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Periodic Solutions of Nonlinear Dynamical Systems
Language: en
Pages: 174
Authors: Eduard Reithmeier
Categories: Mathematics
Type: BOOK - Published: 2006-11-14 - Publisher: Springer

Limit cycles or, more general, periodic solutions of nonlinear dynamical systems occur in many different fields of application. Although, there is extensive literature on periodic solutions, in particular on existence theorems, the connection to physical and technical applications needs to be improved. The bifurcation behavior of periodic solutions by means of parameter variations plays an important role in transition to chaos, so numerical algorithms are necessary to compute periodic solutions and investigate their stability on a numerical basis. From the technical point of view, dynamical systems with discontinuities are of special interest. The discontinuities may occur with respect to the variables describing the configuration space manifold or/and with respect to the variables of the vector-field of the dynamical system. The multiple shooting method is employed in computing limit cycles numerically, and is modified for systems with discontinuities. The theory is supported by numerous examples, mainly from the field of nonlinear vibrations. The text addresses mathematicians interested in engineering problems as well as engineers working with nonlinear dynamics.
Energy Flow Theory of Nonlinear Dynamical Systems with Applications
Language: en
Pages: 299
Authors: Jing Tang Xing
Categories: Technology & Engineering
Type: BOOK - Published: 2015-05-28 - Publisher: Springer

This monograph develops a generalised energy flow theory to investigate non-linear dynamical systems governed by ordinary differential equations in phase space and often met in various science and engineering fields. Important nonlinear phenomena such as, stabilities, periodical orbits, bifurcations and chaos are tack-led and the corresponding energy flow behaviors are revealed using the proposed energy flow approach. As examples, the common interested nonlinear dynamical systems, such as, Duffing’s oscillator, Van der Pol’s equation, Lorenz attractor, Rössler one and SD oscillator, etc, are discussed. This monograph lights a new energy flow research direction for nonlinear dynamics. A generalised Matlab code with User Manuel is provided for readers to conduct the energy flow analysis of their nonlinear dynamical systems. Throughout the monograph the author continuously returns to some examples in each chapter to illustrate the applications of the discussed theory and approaches. The book can be used as an undergraduate or graduate textbook or a comprehensive source for scientists, researchers and engineers, providing the statement of the art on energy flow or power flow theory and methods.
Mathematical Modeling and Applications in Nonlinear Dynamics
Language: en
Pages: 205
Authors: Albert C.J. Luo, Hüseyin Merdan
Categories: Technology & Engineering
Type: BOOK - Published: 2016-01-28 - Publisher: Springer

The book covers nonlinear physical problems and mathematical modeling, including molecular biology, genetics, neurosciences, artificial intelligence with classical problems in mechanics and astronomy and physics. The chapters present nonlinear mathematical modeling in life science and physics through nonlinear differential equations, nonlinear discrete equations and hybrid equations. Such modeling can be effectively applied to the wide spectrum of nonlinear physical problems, including the KAM (Kolmogorov-Arnold-Moser (KAM)) theory, singular differential equations, impulsive dichotomous linear systems, analytical bifurcation trees of periodic motions, and almost or pseudo- almost periodic solutions in nonlinear dynamical systems.
Nonlinear Dynamics in Engineering Systems
Language: en
Pages: 362
Authors: Werner Schiehlen
Categories: Science
Type: BOOK - Published: 2012-12-06 - Publisher: Springer Science & Business Media

The International Union of Theoretical and Applied Mechanics (IUTAM) initiated and sponsored an International Symposium on Nonlinear Dynamics in Engineering Systems held in 1989 in Stuttgart, FRG. The Symposium was intended to bring together scientists working in different fields of dynamics to exchange ideas and to discuss new trends with special emphasis on nonlinear dynamics in engineering systems. A Scientific Committee was appointed by the Bureau of IUTAM with the following members: S. Arimoto (Japan), F.L. Chernousko (USSR), P.J. Holmes (USA), C.S. Hsu (USA), G. looss (France), F.C. Moon (USA), W. Schiehlen (FRG), Chairman, G. Schmidt (GDR), W. Szemplinska-Stupnicka (Poland), J.M.T. Thompson (UK), H. Troger (Austria). This committee selected the participants to be invited and the papers to be presented at the Symposium. As a result of this procedure 78 active scientific participants from 22 countries followed the invitation, and 44 papers were presented in lecture and poster sessions. They are collected in this volume. At the Symposium an exhibition with experiments took place and the movie "An Introduction to the Analysis of Chaotic Dynamics" by E.J. Kreuzer et.al. was presented. The scientific lectures were devoted to the following topics: o Dynamic Structural Engineering Problems, o Analysis of Nonlinear Dynamic Systems, o Bifurcation Problems, o Chaotic Dynamics and Control Problems, o Miscellaneous Problems, o Experimental and Theoretical Investigations, o Chaotic Oscillations of Engineering Systems, o Characterization of Nonlinear Dynamic Systems, o Nonlinear Stochastic Systems.
Galloping Instability to Chaos of Cables
Language: en
Pages: 208
Authors: Albert C. J. Luo, Bo Yu
Categories: Technology & Engineering
Type: BOOK - Published: 2018-01-16 - Publisher: Springer

