Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations
Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations

Author: Valery V. Kozlov

Publisher: Springer Science & Business Media

Published: 2013-01-13

Total Pages: 278

ISBN-13: 3642338178

DOWNLOAD EBOOK

The book is dedicated to the construction of particular solutions of systems of ordinary differential equations in the form of series that are analogous to those used in Lyapunov’s first method. A prominent place is given to asymptotic solutions that tend to an equilibrium position, especially in the strongly nonlinear case, where the existence of such solutions can’t be inferred on the basis of the first approximation alone. The book is illustrated with a large number of concrete examples of systems in which the presence of a particular solution of a certain class is related to special properties of the system’s dynamic behavior. It is a book for students and specialists who work with dynamical systems in the fields of mechanics, mathematics, and theoretical physics.

Strong Nonlinear Oscillators
Strong Nonlinear Oscillators

Author: Livija Cveticanin

Publisher: Springer

Published: 2017-05-29

Total Pages: 320

ISBN-13: 3319588265

DOWNLOAD EBOOK

This textbook presents the motion of pure nonlinear oscillatory systems and various solution procedures which give the approximate solutions of the strong nonlinear oscillator equations. It presents the author’s original method for the analytical solution procedure of the pure nonlinear oscillator system. After an introduction, the physical explanation of the pure nonlinearity and of the pure nonlinear oscillator is given. The analytical solution for free and forced vibrations of the one-degree-of-freedom strong nonlinear system with constant and time variable parameters is considered. In this second edition of the book, the number of approximate solving procedures for strong nonlinear oscillators is enlarged and a variety of procedures for solving free strong nonlinear oscillators is suggested. A method for error estimation is also given which is suitable to compare the exact and approximate solutions. Besides the oscillators with one degree-of-freedom, the one and two mass oscillatory systems with two-degrees-of-freedom and continuous oscillators are considered. The chaos and chaos suppression in ideal and non-ideal mechanical systems is explained. In this second edition more attention is given to the application of the suggested methodologies and obtained results to some practical problems in physics, mechanics, electronics and biomechanics. Thus, for the oscillator with two degrees-of-freedom, a generalization of the solving procedure is performed. Based on the obtained results, vibrations of the vocal cord are analyzed. In the book the vibration of the axially purely nonlinear rod as a continuous system is investigated. The developed solving procedure and the solutions are applied to discuss the muscle vibration. Vibrations of an optomechanical system are analyzed using the oscillations of an oscillator with odd or even quadratic nonlinearities. The extension of the forced vibrations of the system is realized by introducing the Ateb periodic excitation force which is the series of a trigonometric function. The book is self-consistent and suitable for researchers and as a textbook for students and also professionals and engineers who apply these techniques to the field of nonlinear oscillations.

Recent Trends in Nonlinear Partial Differential Equations I
Recent Trends in Nonlinear Partial Differential Equations I

Author: James B. Serrin

Publisher: American Mathematical Soc.

Published: 2013-07-22

Total Pages: 323

ISBN-13: 082188736X

DOWNLOAD EBOOK

This book is the first of two volumes which contain the proceedings of the Workshop on Nonlinear Partial Differential Equations, held from May 28-June 1, 2012, at the University of Perugia in honor of Patrizia Pucci's 60th birthday. The workshop brought t

Principles of Discontinuous Dynamical Systems
Principles of Discontinuous Dynamical Systems

Author: Marat Akhmet

Publisher: Springer Science & Business Media

Published: 2010-08-26

Total Pages: 185

ISBN-13: 1441965815

DOWNLOAD EBOOK

Discontinuous dynamical systems have played an important role in both theory and applications during the last several decades. This is still an area of active research and techniques to make the applications more effective are an ongoing topic of interest. Principles of Discontinuous Dynamical Systems is devoted to the theory of differential equations with variable moments of impulses. It introduces a new strategy of implementing an equivalence to systems whose solutions have prescribed moments of impulses and utilizing special topologies in spaces of piecewise continuous functions. The achievements obtained on the basis of this approach are described in this book. The text progresses systematically, by covering preliminaries in the first four chapters. This is followed by more complex material and special topics such as Hopf bifurcation, Devaney's chaos, and the shadowing property are discussed in the last two chapters. This book is suitable for researchers and graduate students in mathematics and also in diverse areas such as biology, computer science, and engineering who deal with real world problems.

