Averaging Methods in Nonlinear Dynamical Systems

Averaging Methods in Nonlinear Dynamical Systems PDF

Author: Jan A. Sanders

Publisher: Springer Science & Business Media

Published: 2013-04-17

Total Pages: 259

ISBN-13: 1475745753

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In this book we have developed the asymptotic analysis of nonlinear dynamical systems. We have collected a large number of results, scattered throughout the literature and presented them in a way to illustrate both the underlying common theme, as well as the diversity of problems and solutions. While most of the results are known in the literature, we added new material which we hope will also be of interest to the specialists in this field. The basic theory is discussed in chapters two and three. Improved results are obtained in chapter four in the case of stable limit sets. In chapter five we treat averaging over several angles; here the theory is less standardized, and even in our simplified approach we encounter many open problems. Chapter six deals with the definition of normal form. After making the somewhat philosophical point as to what the right definition should look like, we derive the second order normal form in the Hamiltonian case, using the classical method of generating functions. In chapter seven we treat Hamiltonian systems. The resonances in two degrees of freedom are almost completely analyzed, while we give a survey of results obtained for three degrees of freedom systems. The appendices contain a mix of elementary results, expansions on the theory and research problems.

Nonlinear Differential Equations and Dynamical Systems

Nonlinear Differential Equations and Dynamical Systems PDF

Author: Ferdinand Verhulst

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 287

ISBN-13: 3642971490

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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.

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields PDF

Author: John Guckenheimer

Publisher: Springer Science & Business Media

Published: 2013-11-21

Total Pages: 475

ISBN-13: 1461211409

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An application of the techniques of dynamical systems and bifurcation theories to the study of nonlinear oscillations. Taking their cue from Poincare, the authors stress the geometrical and topological properties of solutions of differential equations and iterated maps. Numerous exercises, some of which require nontrivial algebraic manipulations and computer work, convey the important analytical underpinnings of problems in dynamical systems and help readers develop an intuitive feel for the properties involved.

Nonlinear Dynamics and Chaos

Nonlinear Dynamics and Chaos PDF

Author: Steven H. Strogatz

Publisher: CRC Press

Published: 2018-05-04

Total Pages: 532

ISBN-13: 0429961111

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This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.

Multiscale Methods

Multiscale Methods PDF

Author: Grigoris Pavliotis

Publisher: Springer Science & Business Media

Published: 2008-01-18

Total Pages: 314

ISBN-13: 0387738290

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This introduction to multiscale methods gives you a broad overview of the methods’ many uses and applications. The book begins by setting the theoretical foundations of the methods and then moves on to develop models and prove theorems. Extensive use of examples shows how to apply multiscale methods to solving a variety of problems. Exercises then enable you to build your own skills and put them into practice. Extensions and generalizations of the results presented in the book, as well as references to the literature, are provided in the Discussion and Bibliography section at the end of each chapter.With the exception of Chapter One, all chapters are supplemented with exercises.

Advances in Nonlinear Dynamics: Methods and Applications

Advances in Nonlinear Dynamics: Methods and Applications PDF

Author: Anil K. Bajaj

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 202

ISBN-13: 9401103674

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This is the second and final issue of the collection of papers that were contributed by friends and colleagues of (Late) Professor P. R. "Pat" Sethna of the University of Minnesota to commemorate his 70th birthday on May 26, 1993. The first set of contributions was published in Nonlinear Dynamics as the last issue (no. 6) of Vol. 4 in 1993. As circumstances would have it, Professor Sethna was diagnosed with cancer in the fall of 1992 and, after an extended battle with the disease, he passed away on November 4, 1993, just a few days before the first set of contributed papers appeared in print. It is gratifying to report that the organizers of these vi Foreword commemorative issues in Nonlinear Dynamics were able to present to Professor Sethna, on the occasion of his 70th birthday, complete details of the planned commemorative issues. This second set of contributions is dedicated, in memoriam, to Professor P. R. Sethna. As many of you are well aware, Professor Sethna was an active researcher in the field of nonlinear vibrations and dynamics for nearly forty years, making many fundamental and significant contributions to both the theoretical and applied aspects of this field. He was also recognized for his outstanding leadership and administrative abilities, amply demonstrated through his position as the Head of the Department of Aerospace Engineering and Mechanics at the University of Minnesota for twenty-six years (1966-1992).

Nonlinear Dynamics

Nonlinear Dynamics PDF

Author: Ard‚shir Guran

Publisher: World Scientific

Published: 1997

Total Pages: 254

ISBN-13: 9789810229825

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This book is a collection of papers on the subject of nonlinear dynamics and its applications written by experts in this field. It offers the reader a sampling of exciting research areas in this fast-growing field. The topics covered include chaos, tools to analyze motions, fractal boundaries, dynamics of the Fitzhugh-Nagumo equation, structural control, separation of contaminations from signal of interest, parametric excitation, stochastic bifurcation, mode localization in repetitive structures, Toda lattice, transition from soliton to chaotic motion, nonlinear normal modes, noise perturbations of nonlinear dynamical systems, and phase locking of coupled limit cycle oscillators. Mathematical methods include Lie transforms, Monte Carlo simulations, stochastic calculus, perturbation methods and proper orthogonal decomposition. Applications include gyrodynamics, tether connected satellites, shell buckling, nonlinear circuits, volume oscillations of a large lake, systems with stick-slip friction, imperfect or disordered structures, overturning of rigid blocks, central pattern generators, flow induced oscillations, shape control and vibration suppression of elastic structures.All of these diverse contributions have a common thread: the world of nonlinear behavior. Although linear dynamics is an invaluable tool, there are many problems where nonlinear effects are essential. Some examples include bifurcation of solutions, stability of motion, the effects of large displacements, and subharmonic resonance. This book shows how nonlinear dynamics is currently being utilized and investigated. It will be of interest to engineers, applied mathematicians and physicists.

Nonlinear Dynamics

Nonlinear Dynamics PDF

Author: H.G Solari

Publisher: Routledge

Published: 2019-01-22

Total Pages: 369

ISBN-13: 1351428306

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Nonlinear Dynamics: A Two-Way Trip from Physics to Math provides readers with the mathematical tools of nonlinear dynamics to tackle problems in all areas of physics. The selection of topics emphasizes bifurcation theory and topological analysis of dynamical systems. The book includes real-life problems and experiments as well as exercises and work

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