Ebook: Nonlinear Dynamics and Chaotic Phenomena: An Introduction
Author: Bhimsen K. Shivamoggi (auth.)
- Tags: Mechanics, Vibration Dynamical Systems Control, Classical Continuum Physics, Nuclear Physics Heavy Ions Hadrons
- Series: Fluid Mechanics and Its Applications 42
- Year: 1997
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- pdf
FolJowing the formulation of the laws of mechanics by Newton, Lagrange sought to clarify and emphasize their geometrical character. Poincare and Liapunov successfuIJy developed analytical mechanics further along these lines. In this approach, one represents the evolution of all possible states (positions and momenta) by the flow in phase space, or more efficiently, by mappings on manifolds with a symplectic geometry, and tries to understand qualitative features of this problem, rather than solving it explicitly. One important outcome of this line of inquiry is the discovery that vastly different physical systems can actually be abstracted to a few universal forms, like Mandelbrot's fractal and Smale's horse-shoe map, even though the underlying processes are not completely understood. This, of course, implies that much of the observed diversity is only apparent and arises from different ways of looking at the same system. Thus, modern nonlinear dynamics 1 is very much akin to classical thermodynamics in that the ideas and results appear to be applicable to vastly different physical systems. Chaos theory, which occupies a central place in modem nonlinear dynamics, refers to a deterministic development with chaotic outcome. Computers have contributed considerably to progress in chaos theory via impressive complex graphics. However, this approach lacks organization and therefore does not afford complete insight into the underlying complex dynamical behavior. This dynamical behavior mandates concepts and methods from such areas of mathematics and physics as nonlinear differential equations, bifurcation theory, Hamiltonian dynamics, number theory, topology, fractals, and others.
This book has grown out of the author's lecture notes for an interdisciplinary graduate-level course on nonlinear dynamics. The author describes in a clear and coherent way the basic concepts, language and results of nonlinear dynamical systems. In order to allow for an interdisciplinary readership, an informal style has been adopted and the mathematical formalism kept to a minimum.
The book starts with a discussion of nonlinear differential equations, bifurcation theory and Hamiltonian dynamics. It then embarks on a systematic discussion of the traditional topics of modern nonlinear dynamics - integrable systems, Poincar? maps, chaos, fractals and strange attractors. Baker's transformation, the logistic map and the Lorenz system are discussed in detail. Finally, there are systematic discussions of the application of fractals to turbulence in fluids, and the Painlev? property of nonlinear differential equations. Exercises are given at the end of each chapter.
This book is accessible to first-year graduate students in applied mathematics, physics and engineering, and is useful to any theoretically inclined researcher in physical sciences and engineering. Among the unique features of this book are:
This book has grown out of the author's lecture notes for an interdisciplinary graduate-level course on nonlinear dynamics. The author describes in a clear and coherent way the basic concepts, language and results of nonlinear dynamical systems. In order to allow for an interdisciplinary readership, an informal style has been adopted and the mathematical formalism kept to a minimum.
The book starts with a discussion of nonlinear differential equations, bifurcation theory and Hamiltonian dynamics. It then embarks on a systematic discussion of the traditional topics of modern nonlinear dynamics - integrable systems, Poincar? maps, chaos, fractals and strange attractors. Baker's transformation, the logistic map and the Lorenz system are discussed in detail. Finally, there are systematic discussions of the application of fractals to turbulence in fluids, and the Painlev? property of nonlinear differential equations. Exercises are given at the end of each chapter.
This book is accessible to first-year graduate students in applied mathematics, physics and engineering, and is useful to any theoretically inclined researcher in physical sciences and engineering. Among the unique features of this book are:
Content:
Front Matter....Pages i-xiii
Introduction to Chaotic Behavior in Nonlinear Dynamics....Pages 1-11
Nonlinear Differential Equations....Pages 13-60
Bifurcation Theory....Pages 61-91
Hamiltonian Dynamics....Pages 93-126
Integrable Systems....Pages 127-195
Chaos in Conservative Systems....Pages 197-245
Chaos in Dissipative Systems....Pages 247-318
Fractals and Multi-Fractals in Turbulence....Pages 319-351
Singularity Analysis and the Painleve’ Property of Dynamical Systems....Pages 353-373
Back Matter....Pages 375-410
This book has grown out of the author's lecture notes for an interdisciplinary graduate-level course on nonlinear dynamics. The author describes in a clear and coherent way the basic concepts, language and results of nonlinear dynamical systems. In order to allow for an interdisciplinary readership, an informal style has been adopted and the mathematical formalism kept to a minimum.
The book starts with a discussion of nonlinear differential equations, bifurcation theory and Hamiltonian dynamics. It then embarks on a systematic discussion of the traditional topics of modern nonlinear dynamics - integrable systems, Poincar? maps, chaos, fractals and strange attractors. Baker's transformation, the logistic map and the Lorenz system are discussed in detail. Finally, there are systematic discussions of the application of fractals to turbulence in fluids, and the Painlev? property of nonlinear differential equations. Exercises are given at the end of each chapter.
This book is accessible to first-year graduate students in applied mathematics, physics and engineering, and is useful to any theoretically inclined researcher in physical sciences and engineering. Among the unique features of this book are:
Content:
Front Matter....Pages i-xiii
Introduction to Chaotic Behavior in Nonlinear Dynamics....Pages 1-11
Nonlinear Differential Equations....Pages 13-60
Bifurcation Theory....Pages 61-91
Hamiltonian Dynamics....Pages 93-126
Integrable Systems....Pages 127-195
Chaos in Conservative Systems....Pages 197-245
Chaos in Dissipative Systems....Pages 247-318
Fractals and Multi-Fractals in Turbulence....Pages 319-351
Singularity Analysis and the Painleve’ Property of Dynamical Systems....Pages 353-373
Back Matter....Pages 375-410
....