General References

It turns out that an eerie type of chaos can lurk just behind a facade of order - and yet, deep inside the chaos lurks an even eerier type of order - Douglas Hofstadter.

I realize that it is difficult and frustrating when one tries to tackle a new subject, especially one as multifaceted and cross-disciplinary as the field of chaotic dynamics. I know; I was there once - awed by the immensity and complexity of the subject. Heck, I am still there, wide-eyed and all! In any case, I hope the following list of books will help you on your way to being enlightened. Feel free to email me any worthwhile additions and/or corrections at lpoon@chaos.umd.edu.

Help! I try to keep up with the influx of new books, but I can't do it by myself. After several requests for inclusion of more specialized texts, I have hit upon a possible solution. Instead of trying to keep up with all the new chaos books, I welcome submissions from people who feel strongly and positively about any particular book. I would appreciate it if the submission includes a short review that points out the various aspects of the book (eg. good points, bad points, intended audience, etc.) Submissions in the more specialized areas are especially welcome. I can't promise to accept all submissions, but if I do, I will give the proper acknowledgments. Direct your submissions to lpoon@chaos.umd.edu. Thanks!

For your convenience, I have further subdivided the works into several broad categories:

• Good Starting Points
• Mathematically Oriented
• Numerical Considerations
• Collections
• Popular Works
• Specialized Areas

Note: denotes books that I found particularly illuminating.

Alligood, K.T., Sauer, T., and Yorke, J.A. CHAOS: An Introduction to Dynamical Systems, Springer-Verlag, expected in June of 1996.

A very readable introductory text that is designed for students in mathematics and the sciences (undergraduate to beginning graduate level). The book gives a clear and intuitive presentation of the topics and also illustrates the concepts using selected physical experiments in the "Lab Visits" sections. All the important concepts in discrete (iterated maps) and continuous (differential equations) dynamical systems are covered in a careful and detailed manner. Unlike other introductory texts, it does not shy away from presenting deep and important theorems like the Poincaré-Bendixson Theorem, Stable Manifold Theorem, and the Cascade Theorem. In addition to explaining its importance, it delves into the heart of the theorems by looking at the proofs. There is nothing like working through a difficult problem by oneself to get a handle on the concepts, and the "Challenge" sections in the book do just that by letting the reader tackle challenging problems from dynamics. Don't worry, one is guided along the way with many helpful hints. Of course, the importance of chaos would not have been widely appreciated if not for computer simulations, thus the book has "Computer Experiments" sections which guide the reader in exploring the dynamics on computers. Undergraduate Level.

Baker, G. L., and Gollub, J. P., Chaotic Dynamics, Cambridge University Press, 1990.

An introductory text that is designed for undergraduates and science teachers. Using a forced damped pendulum as a model, many basic features of chaotic dynamics are presented though not in-depth. Lower Undergraduate Level.

Devaney, Robert L., Chaos, Fractals and Dynamics:Computer Experiments in Mathematics, Addison-Wesley, 1990.

This is a nice little book whose target audience is high school students, beginning undergraduates, and science teachers. Using many examples , Devaney leads the reader through ideas of iteration and dynamical system. The latter half of the book deals with the connection between chaos and the science of fractals. The readers are encouraged to explore the various facets of chaos and fractals using simple computer programs (many sample source codes are peppered throughout the book). In addition to the book, there is a couple of companion video tapes that are also highly instructive and serve as good introduction to chaos and fractals. The first video, Chaos, Fractals and Dynamics, bears the same name as the book and brings to life many of the examples mentioned in the book in a manner that is not possible via the static book format. The visual highlight occurs when Devaney shows beautiful animations of the Julia set. If this doesn't get the students excited, I don't know what will! The second video, Transition to Chaos, describes the period-doubling approach to chaos using the now standard quadratic map. There is a nice discussion about the relationship between the period-doubling bifurcation and the Mandelbrot and Julia sets. The book and the tapes are offered by Science TV as a set, more precisely, the Chaos Set. High School and Lower Undergraduate Level.

