ambient manifold (for instance, - perbolic manifolds do no admit chaotic behavior, while projective manifolds have a variety of different chaotic pictures). The techniques used to tackle such pr- lems are of variouskinds: complexanalysis, methodsof real analysis, pluripotential theory, algebraic geometry, differential geometry, topology. To cover all the possible points of view of the subject in a unique occasion has become almost impossible, and the CIME session in Cetraro on Holomorphic Dynamical Systems was not an exception.

Author : Michael Shub
Publisher : Springer Science & Business Media
Page : 159 pages
File Size : 50,6 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9781475719475

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Global Stability of Dynamical Systems by Michael Shub Pdf

These notes are the result of a course in dynamical systems given at Orsay during the 1976-77 academic year. I had given a similar course at the Gradu ate Center of the City University of New York the previous year and came to France equipped with the class notes of two of my students there, Carol Hurwitz and Michael Maller. My goal was to present Smale's n-Stability Theorem as completely and compactly as possible and in such a way that the students would have easy access to the literature. I was not confident that I could do all this in lectures in French, so I decided to distribute lecture notes. I wrote these notes in English and Remi Langevin translated them into French. His work involved much more than translation. He consistently corrected for style, clarity, and accuracy. Albert Fathi got involved in reading the manuscript. His role quickly expanded to extensive rewriting and writing. Fathi wrote (5. 1) and (5. 2) and rewrote Theorem 7. 8 when I was in despair of ever getting it right with all the details. He kept me honest at all points and played a large role in the final form of the manuscript. He also did the main work in getting the manuscript ready when I had left France and Langevin was unfortunately unavailable. I ran out of steam by the time it came to Chapter 10. M.