Arnold's Problems
Arnold's Problems

Author: Vladimir I. Arnold

Publisher: Springer Science & Business Media

Published: 2004-06-24

Total Pages: 664

ISBN-13: 9783540206149

DOWNLOAD EBOOK

Vladimir Arnold is one of the most outstanding mathematicians of our time Many of these problems are at the front line of current research

Nonlinear Differential Equations and Dynamical Systems
Nonlinear Differential Equations and Dynamical Systems

Author: Ferdinand Verhulst

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 287

ISBN-13: 3642971490

DOWNLOAD EBOOK

Bridging the gap between elementary courses and the research literature in this field, the book covers the basic concepts necessary to study differential equations. Stability theory is developed, starting with linearisation methods going back to Lyapunov and Poincaré, before moving on to the global direct method. The Poincaré-Lindstedt method is introduced to approximate periodic solutions, while at the same time proving existence by the implicit function theorem. The final part covers relaxation oscillations, bifurcation theory, centre manifolds, chaos in mappings and differential equations, and Hamiltonian systems. The subject material is presented from both the qualitative and the quantitative point of view, with many examples to illustrate the theory, enabling the reader to begin research after studying this book.

Mathematical Aspects of Classical and Celestial Mechanics
Mathematical Aspects of Classical and Celestial Mechanics

Author: Vladimir I. Arnold

Publisher: Springer Science & Business Media

Published: 2007-07-05

Total Pages: 505

ISBN-13: 3540489266

DOWNLOAD EBOOK

The main purpose of the book is to acquaint mathematicians, physicists and engineers with classical mechanics as a whole, in both its traditional and its contemporary aspects. As such, it describes the fundamental principles, problems, and methods of classical mechanics, with the emphasis firmly laid on the working apparatus, rather than the physical foundations or applications. Chapters cover the n-body problem, symmetry groups of mechanical systems and the corresponding conservation laws, the problem of the integrability of the equations of motion, the theory of oscillations and perturbation theory.

The Strong Nonlinear Limit-point/limit-circle Problem
The Strong Nonlinear Limit-point/limit-circle Problem

Author: John R Graef

Publisher: World Scientific

Published: 2017-10-06

Total Pages: 325

ISBN-13: 9813226390

DOWNLOAD EBOOK

The limit-point/limit-circle problem had its beginnings more than 100 years ago with the publication of Hermann Weyl's classic paper in Mathematische Annalen in 1910 on linear differential equations. This concept was extended to second-order nonlinear equations in the late 1970's and later, to higher order nonlinear equations. This monograph traces the development of what is known as the strong nonlinear limit-point and limit-circle properties of solutions. In addition to bringing together all such results into one place, some new directions that the study has taken as well as some open problems for future research are indicated.

Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities
Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities

Author: Marat Akhmet

Publisher: Springer

Published: 2017-01-23

Total Pages: 175

ISBN-13: 9811031800

DOWNLOAD EBOOK

This book focuses on bifurcation theory for autonomous and nonautonomous differential equations with discontinuities of different types – those with jumps present either in the right-hand side, or in trajectories or in the arguments of solutions of equations. The results obtained can be applied to various fields, such as neural networks, brain dynamics, mechanical systems, weather phenomena and population dynamics. Developing bifurcation theory for various types of differential equations, the book is pioneering in the field. It presents the latest results and provides a practical guide to applying the theory to differential equations with various types of discontinuity. Moreover, it offers new ways to analyze nonautonomous bifurcation scenarios in these equations. As such, it shows undergraduate and graduate students how bifurcation theory can be developed not only for discrete and continuous systems, but also for those that combine these systems in very different ways. At the same time, it offers specialists several powerful instruments developed for the theory of discontinuous dynamical systems with variable moments of impact, differential equations with piecewise constant arguments of generalized type and Filippov systems.

Asymptotic Analysis and the Numerical Solution of Partial Differential Equations
Asymptotic Analysis and the Numerical Solution of Partial Differential Equations

Author: Hans G. Kaper

Publisher: CRC Press

Published: 1991-02-25

Total Pages: 288

ISBN-13: 082478538X

DOWNLOAD EBOOK

Integrates two fields generally held to be incompatible, if not downright antithetical, in 16 lectures from a February 1990 workshop at the Argonne National Laboratory, Illinois. The topics, of interest to industrial and applied mathematicians, analysts, and computer scientists, include singular per