Toward Analytical Chaos In Nonlinear Systems

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Toward Analytical Chaos In Nonlinear Systems

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Author : Albert C. J. Luo
language : en
Publisher: John Wiley & Sons
Release Date : 2014-05-27

Toward Analytical Chaos In Nonlinear Systems written by Albert C. J. Luo and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05-27 with Technology & Engineering categories.

Exact analytical solutions to periodic motions in nonlinear dynamical systems are almost not possible. Since the 18th century, one has extensively used techniques such as perturbation methods to obtain approximate analytical solutions of periodic motions in nonlinear systems. However, the perturbation methods cannot provide the enough accuracy of analytical solutions of periodic motions in nonlinear dynamical systems. So the bifurcation trees of periodic motions to chaos cannot be achieved analytically. The author has developed an analytical technique that is more effective to achieve periodic motions and corresponding bifurcation trees to chaos analytically. Toward Analytical Chaos in Nonlinear Systems systematically presents a new approach to analytically determine periodic flows to chaos or quasi-periodic flows in nonlinear dynamical systems with/without time-delay. It covers the mathematical theory and includes two examples of nonlinear systems with/without time-delay in engineering and physics. From the analytical solutions, the routes from periodic motions to chaos are developed analytically rather than the incomplete numerical routes to chaos. The analytical techniques presented will provide a better understanding of regularity and complexity of periodic motions and chaos in nonlinear dynamical systems. Key features: Presents the mathematical theory of analytical solutions of periodic flows to chaos or quasieriodic flows in nonlinear dynamical systems Covers nonlinear dynamical systems and nonlinear vibration systems Presents accurate, analytical solutions of stable and unstable periodic 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 a comprehensive reference for researchers and practitioners across engineering, mathematics and physics disciplines, and is also a useful source of information for graduate and senior undergraduate students in these areas.

Analytical Routes To Chaos In Nonlinear Engineering

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Author : Albert C. J. Luo
language : en
Publisher: John Wiley & Sons
Release Date : 2014-05-23

Analytical Routes To Chaos In Nonlinear Engineering written by Albert C. J. Luo and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05-23 with Technology & Engineering categories.

Nonlinear problems are of interest to engineers, physicists and mathematicians and many other scientists because most systems are inherently nonlinear in nature. As nonlinear equations are difficult to solve, nonlinear systems are commonly approximated by linear equations. This works well up to some accuracy and some range for the input values, but some interesting phenomena such as chaos and singularities are hidden by linearization and perturbation analysis. It follows that some aspects of the behavior of a nonlinear system appear commonly to be chaotic, unpredictable or counterintuitive. Although such a chaotic behavior may resemble a random behavior, it is absolutely deterministic. Analytical Routes to Chaos in Nonlinear Engineering discusses analytical solutions of periodic motions to chaos or quasi-periodic motions in nonlinear dynamical systems in engineering and considers engineering applications, design, and control. It systematically discusses complex nonlinear phenomena in engineering nonlinear systems, including the periodically forced Duffing oscillator, nonlinear self-excited systems, nonlinear parametric systems and nonlinear rotor systems. Nonlinear models used in engineering are also presented and a brief history of the topic is provided. Key features: Considers engineering applications, design and control Presents analytical techniques to show how to find the periodic motions to chaos in nonlinear dynamical systems Systematically discusses complex nonlinear phenomena in engineering nonlinear systems Presents extensively used nonlinear models in engineering Analytical Routes to Chaos in Nonlinear Engineering is a practical reference for researchers and practitioners across engineering, mathematics and physics disciplines, and is also a useful source of information for graduate and senior undergraduate students in these areas.

Bifurcation Dynamics Of A Damped Parametric Pendulum

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Author : Yu Guo
language : en
Publisher: Morgan & Claypool Publishers
Release Date : 2019-12-02

Bifurcation Dynamics Of A Damped Parametric Pendulum written by Yu Guo and has been published by Morgan & Claypool Publishers this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-12-02 with Technology & Engineering categories.

