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continuous systems using ordinary differential equations ",
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The author has gone for breadth of coverage rather than fine detail and \
theorems with proof are kept at a minimum. The material is not clouded by \
functional analytic and group theoretical definitions, and so is intelligible \
to readers with a general mathematical background. Some of the topics covered \
are scarcely covered elsewhere. Most of the material in Chapters 9, 10, 14, \
16 and 17 is at postgraduate level and has been influenced by the author's \
own research interests. It has been found that these chapters are especially \
useful as reference material for senior undergraduate project work. The book \
has a very hands-on approach and takes the reader from the basic theory right \
through to recently published research material.",
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researchers from all sorts of disciplines. It is a symbolic, numerical and \
graphical manipulation package which makes it ideal for the study of \
nonlinear dynamical systems. ",
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Cell[TextData[StyleBox["Chapter 0 provides an introduction to the high-level \
computer language Mathematica, developed by Wolfram Research. The reader is \
shown how to use both text based input commands and palettes. Students should \
be able to complete tutorials one and two in under two hours depending upon \
their past experience. New users will find that the tutorials enable them to \
become familiar with Mathematica within a few hours. Both engineering and \
mathematics students appreciate this method of teaching, and the author has \
found that it generally works well with a ratio of one staff member to about \
20 students in a computer laboratory. Those moderately familiar with the \
package and even the expert users will find Chapter 0 to be a useful source \
of reference. Simple Mathematica programs with output are introduced. \
Mathematica program files in the rest of the book are listed at the end of \
each chapter to avoid unnecessary cluttering in the text. The author suggests \
that the reader should save the relevant example programs listed throughout \
the book in separate notebooks. These programs can then be edited accordingly \
when attempting the exercises at the end of each chapter. The Mathematica \
commands, notebooks, programs and output can also be viewed in color over the \
Web at Mathematica's Information Center ",
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solving systems or producing eye-catching graphics. The author has used \
Mathematica 5.2 in the preparation of the material. However, the Mathematica \
programs have been kept as simple as possible and should also run under later \
versions of the package. One of the advantages of using the Information \
Center rather than an attached CD is that programs can be updated as new \
versions of Mathematica are released.",
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Cell[TextData[StyleBox["The first few chapters of the book cover some theory \
of ordinary differential equations and applications to models in the real \
world are given. The theory of differential equations applied to chemical \
kinetics and electric circuits is introduced in some detail. Chapter 1 ends \
with the existence and uniqueness theorem for the solutions of certain types \
of differential equation. A variety of numerical procedures are available in \
Mathematica when solving stiff and nonstiff systems when an analytic solution \
does not exist or is extremely difficult to find. The theory behind the \
construction of phase plane portraits for two-dimensional systems is dealt \
with in Chapter 2. Applications are taken from chemical kinetics, economics, \
electronics, epidemiology, mechanics, population dynamics; and modeling the \
populations of interacting species are discussed in some detail in Chapter 3. \
Limit cycles, or isolated periodic solutions, are introduced in Chapter 4. \
Since we live in a periodic world, these are the most common type of solution \
found when modeling nonlinear dynamical systems. They appear extensively when \
modeling both the technological and natural sciences. Hamiltonian, or \
conservative, systems and stability are discussed in Chapter 5, and Chapter 6 \
is concerned with how planar systems vary depending upon a parameter. \
Bifurcation, bistability, multistability, and normal forms are discussed.",
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exhibit strange attractors and chaotic dynamics. One can plot the \
three-dimensional objects in Mathematica and graph time series plots to get a \
better understanding of the dynamics involved. Once again the theory can be \
applied to chemical kinetics (including stiff systems), electric circuits, \
and epidemiology; a simplified model for the weather is also briefly \
discussed. The next chapter deals with Poincar\[EAcute] first return maps \
that can be used to untangle complicated interlacing trajectories in \
higher-dimensional spaces. A periodically driven nonlinear pendulum is also \
investigated by means of a nonautonomous differential equation. Both local \
and global bifurcations are investigated in Chapter 9. The main results and \
statement of the famous second part of David Hilbert's sixteenth problem are \
listed in Chapter 10. In order to understand these results, Poincar\[EAcute] \
compactification is introduced. The study of continuous systems ends with one \
of the authors specialities---limit cycles of Li\[EAcute]nard systems. There \
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model used to investigate the population of a single species split into \
