© 2010

Nonlinear Dynamics

Between Linear and Impact Limits

  • First systematic description for the non smooth time transformations and related analytical and numerical algorithms

  • Bridges the gap between linear and strongly nonlinear dynamics by showing how to switch the physical basis when approaching the area of severe (impact) nonlinearities

  • Presents a unified physical basis for analyses of vibrations with essentially non-harmonic or discontinuous temporal shapes


Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 52)

Table of contents

  1. Front Matter
  2. Valery N. Pilipchuk
    Pages 1-36
  3. Valery N. Pilipchuk
    Pages 37-49
  4. Valery N. Pilipchuk
    Pages 51-91
  5. Valery N. Pilipchuk
    Pages 93-129
  6. Valery N. Pilipchuk
    Pages 131-144
  7. Valery N. Pilipchuk
    Pages 145-178
  8. Valery N. Pilipchuk
    Pages 195-239
  9. Valery N. Pilipchuk
    Pages 241-244
  10. Valery N. Pilipchuk
    Pages 245-264
  11. Valery N. Pilipchuk
    Pages 265-273
  12. Valery N. Pilipchuk
    Pages 275-294
  13. Valery N. Pilipchuk
    Pages 295-303
  14. Valery N. Pilipchuk
    Pages 305-337
  15. Back Matter

About this book


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.


NSTT Non-Smooth Processes Non-Smooth Temporal Transformations Nonlinear Dynamics Nonlinear Processes rotation vibration complexity

Authors and affiliations

  1. 1.Professor (Research) Mechanical EngineeringWayne State UniversityDetroitUSA

Bibliographic information

  • Book Title Nonlinear Dynamics
  • Book Subtitle Between Linear and Impact Limits
  • Authors Valery N. Pilipchuk
  • Series Title Lecture Notes in Applied and Computational Mechanics
  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 2010
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Engineering Engineering (R0)
  • Hardcover ISBN 978-3-642-12798-4
  • Softcover ISBN 978-3-642-26353-8
  • eBook ISBN 978-3-642-12799-1
  • Series ISSN 1613-7736
  • Series E-ISSN 1860-0816
  • Edition Number 1
  • Number of Pages XI, 364
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Classical Mechanics
    Vibration, Dynamical Systems, Control
    Engineering, general
  • Buy this book on publisher's site
Industry Sectors
Energy, Utilities & Environment


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)