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.

Galloping Instability To Chaos Of Cables

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Author : Albert C. J. Luo
language : en
Publisher: Springer
Release Date : 2018-01-16

Galloping Instability To Chaos Of Cables written by Albert C. J. Luo and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-01-16 with Technology & Engineering categories.

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.

Dynamics With Chaos And Fractals

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Author : Marat Akhmet
language : en
Publisher: Springer Nature
Release Date : 2020-01-01

Dynamics With Chaos And Fractals written by Marat Akhmet and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-01-01 with Mathematics categories.

The book is concerned with the concepts of chaos and fractals, which are within the scopes of dynamical systems, geometry, measure theory, topology, and numerical analysis during the last several decades. It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. This is the first time in the literature that the description of chaos is initiated from a single motion. Chaos is now placed on the line of oscillations, and therefore, it is a subject of study in the framework of the theories of dynamical systems and differential equations, as in this book. The techniques introduced in the book make it possible to develop continuous and discrete dynamics which admit fractals as points of trajectories as well as orbits themselves. To provide strong arguments for the genericity of chaos in the real and abstract universe, the concept of abstract similarity is suggested.

Towards Analytical Chaotic Evolutions In Brusselators

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Author : Albert C.J. Luo
language : en
Publisher: Springer Nature
Release Date : 2022-05-31

Towards Analytical Chaotic Evolutions In Brusselators written by Albert C.J. Luo and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-05-31 with Technology & Engineering categories.

The Brusselator is a mathematical model for autocatalytic reaction, which was proposed by Ilya Prigogine and his collaborators at the Université Libre de Bruxelles. The dynamics of the Brusselator gives an oscillating reaction mechanism for an autocatalytic, oscillating chemical reaction. The Brusselator is a slow-fast oscillating chemical reaction system. The traditional analytical methods cannot provide analytical solutions of such slow-fast oscillating reaction, and numerical simulations cannot provide a full picture of periodic evolutions in the Brusselator. In this book, the generalized harmonic balance methods are employed for analytical solutions of periodic evolutions of the Brusselator with a harmonic diffusion. The bifurcation tree of period-1 motion to chaos of the Brusselator is presented through frequency-amplitude characteristics, which be measured in frequency domains. Two main results presented in this book are: • analytical routes of periodical evolutions tochaos and • independent period-(2 + 1) evolution to chaos. This book gives a better understanding of periodic evolutions to chaos in the slow-fast varying Brusselator system, and the bifurcation tree of period-1 evolution to chaos is clearly demonstrated, which can help one understand routes of periodic evolutions to chaos in chemical reaction oscillators. The slow-fast varying systems extensively exist in biological systems and disease dynamical systems. The methodology presented in this book can be used to investigate the slow-fast varying oscillating motions in biological systems and disease dynamical systems for a better understanding of how infectious diseases spread.

Periodic Motions To Chaos In A Spring Pendulum System

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Author : Yu Guo
language : en
Publisher: Springer Nature
Release Date : 2023-02-06

Periodic Motions To Chaos In A Spring Pendulum System written by Yu Guo and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-02-06 with Technology & Engineering categories.

This book builds on the fundamental understandings, learned in undergraduate engineering and physics in principles of dynamics and control of mechanical systems. The design of real-world mechanical systems and devices becomes far more complex than the spring-pendulum system to which most engineers have been exposed. The authors provide one of the simplest models of nonlinear dynamical systems for learning complex nonlinear dynamical systems. The book addresses the complex challenges of the necessary modeling for the design of machines. The book addresses the methods to create a mechanical system with stable and unstable motions in environments influenced by an array of motion complexity including varied excitation frequencies ranging from periodic motions to chaos. Periodic motions to chaos, in a periodically forced nonlinear spring pendulum system, are presented through the discrete mapping method, and the corresponding stability and bifurcations of periodic motions on the bifurcation trees are presented. Developed semi-analytical solutions of periodical motions to chaos help the reader to understand complex nonlinear dynamical behaviors in nonlinear dynamical systems. Especially, one can use unstable motions rather than stable motions only.

