Integrability And Nonintegrability Of Dynamical Systems

Author : Alain Goriely
Publisher : World Scientific
Page : 438 pages
File Size : 54,9 Mb
Release : 2001
Category : Science
ISBN : 981281194X

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Integrability and Nonintegrability of Dynamical Systems by Alain Goriely Pdf

This invaluable book examines qualitative and quantitative methods for nonlinear differential equations, as well as integrability and nonintegrability theory. Starting from the idea of a constant of motion for simple systems of differential equations, it investigates the essence of integrability, its geometrical relevance and dynamical consequences. Integrability theory is approached from different perspectives, first in terms of differential algebra, then in terms of complex time singularities and finally from the viewpoint of phase geometry (for both Hamiltonian and non-Hamiltonian systems). As generic systems of differential equations cannot be exactly solved, the book reviews the different notions of nonintegrability and shows how to prove the nonexistence of exact solutions and/or a constant of motion. Finally, nonintegrability theory is linked to dynamical systems theory by showing how the property of complete integrability, partial integrability or nonintegrability can be related to regular and irregular dynamics in phase space. Contents: Integrability: An Algebraic Approach; Integrability: An Analytic Approach; Polynomial and Quasi-Polynomial Vector Fields; Nonintegrability; Hamiltonian Systems; Nearly Integrable Dynamical Systems; Open Problems. Readership: Mathematical and theoretical physicists and astronomers and engineers interested in dynamical systems.

Author : V.I. Arnol'd,S.P. Novikov
Publisher : Springer Science & Business Media
Page : 360 pages
File Size : 51,6 Mb
Release : 1993-12-06
Category : Mathematics
ISBN : 3540181768

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Dynamical Systems VII by V.I. Arnol'd,S.P. Novikov Pdf

A collection of five surveys on dynamical systems, indispensable for graduate students and researchers in mathematics and theoretical physics. Written in the modern language of differential geometry, the book covers all the new differential geometric and Lie-algebraic methods currently used in the theory of integrable systems.

Author : Xiang Zhang
Publisher : Springer
Page : 380 pages
File Size : 46,6 Mb
Release : 2017-03-30
Category : Mathematics
ISBN : 9789811042263

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Integrability of Dynamical Systems: Algebra and Analysis by Xiang Zhang Pdf

This is the first book to systematically state the fundamental theory of integrability and its development of ordinary differential equations with emphasis on the Darboux theory of integrability and local integrability together with their applications. It summarizes the classical results of Darboux integrability and its modern development together with their related Darboux polynomials and their applications in the reduction of Liouville and elementary integrabilty and in the center—focus problem, the weakened Hilbert 16th problem on algebraic limit cycles and the global dynamical analysis of some realistic models in fields such as physics, mechanics and biology. Although it can be used as a textbook for graduate students in dynamical systems, it is intended as supplementary reading for graduate students from mathematics, physics, mechanics and engineering in courses related to the qualitative theory, bifurcation theory and the theory of integrability of dynamical systems.

Author : Juan J. Morales Ruiz
Publisher : Birkhäuser
Page : 177 pages
File Size : 47,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034887182

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Differential Galois Theory and Non-Integrability of Hamiltonian Systems by Juan J. Morales Ruiz Pdf

This book is devoted to the relation between two different concepts of integrability: the complete integrability of complex analytical Hamiltonian systems and the integrability of complex analytical linear differential equations. For linear differential equations, integrability is made precise within the framework of differential Galois theory. The connection of these two integrability notions is given by the variational equation (i.e. linearized equation) along a particular integral curve of the Hamiltonian system. The underlying heuristic idea, which motivated the main results presented in this monograph, is that a necessary condition for the integrability of a Hamiltonian system is the integrability of the variational equation along any of its particular integral curves. This idea led to the algebraic non-integrability criteria for Hamiltonian systems. These criteria can be considered as generalizations of classical non-integrability results by Poincaré and Lyapunov, as well as more recent results by Ziglin and Yoshida. Thus, by means of the differential Galois theory it is not only possible to understand all these approaches in a unified way but also to improve them. Several important applications are also included: homogeneous potentials, Bianchi IX cosmological model, three-body problem, Hénon-Heiles system, etc. The book is based on the original joint research of the author with J.M. Peris, J.P. Ramis and C. Simó, but an effort was made to present these achievements in their logical order rather than their historical one. The necessary background on differential Galois theory and Hamiltonian systems is included, and several new problems and conjectures which open new lines of research are proposed. - - - The book is an excellent introduction to non-integrability methods in Hamiltonian mechanics and brings the reader to the forefront of research in the area. The inclusion of a large number of worked-out examples, many of wide applied interest, is commendable. There are many historical references, and an extensive bibliography. (Mathematical Reviews) For readers already prepared in the two prerequisite subjects [differential Galois theory and Hamiltonian dynamical systems], the author has provided a logically accessible account of a remarkable interaction between differential algebra and dynamics. (Zentralblatt MATH)