This book provides students and researchers with a systematic solution for fluid-induced structural vibrations, galloping instability and the chaos of cables. They will also gain a better understanding of stable and unstable periodic motions and chaos in fluid-induced structural vibrations. Further, the results presented here will help engineers effectively design and analyze fluid-induced vibrations.
Discretization and Implicit Mapping Dynamics
Language: en
Pages: 310
Authors: Albert C. J. Luo
Categories: Science
Type: BOOK - Published: 2015-07-30 - Publisher: Springer

This unique book presents the discretization of continuous systems and implicit mapping dynamics of periodic motions to chaos in continuous nonlinear systems. The stability and bifurcation theory of fixed points in discrete nonlinear dynamical systems is reviewed, and the explicit and implicit maps of continuous dynamical systems are developed through the single-step and multi-step discretizations. The implicit dynamics of period-m solutions in discrete nonlinear systems are discussed. The book also offers a generalized approach to finding analytical and numerical solutions of stable and unstable periodic flows to chaos in nonlinear systems with/without time-delay. The bifurcation trees of periodic motions to chaos in the Duffing oscillator are shown as a sample problem, while the discrete Fourier series of periodic motions and chaos are also presented. The book offers a valuable resource for university students, professors, researchers and engineers in the fields of applied mathematics, physics, mechanics, control systems, and engineering.
Bifurcation Dynamics of a Damped Parametric Pendulum
Language: en
Pages: 98
Authors: Yu Guo, Albert C.J. Luo
Categories: Technology & Engineering
Type: BOOK - Published: 2019-12-02 - Publisher: Morgan & Claypool Publishers

The inherent complex dynamics of a parametrically excited pendulum is of great interest in nonlinear dynamics, which can help one better understand the complex world. Even though the parametrically excited pendulum is one of the simplest nonlinear systems, until now, complex motions in such a parametric pendulum cannot be achieved. In this book, the bifurcation dynamics of periodic motions to chaos in a damped, parametrically excited pendulum is discussed. Complete bifurcation trees of periodic motions to chaos in the parametrically excited pendulum include: period-1 motion (static equilibriums) to chaos, and period-m motions to chaos (m = 1, 2, ···, 6, 8, ···, 12). The aforesaid bifurcation trees of periodic motions to chaos coexist in the same parameter ranges, which are very difficult to determine through traditional analysis. Harmonic frequency-amplitude characteristics of such bifurcation trees are also presented to show motion complexity and nonlinearity in such a parametrically excited pendulum system. The non-travelable and travelable periodic motions on the bifurcation trees are discovered. Through the bifurcation trees of travelable and non-travelable periodic motions, the travelable and non-travelable chaos in the parametrically excited pendulum can be achieved. Based on the traditional analysis, one cannot achieve the adequate solutions presented herein for periodic motions to chaos in the parametrically excited pendulum. The results in this book may cause one rethinking how to determine motion complexity in nonlinear dynamical systems.
Toward Analytical Chaos in Nonlinear Systems
Language: en
Pages: 272
Authors: Albert C. J. Luo
Categories: Technology & Engineering
Type: BOOK - Published: 2014-05-27 - Publisher: John Wiley & Sons

Exact analytical solutions to periodic motions in nonlineardynamical systems are almost not possible. Since the 18th century,one has extensively used techniques such as perturbation methods toobtain approximate analytical solutions of periodic motions innonlinear systems. However, the perturbation methods cannot providethe enough accuracy of analytical solutions of periodic motions innonlinear dynamical systems. So the bifurcation trees of periodicmotions to chaos cannot be achieved analytically. The authorhas developed an analytical technique that is more effective toachieve periodic motions and corresponding bifurcation trees tochaos analytically. Toward Analytical Chaos in Nonlinear Systemssystematically presents a new approach to analytically determineperiodic flows to chaos or quasi-periodic flows in nonlineardynamical systems with/without time-delay. It covers themathematical theory and includes two examples of nonlinear systemswith/without time-delay in engineering and physics. From theanalytical solutions, the routes from periodic motions to chaos aredeveloped analytically rather than the incomplete numerical routesto chaos. The analytical techniques presented will provide abetter understanding of regularity and complexity of periodicmotions and chaos in nonlinear dynamical systems. Key features: Presents the mathematical theory of analytical solutions ofperiodic flows to chaos or quasieriodic flows in nonlineardynamical systems Covers nonlinear dynamical systems and nonlinear vibrationsystems Presents accurate, analytical solutions of stable and unstableperiodic flows for popular nonlinear systems Includes two complete sample systems Discusses time-delayed, nonlinear systems and time-delayed,nonlinear vibrational systems Includes real world examples Toward Analytical Chaos in Nonlinear Systems is acomprehensive reference for researchers and practitioners acrossengineering, mathematics and physics disciplines, and is also auseful source of information for graduate and senior undergraduatestudents in these areas.
Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems
Language: en
Pages: 336
Authors: Jan Awrejcewicz
Categories: Technology & Engineering
Type: BOOK - Published: 2008-12-26 - Publisher: Springer Science & Business Media

