Good overview of nonlinear dynamics

This book is an overview of all of the components of nonlinear dynamics. Nonlinear dynamics is a field of study that enables well-constructed predictive modeling of systems that might be difficult to solve otherwise. Such continuous systems were first widely modeled by ordinary and differential equations, but with the passage of time there are now tools and mathematical models at our disposal that make for a much more concise model of many systems. This workbook tries to touch on all of those mathematical tools. The first six chapters of the book has to do with modeling such complex systems in general, and the rest of the book is a survey of the tools needed to perform complex modeling. The book's format is that of briefly explaining a concept in a few pages, and then presenting a computer program that demonstrates the concept just explained. The explanations are very clear and concise, there are plenty of equations shown, and the accompanying code is well commented. If you want to really drill deeply into any of the concepts then you are going to need some other books. I suggest that for further reading for the mathematically inclined that you pick up "Chaos: An Introduction to Dynamical Systems" by Kathleen Alligood. For scientists that want to see specific problems that can be solved by dynamical systems I suggest the excellent "Nonlinear Dynamics and Chaos: With Applications in Physics, Biology, Chemistry, and Engineering" by Strogatz. The only real complaint I have against this book is that there is uneven coverage of different tools. For example, the author has a great deal to say about neural networks and fuzzy logic, but has very short chapters covering discrete wavelets and cellular automata. More material would have been great, since it is hard to find good books on discrete wavelets and cellular automata in particular. Some readers may also be annoyed that much of the book are code listings of the various demonstration programs. Overall, I would highly recommend it as one of several books that anyone interested in dynamical systems should definitely own. In particular, those individuals interested in the techniques of algorithmic composition of music might find this book a good jumping off point for studying the tools and techniques that make such compositions possible.

Helpful in some sections

The topics covered in this book are all important from the standpoint of applications in physics, engineering, computer science, financial engineering, and computational biology. It is written for the person just getting started in these topics, and the author does a fairly good job of discussing them. Readers should not expect, and they will not get, in-depth discussions on these topics, as this would swell the book to 10 times the size. They will however get preparation for moving on to more advanced and complete treatments. Nonlinear and chaotic maps are considered in chapter 1, with elementary definitions given and six different examples of maps discussed. In discussing the calculation of numerical trajectories of maps, the author deals with the problem of large initial values for the maps and how to implement these in SymbolicC++ and Java. He also shows how to write/read data to a file using C, C++, and JAVA. The exception handling capability of JAVA comes out nicely, but no performance comparison between the three languages for simulating the maps is given Kutta technique in a C++ program, along with the Lotka-Volterra system from mathematical biology. Hamiltonian mechanics is discussed in chapter 4, with the important Henon-Heiles model from astrophysics is discussed and JAVA programs given for studying its behavior using the Poincare section technique. Newcomers to this technique will appreciate seeing it done here explicitly. Integrability of Hamiltonian systems using the Lax representation and Floquet theory are also treated, but only at a very rudimentary level. Dissipation is included in the next chapter, and the author discusses the classification of fixed points according to their stability. Lyapunov exponents are again brought into the picture, and the phenomenon of hyperchaos is discussed. Some bifurcation theory is introduced with an example of the Hopf bifurcation. Chapter 6 studies nonlinear driven systems, with the Duffing oscillator treated, and the author gives a useful program for calculating the autocorrelation function of this system. The controlling of chaos with feedback and non-feedback controls is the subject of the next chapter, mostly in the context of difference maps. Fractals finally get introduced in chapter 8, with iterated function systems defined but proofs of their properties omitted. The author gives programs for calculating various popular fractals, such as the dragon, Sierpinski gasket, Koch curve, the Mandelbrot set, and the Julia set. The main disappointment in this chapter is that the author does not give programs for calculating the Hausdorff dimension or capacity, quantities that are notoriously difficult to get a meaningful computational handle on. The author switches gears in the next chapter and discusses cellular automata, which have recently made a comeback, especially in research on quantum computation. The discussion is too brief however, and does not allow the reader to gain an appreciation of the properties of these important objects. Chapter 10 gives a brief overview of some techniques for solving differential equations, such as the Euler method and the Lie series technique. The latter is not commonly treated in beginning books so its inclusion here is helpful. Symplectic integration is also discussed briefly, but the author does not discuss how to check the integrators using backward integration, which is commonly used in conservative systems modeled written in the book, and the newcomer to the field will get a fairly decent introduction to the subject. The supplied programs serve to illustrate some of the important concepts in neural networks, such as the Hopfield model, the Kohonen network, the perceptron learning algorithm, and the back-propagation algorithm. Chapter 12 is an introduction to genetic algorithms, and I find this one particularly nice also, as it does give a rudimentary introduction to what evolutionary algorithms are all about, and gives some elementary genetic programs that find the maximum of one- and two-dimensional maps. He also discusses simulated annealing, and gives a useful program that allows the reader to see clearly how this technique works. The last chapter covers fuzzy sets and fuzzy logic, which has also taken on importance in recent years, especially in data mining and financial engineering. The programs given to illustrate the concepts are particularly interesting from the standpoint of coding in C++, as the author uses friend functions and operating overloading in some of them. The reader gets a good overview of fuzzy reasoning and fuzzy rule-based systems.

Useful information, poor presentation

The information was useful & approprate to the topic. I'd rank it as an average quality refence but a very poor text book.The text is poorly written. The code is simple and easy to understand, but not very object oriented. There is not enough explanation of the code. The code is not electronically available.The treatment was very mathematical but lacking in explanation & application examples. There were plenty of deffinitions, but not enough examples.