Nonlinear Dynamics: Between Linear and Impact Limits

ISBN-13: 9783642263538 / Angielski / Miękka / 2012 / 364 str.

Valery N. Pilipchuk

Opis

Nonlinear Dynamics represents a wide interdisciplinary area of research dealing with a variety of unusual physical phenomena by means of nonlinear differential equations, discrete mappings, and related mathematical algorithms. However, with no real substitute for the linear superposition principle, the methods of Nonlinear Dynamics appeared to be very diverse, individual and technically complicated. This book makes an attempt to find a common ground for nonlinear dynamic analyses based on the existence of strongly nonlinear but quite simple counterparts to the linear models and tools. It is shown that, since the subgroup of rotations, harmonic oscillators, and the conventional complex analysis generate linear and weakly nonlinear approaches, then translations and reflections, impact oscillators, and hyperbolic (Clifford s) algebras must give rise to some quasi impact methodology. Such strongly nonlinear methods are developed in several chapters of this book based on the idea of non-smooth time substitutions. Although most of the illustrations are based on mechanical oscillators, the area of applications may include also electric, electro-mechanical, electrochemical and other physical models generating strongly anharmonic temporal signals or spatial distributions. Possible applications to periodic elastic structures with non-smooth or discontinuous characteristics are outlined in the final chapter of the book.

Recenzje

From the reviews:

"This book is based on a series of papers which the author published earlier. The main subject of this book is the concept of non-smooth time transformations (NSTT) to describe periodic solutions of essentially nonlinear differential equations. ... This book is suitable for scientists interested in applied mathematics, in particular those interested in constructing (approximations of) periodic solutions for differential equations." (Wim T. van Horssen, Mathematical Reviews, Issue 2012 j)

"The content of this book is built on a new physical idea that the effectiveness of linear and weakly nonlinear dynamic theories is due to the spatio-temporal nature of harmonic motions associated with subgroup of rigid-body rotations. ... this original book will be of interest to the experts, professors and post-graduate students in various areas of nonlinear physics, fundamental and engineering mechanics, applied mathematics, and other fields of research dealing with nonlinear dynamic models, non-smooth or discontinuous processes." (Anatoly Martynyuk, Zentralblatt MATH, Vol. 1202, 2011)

Spis treści

Smooth Oscillating Processes.- Nonsmooth Processes as Asymptotic Limits.- Nonsmooth Temporal Transformations (NSTT).- Sawtooth Power Series.- NSTT for Linear and Piecewise-Linear Systems.- Periodic and Transient Nonlinear Dynamics under Discontinuous Loading.- Strongly Nonlinear Vibrations.- Strongly Nonlinear Waves.- Impact Modes and Parameter Variations.- Principal Trajectories of Forced Vibrations.- NSTT and Shooting Method for Periodic Motions.- Essentially Non-periodic Processes.- Spatially-Oscillating Structures.