This volume contains the invited papers presented at the 9th International Conference "Dynamical Systems — Theory and Applications" held in Lódz, Poland, December 17-20, 2007, dealing with nonlinear dynamical systems. The conference brought together a large group of outstanding scientists and engineers, who deal with various problems of dynamics encountered both in engineering and in daily life. Topics covered include, among others, bifurcations and chaos in mechanical systems; control in dynamical systems; asymptotic methods in nonlinear dynamics; stability of dynamical systems; lumped and continuous systems vibrations; original numerical methods of vibration analysis; and man-machine interactions. Thus, the reader is given an overview of the most recent developments of dynamical systems and can follow the newest trends in this field of science. This book will be of interest to to pure and applied scientists working in the field of nonlinear dynamics.
Finite Element Methods for Periodic Solutions of Dynamical Systems
Language: en
Pages: 238
Authors: Mohamed Shendy El-Mandouh
Categories: Finite element method
Type: BOOK - Published: 1980 - Publisher:

Books about Finite Element Methods for Periodic Solutions of Dynamical Systems
Frontiers in the Study of Chaotic Dynamical Systems with Open Problems
Language: en
Pages: 268
Authors: Elhadj Zeraoulia, Julien Clinton Sprott
Categories: Science
Type: BOOK - Published: 2011-03-08 - Publisher: World Scientific

This collection of review articles is devoted to new developments in the study of chaotic dynamical systems with some open problems and challenges. The papers, written by many of the leading experts in the field, cover both the experimental and theoretical aspects of the subject. This edited volume presents a variety of fascinating topics of current interest and problems arising in the study of both discrete and continuous time chaotic dynamical systems. Exciting new techniques stemming from the area of nonlinear dynamical systems theory are currently being developed to meet these challenges. Presenting the state-of-the-art of the more advanced studies of chaotic dynamical systems, Frontiers in the Study of Chaotic Dynamical Systems with Open Problems is devoted to setting an agenda for future research in this exciting and challenging field. Contents:Problems with Lorenz's Modeling and the Algorithm of Chaos Doctrine (S OuYang & Y Lin)Nonexistence of Chaotic Solutions of Nonlinear Differential Equations (L S Yao)Some Open Problems in the Dynamics of Quadratic and Higher Degree Polynomial ODE Systems (F Zhang & J Heidel)On Chaotic and Hyperchaotic Complex Nonlinear Dynamical Systems (G M Mahmoud)On the Study of Chaotic Systems with Non-Horseshoe Template (A Ray et al.)Instability of Solutions of Fourth and Fifth Order Delay Differential Equations (C Tunç)Some Conjectures About the Synchronizability and the Topology of Networks (A Caneco et al.)Wavelet Study of Dynamical Systems Using Partial Differential Equations (E B Postnikov)Combining the Dynamics of Discrete Dynamical Systems (J S Cánovas)Code Structure for Pairs of Linear Maps with Some Open Problems (P Troshin)Recent Advances in Open Billiards with Some Open Problems (C P Dettmann)Open Problems in the Dynamics of the Expression of Gene Interaction Networks (L S Liebovitch & V Naudot)How to Transform a Type of Chaos in Dynamical Systems? (E Zeraoulia & J C Sprott) Readership: Graduate students and researchers interested in chaotic dynamical systems. Keywords:Nonlinear Differential Equations;Lorenz Modeling;ODE;Discrete Dynamical Systems;Chaotic Systems
Nonlinear Dynamics in Circuits
Language: en
Pages: 336
Authors: Thomas L. Carroll, Louis M. Pecora
Categories: Science
Type: BOOK - Published: 1995 - Publisher: World Scientific

This volume describes the use of simple analog circuits to study nonlinear dynamics, chaos and stochastic resonance. The circuit experiments that are described are mostly easy and inexpensive to reproduce, and yet these experiments come from the forefront of nonlinear dynamics research. The individual chapters describe why analog circuits are so useful for studying nonlinear dynamics, and include theoretical as well as experimental results from some of the leading researchers in the field. Most of the articles contain some tutorial sections for the less experienced readers.The audience for this book includes researchers in nonlinear dynamics, chaos and statistical physics as well as electrical engineering, and graduate and advanced undergraduate students in these fields.