Billiards A Genetic Introduction To The Dynamics Of Systems With Impacts
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Billiards A Genetic Introduction To The Dynamics Of Systems With Impacts
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Author : Valeriĭ Viktorovich Kozlov
language : en
Publisher: American Mathematical Soc.
Release Date : 1991-08-05
Billiards A Genetic Introduction To The Dynamics Of Systems With Impacts written by Valeriĭ Viktorovich Kozlov and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991-08-05 with Mathematics categories.
Starting with the work of G D Birkhoff, billiards have been a popular research topic drawing on such areas as ergodic theory, Morse theory, and KAM theory. Billiard systems are also remarkable in that they arise naturally in a number of important problems of mechanics and physics. This book is devoted to mathematical aspects of the theory of dynamical systems of billiard type. Focusing on the genetic approach, the authors strive to clarify the genesis of the basic ideas and concepts of the theory of dynamical systems with impact intereactions and also to demonstrate that these methods are natural and effective. Recent limit theorems, which justify various mathematical models of impact theory, are key features. Questions of existence and stability of periodic trajectories of elastic billiards occupy a special place in the book, and considerable attention is devoted to integrable billiards. A brief survey is given of work on billiards with ergodic behaviour. Each chapter ends with a list of problems.
Advances In Dynamical Systems And Control
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Author : Victor A. Sadovnichiy
language : en
Publisher: Springer
Release Date : 2016-08-16
Advances In Dynamical Systems And Control written by Victor A. Sadovnichiy and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-08-16 with Technology & Engineering categories.
Focused on recent advances, this book covers theoretical foundations as well as various applications. It presents modern mathematical modeling approaches to the qualitative and numerical analysis of solutions for complex engineering problems in physics, mechanics, biochemistry, geophysics, biology and climatology. Contributions by an international team of respected authors bridge the gap between abstract mathematical approaches, such as applied methods of modern analysis, algebra, fundamental and computational mechanics, nonautonomous and stochastic dynamical systems on the one hand, and practical applications in nonlinear mechanics, optimization, decision making theory and control theory on the other. As such, the book will be of interest to mathematicians and engineers working at the interface of these fields.
Introduction To The Modern Theory Of Dynamical Systems
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Author : Anatole Katok
language : en
Publisher: Cambridge University Press
Release Date : 1995
Introduction To The Modern Theory Of Dynamical Systems written by Anatole Katok and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Mathematics categories.
This book provided the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with most of the main areas of mathematics. The authors introduce and rigorously develop the theory while providing researchers interested in applications with fundamental tools and paradigms. The book begins with a discussion of several elementary but fundamental examples. These are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods. The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. The third and fourth parts develop the theories of low-dimensional dynamical systems and hyperbolic dynamical systems in depth. Over 400 systematic exercises are included in the text. The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate up.
Impact Mechanics
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Author : W. J. Stronge
language : en
Publisher: Cambridge University Press
Release Date : 2018-11-15
Impact Mechanics written by W. J. Stronge and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-11-15 with Science categories.
This second edition of Impact Mechanics offers new analytical methods with examples for the dynamics of low-speed impact.
Poincar Andronov Melnikov Analysis For Non Smooth Systems
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Author : Michal Feckan
language : en
Publisher: Academic Press
Release Date : 2016-06-07
Poincar Andronov Melnikov Analysis For Non Smooth Systems written by Michal Feckan and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-06-07 with Mathematics categories.
Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems is devoted to the study of bifurcations of periodic solutions for general n-dimensional discontinuous systems. The authors study these systems under assumptions of transversal intersections with discontinuity-switching boundaries. Furthermore, bifurcations of periodic sliding solutions are studied from sliding periodic solutions of unperturbed discontinuous equations, and bifurcations of forced periodic solutions are also investigated for impact systems from single periodic solutions of unperturbed impact equations. In addition, the book presents studies for weakly coupled discontinuous systems, and also the local asymptotic properties of derived perturbed periodic solutions. The relationship between non-smooth systems and their continuous approximations is investigated as well. Examples of 2-, 3- and 4-dimensional discontinuous ordinary differential equations and impact systems are given to illustrate the theoretical results. The authors use so-called discontinuous Poincaré mapping which maps a point to its position after one period of the periodic solution. This approach is rather technical, but it does produce results for general dimensions of spatial variables and parameters as well as the asymptotical results such as stability, instability, and hyperbolicity. - Extends Melnikov analysis of the classic Poincaré and Andronov staples, pointing to a general theory for freedom in dimensions of spatial variables and parameters as well as asymptotical results such as stability, instability, and hyperbolicity - Presents a toolbox of critical theoretical techniques for many practical examples and models, including non-smooth dynamical systems - Provides realistic models based on unsolved discontinuous problems from the literature and describes how Poincaré-Andronov-Melnikov analysis can be used to solve them - Investigates the relationship between non-smooth systems and their continuous approximations
Introduction To The Perturbation Theory Of Hamiltonian Systems
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Author : Dmitry Treschev
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-10-08
Introduction To The Perturbation Theory Of Hamiltonian Systems written by Dmitry Treschev and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-10-08 with Mathematics categories.
This book is an extended version of lectures given by the ?rst author in 1995–1996 at the Department of Mechanics and Mathematics of Moscow State University. We believe that a major part of the book can be regarded as an additional material to the standard course of Hamiltonian mechanics. In comparison with the original Russian 1 version we have included new material, simpli?ed some proofs and corrected m- prints. Hamiltonian equations ?rst appeared in connection with problems of geometric optics and celestial mechanics. Later it became clear that these equations describe a large classof systemsin classical mechanics,physics,chemistry,and otherdomains. Hamiltonian systems and their discrete analogs play a basic role in such problems as rigid body dynamics, geodesics on Riemann surfaces, quasi-classic approximation in quantum mechanics, cosmological models, dynamics of particles in an accel- ator, billiards and other systems with elastic re?ections, many in?nite-dimensional models in mathematical physics, etc. In this book we study Hamiltonian systems assuming that they depend on some parameter (usually?), where for?= 0 the dynamics is in a sense simple (as a rule, integrable). Frequently such a parameter appears naturally. For example, in celestial mechanics it is accepted to take? equal to the ratio: the mass of Jupiter over the mass of the Sun. In other cases it is possible to introduce the small parameter ar- ?cially.
Nonsmooth Mechanics
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Author : Bernard Brogliato
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Nonsmooth Mechanics written by Bernard Brogliato and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Technology & Engineering categories.
Thank you for opening the second edition of this monograph, which is devoted to the study of a class of nonsmooth dynamical systems of the general form: ::i; = g(x,u) (0. 1) f(x, t) 2: 0 where x E JRn is the system's state vector, u E JRm is the vector of inputs, and the function f (-, . ) represents a unilateral constraint that is imposed on the state. More precisely, we shall restrict ourselves to a subclass of such systems, namely mechanical systems subject to unilateral constraints on the position, whose dynamical equations may be in a first instance written as: ii= g(q,q,u) (0. 2) f(q, t) 2: 0 where q E JRn is the vector of generalized coordinates of the system and u is an in put (or controller) that generally involves a state feedback loop, i. e. u= u(q, q, t, z), with z= Z(z, q, q, t) when the controller is a dynamic state feedback. Mechanical systems composed of rigid bodies interacting fall into this subclass. A general prop erty of systems as in (0. 1) and (0. 2) is that their solutions are nonsmooth (with respect to time): Nonsmoothness arises primarily from the occurence of impacts (or collisions, or percussions) in the dynamical behaviour, when the trajectories attain the surface f(x, t) = O. They are necessary to keep the trajectories within the subspace = {x : f(x, t) 2: O} of the system's state space.
Optimization Of Dynamical Systems With Impulse Controls And Shocks
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Author : Boris Miller
language : en
Publisher: Springer Nature
Release Date : 2024-09-19
Optimization Of Dynamical Systems With Impulse Controls And Shocks written by Boris Miller and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-09-19 with Science categories.
