This book starts with a discussion of nonlinear ordinary differential
equations, bifurcation theory and Hamiltonian dynamics. It then embarks
on a systematic discussion of the traditional topics of modern nonlinear
dynamics -- integrable systems, Poincaré maps, chaos, fractals and
strange attractors. The Baker's transformation, the logistic map and
Lorenz system are discussed in detail in view of their central place in
the subject. There is a detailed discussion of solitons centered around
the Korteweg-deVries equation in view of its central place in integrable
systems. Then, there is a discussion of the Painlevé property of
nonlinear differential equations which seems to provide a test of
integrability. Finally, there is a detailed discussion of the
application of fractals and multi-fractals to fully-developed turbulence
-- a problem whose understanding has been considerably enriched by the
application of the concepts and methods of modern nonlinear dynamics. On
the application side, there is a special emphasis on some aspects of
fluid dynamics and plasma physics reflecting the author's involvement in
these areas of physics. A few exercises have been provided that range
from simple applications to occasional considerable extension of the
theory. Finally, the list of references given at the end of the book
contains primarily books and papers used in developing the lecture
material this volume is based on.
This book has grown out of the author's lecture notes for an
interdisciplinary graduate-level course on nonlinear dynamics. The basic
concepts, language and results of nonlinear dynamical systems are
described in a clear and coherent way. In order to allow for an
interdisciplinary readership, an informal style has been adopted and the
mathematical formalism has been kept to a minimum.
This book is addressed to first-year graduate students in applied
mathematics, physics, and engineering, and is useful also to any
theoretically inclined researcher in the physical sciences and
engineering.
This second edition constitutes an extensive rewrite of the text
involving refinement and enhancement of the clarity and precision,
updating and amplification of several sections, addition of new material
like theory of nonlinear differential equations, solitons, Lagrangian
chaos in fluids, and critical phenomena perspectives on the fluid
turbulence problem and many new exercises.