Author : Muthuswamy Lakshmanan
Publisher : Springer
Page : 232 pages
File Size : 44,5 Mb
Release : 1990
Category : Mathematics
ISBN : UCAL:B4128782

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Symmetries and Singularity Structures by Muthuswamy Lakshmanan Pdf

Symmetries and singularity structures play important roles in the study of nonlinear dynamical systems. It was Sophus Lie who originally stressed the importance of symmetries and invariance in the study of nonlinear differential equations. How ever, the full potentialities of symmetries had been realized only after the advent of solitons in 1965. It is now a well-accepted fact that associated with the infinite number of integrals of motion of a given soliton system, an infinite number of gep. eralized Lie BAcklund symmetries exist. The associated bi-Hamiltonian struc ture, Kac-Moody, Vrrasoro algebras, and so on, have been increasingly attracting the attention of scientists working in this area. Similarly, in recent times the role of symmetries in analyzing both the classical and quantum integrable and nonintegrable finite dimensional systems has been remarkable. On the other hand, the works of Fuchs, Kovalevskaya, Painleve and coworkers on the singularity structures associated with the solutions of nonlinear differen tial equations have helped to classify first and second order nonlinear ordinary differential equations. The recent work of Ablowitz, Ramani and Segur, con jecturing a connection between soliton systems and Painleve equations that are free from movable critical points, has motivated considerably the search for the connection between integrable dynamical systems with finite degrees of freedom and the Painleve property. Weiss, Tabor and Carnevale have extended these ideas to partial differential equations."

Author : Michael Tabor
Publisher : Wiley-Interscience
Page : 392 pages
File Size : 42,5 Mb
Release : 1989-01-18
Category : Mathematics
ISBN : UOM:39015015760393

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Chaos and Integrability in Nonlinear Dynamics by Michael Tabor Pdf

Presents the newer field of chaos in nonlinear dynamics as a natural extension of classical mechanics as treated by differential equations. Employs Hamiltonian systems as the link between classical and nonlinear dynamics, emphasizing the concept of integrability. Also discusses nonintegrable dynamics, the fundamental KAM theorem, integrable partial differential equations, and soliton dynamics.

Author : Edson Denis Leonel
Publisher : Springer Nature
Page : 247 pages
File Size : 40,7 Mb
Release : 2021-08-26
Category : Mathematics
ISBN : 9789811635441

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Scaling Laws in Dynamical Systems by Edson Denis Leonel Pdf

This book discusses many of the common scaling properties observed in some nonlinear dynamical systems mostly described by mappings. The unpredictability of the time evolution of two nearby initial conditions in the phase space together with the exponential divergence from each other as time goes by lead to the concept of chaos. Some of the observables in nonlinear systems exhibit characteristics of scaling invariance being then described via scaling laws. From the variation of control parameters, physical observables in the phase space may be characterized by using power laws that many times yield into universal behavior. The application of such a formalism has been well accepted in the scientific community of nonlinear dynamics. Therefore I had in mind when writing this book was to bring together few of the research results in nonlinear systems using scaling formalism that could treated either in under-graduation as well as in the post graduation in the several exact programs but no earlier requirements were needed from the students unless the basic physics and mathematics. At the same time, the book must be original enough to contribute to the existing literature but with no excessive superposition of the topics already dealt with in other text books. The majority of the Chapters present a list of exercises. Some of them are analytic and others are numeric with few presenting some degree of computational complexity.

