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This book offers a short and concise introduction to the many
facets of chaos theory. While the study of chaotic behavior in
nonlinear, dynamical systems is a well-established research field
with ramifications in all areas of science, there is a lot to be
learnt about how chaos can be controlled and, under appropriate
conditions, can actually be constructive in the sense of becoming a
control parameter for the system under investigation, stochastic
resonance being a prime example. The present work stresses the
latter aspects and, after recalling the paradigm changes introduced
by the concept of chaos, leads the reader skillfully through the
basics of chaos control by detailing the relevant algorithms for
both Hamiltonian and dissipative systems, among others. The main
part of the book is then devoted to the issue of synchronization in
chaotic systems, an introduction to stochastic resonance, and a
survey of ratchet models. In this second, revised and enlarged
edition, two more chapters explore the many interfaces of quantum
physics and dynamical systems, examining in turn statistical
properties of energy spectra, quantum ratchets, and dynamical
tunneling, among others. This text is particularly suitable for
non-specialist scientists, engineers, and applied mathematical
scientists from related areas, wishing to enter the field quickly
and efficiently. From the reviews of the first edition: This book
is an excellent introduction to the key concepts and control of
chaos in (random) dynamical systems [...] The authors find an
outstanding balance between main physical ideas and mathematical
terminology to reach their audience in an impressive and lucid
manner. This book is ideal for anybody who would like to grasp
quickly the main issues related to chaos in discrete and continuous
time. Henri Schurz, Zentralblatt MATH, Vol. 1178, 2010.
This book offers a short and concise introduction to the many
facets of chaos theory. While the study of chaotic behavior in
nonlinear, dynamical systems is a well-established research field
with ramifications in all areas of science, there is a lot to be
learnt about how chaos can be controlled and, under appropriate
conditions, can actually be constructive in the sense of becoming a
control parameter for the system under investigation, stochastic
resonance being a prime example. The present work stresses the
latter aspects and, after recalling the paradigm changes introduced
by the concept of chaos, leads the reader skillfully through the
basics of chaos control by detailing the relevant algorithms for
both Hamiltonian and dissipative systems, among others. The main
part of the book is then devoted to the issue of synchronization in
chaotic systems, an introduction to stochastic resonance, and a
survey of ratchet models. In this second, revised and enlarged
edition, two more chapters explore the many interfaces of quantum
physics and dynamical systems, examining in turn statistical
properties of energy spectra, quantum ratchets, and dynamical
tunneling, among others. This text is particularly suitable for
non-specialist scientists, engineers, and applied mathematical
scientists from related areas, wishing to enter the field quickly
and efficiently. From the reviews of the first edition: This book
is an excellent introduction to the key concepts and control of
chaos in (random) dynamical systems [...] The authors find an
outstanding balance between main physical ideas and mathematical
terminology to reach their audience in an impressive and lucid
manner. This book is ideal for anybody who would like to grasp
quickly the main issues related to chaos in discrete and continuous
time. Henri Schurz, Zentralblatt MATH, Vol. 1178, 2010.
The study of physics has changed in character, mainly due to the
passage from the analyses of linear systems to the analyses of
nonlinear systems. Such a change began, it goes without saying, a
long time ago but the qualitative change took place and boldly
evolved after the understanding of the nature of chaos in nonlinear
s- tems. The importance of these systems is due to the fact that
the major part of physical reality is nonlinear. Linearity appears
as a result of the simpli?cation of real systems, and often, is
hardly achievable during the experimental studies. In this book, we
focus our attention on some general phenomena, naturally linked
with nonlinearity where chaos plays a constructive part. The ?rst
chapter discusses the concept of chaos. It attempts to describe the
me- ing of chaos according to the current understanding of it in
physics and mat- matics. The content of this chapter is essential
to understand the nature of chaos and its appearance in
deterministic physical systems. Using the Turing machine, we
formulate the concept of complexity according to Kolmogorov.
Further, we state the algorithmic theory of Kolmogorov-Martin-Lof ]
randomness, which gives a deep understanding of the nature of
deterministic chaos. Readers will not need any advanced knowledge
to understand it and all the necessary facts and de?nitions will be
explained."
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