Nonlinear Dynamics and Chaotic Phenomena: An Introduction: 42 (Fluid Mechanics and Its Applications, 42) - Hardcover

Shivamoggi, B.K

Hardcover

ISBN 10: 0792347722 ISBN 13: 9780792347729

Publisher: Springer, 1997

Synopsis

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.

About this title

Synopsis

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 non-linear differential equations, bifurcation theory and Hamiltonian dynamics. It then embarks on a systematic discussion of the traditional topics of modern nonlinear dynamics - integrable systems, Poincare 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 Painleve 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: a strong middle ground between elementary undergraduate texts on the one hand, and advanced level monographs on the other the presentation of some original developments a thorough discussion of the application of fractals to turbulence in fluids.

Edition details

• Publisher: Springer
• Publication date: 1997
• Language: English
• ISBN 10: 0792347722
• ISBN 13: 9780792347729
• Binding: Hardcover
• Number of pages: 423