different age classes. Harvesting and culling policies are then investigated \
and optimal solutions are sought. Nonlinear discrete dynamical systems are \
dealt with in Chapter 12. Bifurcation diagrams, chaos, intermittency, \
Lyapunov exponents, periodicity, quasiperiodicity, and universality are some \
of the topics discussed. The theory is then applied to real-world problems \
from a broad range of disciplines including population dynamics, biology, \
economics, nonlinear optics, and neural networks. The next chapter is \
concerned with complex iterative maps, Julia sets and the now famous \
Mandelbrot set are plotted. Basins of attraction are investigated for the \
first time in this text. As a simple introduction to optics, electromagnetic \
waves and Maxwell's equations are studied at the beginning of Chapter 14. \
Complex iterative equations are used to model the propagation of light waves \
through nonlinear optical fibers. A brief history of nonlinear bistable \
optical resonators is discussed and the simple fibre ring resonator is \
analyzed in particular. Chapter 14 is devoted to the study of these optical \
resonators and phenomena such as bistability, chaotic attractors, feedback, \
hysteresis, instability, linear stability analysis, multistability, \
nonlinearity, and steady-states are dealt with. The first and second \
iterative methods are defined in this chapter. Some simple fractals may be \
constructed using pencil and paper in Chapter 15, and the concept of fractal \
dimension is introduced. Fractals may be thought of as identical motifs \
repeated on ever reduced scales. Unfortunately, most of the fractals \
appearing in nature are not homogeneous but are more heterogeneous, hence the \
need for the multifractal theory given later in the chapter. It has been \
found that the distribution of stars and galaxies in our universe are \
multifractal, and there is even evidence of multifractals in rainfall, stock \
markets, and heartbeat rhythms. Applications in materials science, \
geoscience, and image processing are briefly discussed. The next chapter is \
devoted to the new and exciting theory behind chaos control and \
synchronization. For most systems, the maxim used by engineers in the past \
has been \"stability good, chaos bad\", but more and more nowadays this is \
being replaced with \"stability good, chaos better\". There are exciting and \
novel applications in cardiology, communications, engineering, laser \
technology, and space research, for example.",
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networks is presented in Chapter 17. Imagine trying to make a computer mimic \
the human brain. One could ask the question: In the future will it be \
possible for computers to think and even be conscious? The human brain will \
always be more powerful than traditional, sequential, logic-based digital \
computers and scientists are trying to incorporate some features of the brain \
into modern computing. Neural networks perform through learning and no \
underlying equations are required. Mathematicians and computer scientists are \
attempting to mimic the way neurons work together via synapses, indeed, a \
neural network can be thought of as a crude multidimensional model of the \
human brain. The potential for this theory is still largely unexplored, but \
the expectations are high for future applications in a broad range of \
disciplines. Neural networks are already being used in pattern recognition \
(credit card fraud, prediction and forecasting, disease recognition, facial \
and speech recognition), psychological profiling, predicting wave overtopping \
events, and control problems, for example. They also provide a parallel \
architecture allowing for very fast computational and response times. In \
recent years, the disciplines of neural networks and nonlinear dynamics have \
increasingly coalesced and a new branch of science called neurodynamics is \
emerging. Lyapunov functions can be used to determine the stability of \
certain types of neural network. There is also evidence of chaos, feedback, \
nonlinearity, periodicity, and chaos synchronization in the brain. ",
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first section to be used without the package and the second section to be \
used with the Mathematica package in a computer laboratory.\n\nBoth textbooks \
and research papers are presented in the list of references. The textbooks \
can be used to gain more background material, and the research papers have \
been given to encourage further reading and independent study.\n\nThis book \
is informed by the research interests of the author which are currently \
nonlinear ordinary differential equations, nonlinear optics, multifractals, \
and neural networks. Some references include recently published research \
articles by the author.\n\nThe prerequisites for studying dynamical systems \
using this book are undergraduate courses in linear algebra, real and complex \
analysis, calculus and ordinary differential equations; a knowledge of a \
computer language such as C or Fortran would be beneficial but not essential. \
\n\nI would like to express my sincere thanks to Wolfram Research for \
supplying me with the latest versions of Mathematica. Thanks also go to all \
of the reviewers from the first editions of the Maple and MATLAB books. \
Special thanks go to Tom Grasso and Ann Kostant (Executive Editor, \
Mathematics and Physics, Birkh\[ADoubleDot]user). Thanks to Jon Borresen \
(University of Manchester) and Yibin Fu (University of Keele) for reviewing \
the first draft of the book and checking my Mathematica programming skills. \
Finally, thanks to my family and especially my wife Gaynor, and our children, \
Sebastian and Thalia, for their continuing love, inspiration and support.",
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