Gulick, Denny, Encounters with chaos, McGraw-Hill, 1992.

A very readable introduction for students with moderate calculus background. Lower Undergraduate Level.

Gutzwiller, Martin. C., Chaos in Classical and Quantum Mechanics, Springer-Verlag, 1990.

This is a very nice textbook for people interested in the connection between classical mechanics and quantum mechanics. Students of quantum chaos should definitely have a copy of this on their bookshelf! Graduate Level.

Hilborn, Robert C., Chaos and Nonlinear Dynamics, Oxford University Press, 1994.

This is a nice hefty chaos that text does a good job of giving readers physical insights underlying many of the phenonmena and concepts. Another nice feature of the book is the annotated bibliography at the end of each chapter which serves as a nice starting point for further exploration into the world of chaos. And finally, a book that presents a detailed sketch of how Lorenz obtained the Lorenz equations! (Appendix C) Upper Undergraduate Level

Jackson, E. Atlee, Perspectives of Nonlinear Dynamics, Vols. 1 2, Cambridge University Press, 1989.

An ambitious text that is chock-full of topics and diagrams. Its breadth of topics is a drawback when one considers using it as an introductory text since the book is very loosely organized in terms of logical structure. However, the two-volume set makes excellent supplementary texts. In addition, it has a very extensive bibliography section that is also referenced by topics. Graduate Level.

Kaplan, Daniel, and Glass, Leon Understanding Nonlinear Dynamics, Springer-Verlag, 1995.

This is an introductory text that grew out of an undergraduate biology course taught by the authors. Thus, some of the topics covered and many of the illustrative examples of physical experiments are slanted towards the biological sciences. This does not mean that one needs to be a biology student to appreciate the book. In fact, I think the biological examples cited in the text make us appreciate a little more how nonlinear dynamics can play a major role in our lives. There is also a chapter on time-series analysis which might be of interest to people analyzing data. Undergraduate Level.

Lichtenberg, A. J., and Lieberman, M. A., Regular and Chaotic Dynamics, John Wiley & Sons, 1992.

Another classic text that is especially in-depth in the treatment of Hamiltonian dynamics. Graduate Level.

Moon, Francis C., Chaotic and Fractal Dynamics, John Wiley & Sons, 1992.

A book written by an applied scientist and engineer for (curiously enough) applied scientists and engineers. This is one of only a handful of books that actually deals with how an applied scientist goes about identifying and classifying chaos in physical systems. True to its aim, the book is filled with many illustrative examples of chaos in various diverse physical systems including an appendix on table-top experiments with "chaotic toys". While it does not go in-depth into some of the topics as one would like, its breadth and scope more than make up for it. Upper Undergraduate and beginning Graduate Level.

Ott, Edward,  Chaos in Dynamical Systems, Cambridge University Press, 1993.

An excellent text that is written in a very understandable and careful style. It gives the readers a good grasp of the fundamentals by emphasizing main ideas instead of harping on technical definitions. The bibliography at the end of the book is also a good source for readers who want to delve further into the technical literature. Graduate Level.

Peitgen, Heinz-Otto, Jurgens, Hartmut, and Saupe, Dietmar. Chaos and fractals : new frontiers of science, Springer-Verlag, 1992.

The authors are the same people who gave us the nice coffee-table book The Beauty of Fractals. This book gives a very solid and clear introduction on two topics that has captured the imagination of the public. It has a particularly nice discussion about the logistic map with some interesting details that I don't think are found in other references. The computer excercises at the end of each chapter are especially illuminating. Undergraduate Level.

Schuster, Heinz. G., Deterministic Chaos, 2nd ed., VHC Publishers, 1988.

This book is geared towards the physics audience. While it does cover a wide range of topics, it has tendency to be a little too concise. This makes some of the arguments difficult to follow. Graduate Level.