This text explores the state-of-the-art in the rapidly developing theory of impulse control and introduces the theory of singular space-time transformations, a new method for studying shock mechanical systems. Two approaches in the theory of impulse control are presented: The first, more traditional approach defines the impulsive action as a discontinuity of phase coordinates depending on the current time, the state preceding the action, and its magnitude. The second requires the use of modern methods for describing dynamical systems - differential equations with measures. The impulse is treated as an idealization of a very short action of high magnitude, which produces an almost abrupt change of phase coordinates. The relation between these two approaches is also discussed, and several applications, both traditional and emerging, are considered. This text is intended for graduate students and researchers in control engineering and optimal control theory for dynamical systems. Readers are assumed to be familiar with the theory of ODEs, optimal control, and functional analysis, though an appendix is included that covers many of the necessary mathematical concepts.
Quantum Chaos And Mesoscopic Systems
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Author : N.E. Hurt
language : en
Publisher: Springer Science & Business Media
Release Date : 1997-02-28
Quantum Chaos And Mesoscopic Systems written by N.E. Hurt and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-02-28 with Mathematics categories.
4. 2 Variance of Quantum Matrix Elements. 125 4. 3 Berry's Trick and the Hyperbolic Case 126 4. 4 Nonhyperbolic Case . . . . . . . 128 4. 5 Random Matrix Theory . . . . . 128 4. 6 Baker's Map and Other Systems 129 4. 7 Appendix: Baker's Map . . . . . 129 5 Error Terms 133 5. 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . 133 5. 2 The Riemann Zeta Function in Periodic Orbit Theory 135 5. 3 Form Factor for Primes . . . . . . . . . . . . . . . . . 137 5. 4 Error Terms in Periodic Orbit Theory: Co-compact Case. 138 5. 5 Binary Quadratic Forms as a Model . . . . . . . . . . . . 139 6 Co-Finite Model for Quantum Chaology 141 6. 1 Introduction. . . . . . . . 141 6. 2 Co-finite Models . . . . . 141 6. 3 Geodesic Triangle Spaces 144 6. 4 L-Functions. . . . . . . . 145 6. 5 Zelditch's Prime Geodesic Theorem. 146 6. 6 Zelditch's Pseudo Differential Operators 147 6. 7 Weyl's Law Generalized 148 6. 8 Equidistribution Theory . . . . . . . . . 150 7 Landau Levels and L-Functions 153 7. 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . 153 7. 2 Landau Model: Mechanics on the Plane and Sphere. 153 7. 3 Landau Model: Mechanics on the Half-Plane 155 7. 4 Selberg's Spectral Theorem . . . . . . . . . . . 157 7. 5 Pseudo Billiards . . . . . . . . . . . . . . . . . 158 7. 6 Landau Levels on a Compact Riemann Surface 159 7. 7 Automorphic Forms . . . . . 160 7. 8 Maass-Selberg Trace Formula 162 7. 9 Degeneracy by Selberg. . . . 163 7. 10 Hecke Operators . . . . . . . 163 7. 11 Selberg Trace Formula for Hecke Operators 167 7. 12 Eigenvalue Statistics on X . . . . 169 7. 13 Mesoscopic Devices. . . . . . . . 170 7. 14 Hall Conductance on Leaky Tori 170 7.
Discreteness And Continuity In Problems Of Chaotic Dynamics
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Author : Michael L. Blank
language : en
Publisher: American Mathematical Soc.
Release Date : 1997-01-01
Discreteness And Continuity In Problems Of Chaotic Dynamics written by Michael L. Blank and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-01-01 with Mathematics categories.
This book presents the study of ergodic properties of so-called chaotic dynamical systems. One of the central topics is the interplay between deterministic and quasi-stochastic behaviour in chaotic dynamics and between properties of continuous dynamical systems and those of their discrete approximations. Using simple examples, the author describes the main phenomena known in chaotic dynamical systems, studying topics such as the operator approach in chaotic dynamics, stochastic stability, and the so-called coupled systems. The last two chapters are devoted to problems of numerical modeling of chaotic dynamics.