Author : Yvette Kosmann-Schwarzbach,Basil Grammaticos,K. M. Tamizhmani
Publisher : Unknown
Page : 348 pages
File Size : 48,5 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 366214431X

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Integrability of Nonlinear Systems by Yvette Kosmann-Schwarzbach,Basil Grammaticos,K. M. Tamizhmani Pdf

Author : Alexander B. Borisov,Vladimir V. Zverev
Publisher : Walter de Gruyter GmbH & Co KG
Page : 299 pages
File Size : 47,6 Mb
Release : 2016-11-21
Category : Science
ISBN : 9783110430585

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Nonlinear Dynamics by Alexander B. Borisov,Vladimir V. Zverev Pdf

The book provides a concise and rigor introduction to the fundamentals of methods for solving the principal problems of modern non-linear dynamics. This monograph covers the basic issues of the theory of integrable systems and the theory of dynamical chaos both in nonintegrable conservative and in dissipative systems. A distinguishing feature of the material exposition is to add some comments, historical information, brief biographies and portraits of the researchers who made the most significant contribution to science. This allows one to present the material as accessible and attractive to students to acquire indepth scientific knowledge of nonlinear mechanics, feel the atmosphere where those or other important discoveries were made. The book can be used as a textbook for advanced undergraduate and graduate students majoring in high-tech industries and high technology (the science based on high technology) to help them to develop lateral thinking in early stages of training. Contents: Nonlinear Oscillations Integrable Systems Stability of Motion and Structural Stability Chaos in Conservative Systems Chaos and Fractal Attractors in Dissipative Systems Conclusion References Index

Author : J. Hietarinta,N. Joshi,F. W. Nijhoff
Publisher : Cambridge University Press
Page : 461 pages
File Size : 49,6 Mb
Release : 2016-09
Category : Mathematics
ISBN : 9781107042728

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Discrete Systems and Integrability by J. Hietarinta,N. Joshi,F. W. Nijhoff Pdf

A first introduction to the theory of discrete integrable systems at a level suitable for students and non-experts.

Author : Da-hsuan Feng,Bailin Hao,Jian-min Yuan
Publisher : World Scientific
Page : 562 pages
File Size : 53,7 Mb
Release : 1992-09-30
Category : Electronic
ISBN : 9789814635684

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Quantum Non-integrability by Da-hsuan Feng,Bailin Hao,Jian-min Yuan Pdf

Recent developments in nonlinear dynamics has significantly altered our basic understanding of the foundations of classical physics. However, it is quantum mechanics, not classical mechanics, which describes the motion of the nucleons, atoms, and molecules in the microscopic world. What are then the quantum signatures of the ubiquitous chaotic behavior observed in classical physics? In answering this question one cannot avoid probing the deepest foundations connecting classical and quantum mechanics. This monograph reviews some of the most current thinkings and developments in this exciting field of physics.

Author : Andreas Knauf,Yakov G. Sinai
Publisher : Springer Science & Business Media
Page : 114 pages
File Size : 50,9 Mb
Release : 1997-03-20
Category : Science
ISBN : 3764357088

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Classical Nonintegrability, Quantum Chaos by Andreas Knauf,Yakov G. Sinai Pdf

Our DMV Seminar on 'Classical Nonintegrability, Quantum Chaos' intended to introduce students and beginning researchers to the techniques applied in nonin tegrable classical and quantum dynamics. Several of these lectures are collected in this volume. The basic phenomenon of nonlinear dynamics is mixing in phase space, lead ing to a positive dynamical entropy and a loss of information about the initial state. The nonlinear motion in phase space gives rise to a linear action on phase space functions which in the case of iterated maps is given by a so-called transfer operator. Good mixing rates lead to a spectral gap for this operator. Similar to the use made of the Riemann zeta function in the investigation of the prime numbers, dynamical zeta functions are now being applied in nonlinear dynamics. In Chapter 2 V. Baladi first introduces dynamical zeta functions and transfer operators, illustrating and motivating these notions with a simple one-dimensional dynamical system. Then she presents a commented list of useful references, helping the newcomer to enter smoothly into this fast-developing field of research. Chapter 3 on irregular scattering and Chapter 4 on quantum chaos by A. Knauf deal with solutions of the Hamilton and the Schr6dinger equation. Scatter ing by a potential force tends to be irregular if three or more scattering centres are present, and a typical phenomenon is the occurrence of a Cantor set of bounded orbits. The presence of this set influences those scattering orbits which come close.

