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It is written for both senior \ undergraduates and graduate students. The first part of the book deals with \ continuous systems using ordinary differential equations ", FontFamily->"Courier New", FontSize->12, Background->None], StyleBox["(", FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.501961, 0, 0], Background->None], StyleBox["Chapters 1--10", FontFamily->"Courier New", FontSize->12, Background->None], StyleBox[")", FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.501961, 0, 0], Background->None], StyleBox[", the second part is devoted to the study of discrete dynamical \ systems ", FontFamily->"Courier New", FontSize->12, Background->None], StyleBox["(", FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.501961, 0, 0], Background->None], StyleBox["Chapters 11--16", FontFamily->"Courier New", FontSize->12, Background->None], StyleBox[")", FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.501961, 0, 0], Background->None], StyleBox[", and Chapter 17 deals with both continuous and discrete systems. \ 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.", FontFamily->"Courier New", FontSize->12, Background->None] }], "Text", FontSize->14], Cell[TextData[StyleBox["Mathematica is extremely popular with a wide range of \ 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. ", FontFamily->"Courier New"]], "Text"], 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 ", FontFamily->"Courier New"]], "Text"], Cell[BoxData[ RowBox[{"\t\t", StyleBox[ ButtonBox[ StyleBox[\(\(\(http\)\(:\)\) // \(\(\(\(library . wolfram . com/ infocenter\)/Books\)/AppliedMathematics\)\(/\)\)\), FormatType->StandardForm, FontFamily->"Courier New", FontSize->10, Background->None], ButtonData:>{ URL[ "http://library.wolfram.com/infocenter/Books/AppliedMathematics/"]\ , None}, ButtonStyle->"Hyperlink"], FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->16, Background->None]}]], "Text", TextAlignment->Center], Cell[TextData[StyleBox["The Mathematica program files can also be downloaded \ at this site. Throughout this book, Mathematica is viewed as a tool for \ 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.", FontFamily->"Courier New"]], "Text"], 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.", FontFamily->"Courier New"]], "Text"], Cell[TextData[{ StyleBox["The concept of chaos is expanded upon in Chapters 7 and 8, where \ three-dimensional systems and Poincar\[EAcute]", FontFamily->"Courier New"], " ", StyleBox["maps are investigated. These higher dimensional systems can \ 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 \ is some detail on Li\[EAcute]nard systems in particular in this part of the \ book, but they do have a ubiquity for systems in the plane.", FontFamily->"Courier New"] }], "Text"], Cell[TextData[{ StyleBox["Chapters 11-16 deal with discrete dynamical systems. Chapter 11 \ starts with a general introduction to iteration and linear recurrence (or \ difference) equations. The bulk of the chapter is concerned with the Leslie \ 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.", FontFamily->"Courier New"], " " }], "Text"], Cell[TextData[StyleBox["A brief introduction to the enticing field of neural \ 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. ", FontFamily->"Courier New"]], "Text"], Cell[TextData[StyleBox["Chapter 18 lists examination-type questions; the \ 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.", FontFamily->"Courier New"]], "Text"] }, Open ]] }, FrontEndVersion->"5.1 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 685}}, WindowSize->{820, 640}, WindowMargins->{{58, Automatic}, {-28, Automatic}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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