Nonlinear Analysis For Human Movement Variability

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Author : Nicholas Stergiou
language : en
Publisher: CRC Press
Release Date : 2018-09-03

Nonlinear Analysis For Human Movement Variability written by Nicholas Stergiou and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-03 with Medical categories.

How Does the Body’s Motor Control System Deal with Repetition? While the presence of nonlinear dynamics can be explained and understood, it is difficult to be measured. A study of human movement variability with a focus on nonlinear dynamics, Nonlinear Analysis for Human Movement Variability, examines the characteristics of human movement within this framework, explores human movement in repetition, and explains how and why we analyze human movement data. It takes an in-depth look into the nonlinear dynamics of systems within and around us, investigates the temporal structure of variability, and discusses the properties of chaos and fractals as they relate to human movement. Providing a foundation for the use of nonlinear analysis and the study of movement variability in practice, the book describes the nonlinear dynamical features found in complex biological and physical systems, and introduces key concepts that help determine and identify patterns within the fluctuations of data that are repeated over time. It presents commonly used methods and novel approaches to movement analysis that reveal intriguing properties of the motor control system and introduce new ways of thinking about variability, adaptability, health, and motor learning. In addition, this text: Demonstrates how nonlinear measures can be used in a variety of different tasks and populations Presents a wide variety of nonlinear tools such as the Lyapunov exponent, surrogation, entropy, and fractal analysis Includes examples from research on how nonlinear analysis can be used to understand real-world applications Provides numerous case studies in postural control, gait, motor control, and motor development Nonlinear Analysis for Human Movement Variability advances the field of human movement variability research by dissecting human movement and studying the role of movement variability. The book proposes new ways to use nonlinear analysis and investigate the temporal structure of variability, and enables engineers, movement scientists, clinicians, and those in related disciplines to effectively apply nonlinear analysis in practice.

Chaos And Order Miniconference On Proceedings Of The Centre For Mathematical Analysis Australian National University

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Author : Robert L Dewar
language : en
Publisher: World Scientific
Release Date : 1991-01-14

Chaos And Order Miniconference On Proceedings Of The Centre For Mathematical Analysis Australian National University written by Robert L Dewar and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991-01-14 with categories.

This volume is dedicated to Dr Ding Lee for his untiring efforts in promoting the advancement of theoretical and computational acoustics.This proceedings volume provides a forum for active researchers to discuss the state-of-the-art developments and results in theoretical and computational acoustics, covering aero-, seismo- and ocean acoustics and related topics. It discusses multidimensional wave propagation modeling, methods of computational acoustics, wave propagation in rocks, fluid-solid interfaces, nonlinear acoustics, neural networks, real applications and experimental results.

Analytical Mechanics

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Author : Louis N. Hand
language : en
Publisher: Cambridge University Press
Release Date : 1998-11-13

Analytical Mechanics written by Louis N. Hand 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 1998-11-13 with Science categories.

Analytical Mechanics, first published in 1999, provides a detailed introduction to the key analytical techniques of classical mechanics, one of the cornerstones of physics. It deals with all the important subjects encountered in an undergraduate course and prepares the reader thoroughly for further study at graduate level. The authors set out the fundamentals of Lagrangian and Hamiltonian mechanics early on in the book and go on to cover such topics as linear oscillators, planetary orbits, rigid-body motion, small vibrations, nonlinear dynamics, chaos, and special relativity. A special feature is the inclusion of many 'e-mail questions', which are intended to facilitate dialogue between the student and instructor. Many worked examples are given, and there are 250 homework exercises to help students gain confidence and proficiency in problem-solving. It is an ideal textbook for undergraduate courses in classical mechanics, and provides a sound foundation for graduate study.