Author : H Ito
Publisher : World Scientific
Page : 190 pages
File Size : 55,8 Mb
Release : 1994-10-10
Category : Electronic
ISBN : 9789814550697

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Geometry And Analysis In Dynamical Systems - Proceedings Of The Rims Conference by H Ito Pdf

This volume of proceedings consists of 14 invited papers. It aims to understand the geometric and analytical aspects in recent research of dynamical systems. It deals with topics such as complex dynamical systems, electric circuits, reconstruction of bifurcation diagrams, integrable systems, quantum chaos, ergodic theory, foliation, zeta functions, etc.

Author : Archie E. Roy
Publisher : Springer Science & Business Media
Page : 581 pages
File Size : 41,9 Mb
Release : 2012-12-06
Category : Science
ISBN : 9781468459975

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Predictability, Stability, and Chaos in N-Body Dynamical Systems by Archie E. Roy Pdf

The reader will find in this volume the Proceedings of the NATO Advanced Study Institute held in Cortina d'Ampezzo, Italy between August 6 and August 17, 1990 under the title "Predictability, Stability, and Chaos in N-Body Dynamical Systems". The Institute was the latest in a series held at three-yearly inter vals from 1972 to 1987 in dynamical astronomy, theoretical mechanics and celestial mechanics. These previous institutes, held in high esteem by the international community of research workers, have resulted in a series of well-received Proceedings. The 1990 Institute attracted 74 participants from 16 countries, six outside the NATO group. Fifteen series of lectures were given by invited speakers; additionally some 40 valuable presentations were made by the younger participants, most of which are included in these Proceedings. The last twenty years in particular has been a time of increasingly rapid progress in tackling long-standing and also newly-arising problems in dynamics of N-body systems, point-mass and non-point-mass, a rate of progress achieved because of correspondingly rapid developments of new computer hardware and software together with the advent of new analytical techniques. It was a time of exciting progress culminating in the ability to carry out research programmes into the evolution of the outer Solar 8 System over periods of more than 10 years and to study star cluster and galactic models in unprecedented detail.

Author : A.K. Prykarpatsky,I.V. Mykytiuk
Publisher : Springer Science & Business Media
Page : 555 pages
File Size : 46,8 Mb
Release : 2013-04-09
Category : Science
ISBN : 9789401149945

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Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds by A.K. Prykarpatsky,I.V. Mykytiuk Pdf

In recent times it has been stated that many dynamical systems of classical mathematical physics and mechanics are endowed with symplectic structures, given in the majority of cases by Poisson brackets. Very often such Poisson structures on corresponding manifolds are canonical, which gives rise to the possibility of producing their hidden group theoretical essence for many completely integrable dynamical systems. It is a well understood fact that great part of comprehensive integrability theories of nonlinear dynamical systems on manifolds is based on Lie-algebraic ideas, by means of which, in particular, the classification of such compatibly bi Hamiltonian and isospectrally Lax type integrable systems has been carried out. Many chapters of this book are devoted to their description, but to our regret so far the work has not been completed. Hereby our main goal in each analysed case consists in separating the basic algebraic essence responsible for the complete integrability, and which is, at the same time, in some sense universal, i. e. , characteristic for all of them. Integrability analysis in the framework of a gradient-holonomic algorithm, devised in this book, is fulfilled through three stages: 1) finding a symplectic structure (Poisson bracket) transforming an original dynamical system into a Hamiltonian form; 2) finding first integrals (action variables or conservation laws); 3) defining an additional set of variables and some functional operator quantities with completely controlled evolutions (for instance, as Lax type representation).