Strogatz, Steven H., Nonlinear dynamics and Chaos, Addison-Wesley, 1994.

A very good undergraduate text that does a nice job of explaining all the relevant concepts and ideas in the qualitative theory of Ordinary Differential Equations (ODE) like bifurcations and phase plane analysis. In addition, the book is peppered with many interesting applications and models. Check out the amorous ODE model (pg. 138) on how the romance between Romeo and Juliet changes as one varies several "love" parameters. Undergraduate Level.

Tsonis, A. A., Chaos: from Theory to Applications, Plenum Press, 1992. Reviewed by Dimitris Kugiumtzis.

It gets pretty old now but at that time (and still now I believe) it was maybe the only book focusing on the analysis of chaotic time series. It is in fact a very simple and readable introductory book to this subject. I use part of it as textbook in a graduate course on analysis of chaotic time series here at the Dept of Informatics. The book begins with basic mathematical and physical background knowledge on dynamical systems and fractals (part I), then presents shortly main parts of the theory of dynamical systems (part II), and then describes the methodology (up to 1992) related to the analysis of chaotic time series, which is the main subject (and merit) of the book (part III). Especially for this last part, most of the work on this topic up to this date is cited or presented shortly

Arrowsmith, D. K., and Place, C. M. An Introduction to Dynamical Systems, Cambridge University Press, 1990.

A solid introduction to the mathematical aspects of dynamical systems. Graduate Level.

Devaney, Robert L., An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley, 1989.

A good introductory mathematical text at the undergraduate level. It deals with the theory of discrete dynamical systems and include such important topics as structural stability, homoclinic points, bifurcation theory, and the study of the Julia set. Since this is a mathematics text, many proofs are worked out. Undergraduate Level.

Hale, Jack, and Kocak, H. Dynamics and Bifurcations, Springer-Verlag, 1991.

This is a very readable and careful introduction on the theory of dynamical systems and bifurcations. Many illuminating examples are presented to clarify the concepts. Upper Undergraduate Level.

Guckenheimer, John, and Holmes, Philip, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, 1983.

While it doesn't have the jazziest of titles, it is still considered one of the classic texts in the field. You will repeatedly come back to this book to get your mathematical facts straight! Graduate Level.

Wiggins, Stephen, Global bifurcations and chaos : analytical methods, Springer-Verlag, 1988.

Not an easy read by any means, but tons of useful information Graduate Level.

Nusse, H. E., and Yorke, J. A., Dynamics: Numerical Explorations, Springer-Verlag, 1994.

A hands on approach to learning the concepts and the various aspects in computing relevant quantities in chaos. It includes a computer disk with the program "Dynamics" which is compatible with IBM PCs and unix workstations running X-windows. A large variety of chaotic systems, including maps and flows, are included in the program, and it is fairly straightforward to incorporate additional systems into the program (assuming you are comfortable with C). "Dynamics" provides very comprehensive analysis tools that are always available for all the systems: Lyapunov exponents, dimension calculations, periodic orbit finders, stable and unstable manifolds, bifurcation diagrams, basin analysis, straddle trajectories, etc.

Parker, Thomas S., and Chua, Leon, Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, 1989.

This book explains many of the concepts and techniques that are useful for doing numerical simulations of chaotic systems. In many cases, the suggested algorithms are presented explicitly using pseudo-codes. Particularly good are the discussions about integrating ODEs.

Edited by Hao, Bai-Lin, Chaos II, World Scientific, 1990.

A very comprehensive set of reprints. Highly recommended for sources of original literature.

Edited by Cvitanovic, Predrag, Universality in Chaos, 2nd ed., Adam Hilger, 1989.

Another good collection of reprints. There are many articles that are common to Hao's and Cvitanovic's books, so you need to make a choice as to which to sit on your shelf.

Edited by MacKay, R. S. and J. D. Meiss, Hamiltonian Dynamical System, