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Chaos is the idea that a system will produce very different
long-term behaviors when the initial conditions are perturbed only
slightly. Chaos is used for novel, time- or energy-critical
interdisciplinary applications. Examples include high-performance
circuits and devices, liquid mixing, chemical reactions, biological
systems, crisis management, secure information processing, and
critical decision-making in politics, economics, as well as
military applications, etc. This book presents the latest
investigations in the theory of chaotic systems and their dynamics.
The book covers some theoretical aspects of the subject arising in
the study of both discrete and continuous-time chaotic dynamical
systems. This book presents the state-of-the-art of the more
advanced studies of chaotic dynamical systems.
This book is a comprehensive collection of known results about the
Lozi map, a piecewise-affine version of the Henon map. Henon map is
one of the most studied examples in dynamical systems and it
attracts a lot of attention from researchers, however it is
difficult to analyze analytically. Simpler structure of the Lozi
map makes it more suitable for such analysis. The book is not only
a good introduction to the Lozi map and its generalizations, it
also summarizes of important concepts in dynamical systems theory
such as hyperbolicity, SRB measures, attractor types, and more.
Robust chaos is defined by the absence of periodic windows and
coexisting attractors in some neighborhoods in the parameter space
of a dynamical system. This unique book explores the definition,
sources, and roles of robust chaos. The book is written in a
reasonably self-contained manner and aims to provide students and
researchers with the necessary understanding of the subject. Most
of the known results, experiments, and conjectures about chaos in
general and about robust chaos in particular are collected here in
a pedagogical form. Many examples of dynamical systems, ranging
from purely mathematical to natural and social processes displaying
robust chaos, are discussed in detail. At the end of each chapter
is a set of exercises and open problems (more than 260 in the whole
book) intended to reinforce the ideas and provide additional
experiences for both readers and researchers in nonlinear science
in general, and chaos theory in particular.
This collection of review articles is devoted to new developments
in the study of chaotic dynamical systems with some open problems
and challenges. The papers, written by many of the leading experts
in the field, cover both the experimental and theoretical aspects
of the subject. This edited volume presents a variety of
fascinating topics of current interest and problems arising in the
study of both discrete and continuous time chaotic dynamical
systems. Exciting new techniques stemming from the area of
nonlinear dynamical systems theory are currently being developed to
meet these challenges. Presenting the state-of-the-art of the more
advanced studies of chaotic dynamical systems, Frontiers in the
Study of Chaotic Dynamical Systems with Open Problems is devoted to
setting an agenda for future research in this exciting and
challenging field.
This book is based on research on the rigorous proof of chaos and
bifurcations in 2-D quadratic maps, especially the invertible case
such as the H non map, and in 3-D ODE's, especially piecewise
linear systems such as the Chua's circuit. In addition, the book
covers some recent works in the field of general 2-D quadratic
maps, especially their classification into equivalence classes, and
finding regions for chaos, hyperchaos, and non-chaos in the space
of bifurcation parameters. Following the main introduction to the
rigorous tools used to prove chaos and bifurcations in the two
representative systems, is the study of the invertible case of the
2-D quadratic map, where previous works are oriented toward H non
mapping. 2-D quadratic maps are then classified into 30 maps with
well-known formulas. Two proofs on the regions for chaos,
hyperchaos, and non-chaos in the space of the bifurcation
parameters are presented using a technique based on the
second-derivative test and bounds for Lyapunov exponents. Also
included is the proof of chaos in the piecewise linear Chua's
system using two methods, the first of which is based on the
construction of Poincar map, and the second is based on a
computer-assisted proof. Finally, a rigorous analysis is provided
on the bifurcational phenomena in the piecewise linear Chua's
system using both an analytical 2-D mapping and a 1-D approximated
Poincar mapping in addition to other analytical methods.
Chaos is the idea that a system will produce very different
long-term behaviors when the initial conditions are perturbed only
slightly. Chaos is used for novel, time- or energy-critical
interdisciplinary applications. Examples include high-performance
circuits and devices, liquid mixing, chemical reactions, biological
systems, crisis management, secure information processing, and
critical decision-making in politics, economics, as well as
military applications, etc. This book presents the latest
investigations in the theory of chaotic systems and their dynamics.
The book covers some theoretical aspects of the subject arising in
the study of both discrete and continuous-time chaotic dynamical
systems. This book presents the state-of-the-art of the more
advanced studies of chaotic dynamical systems.
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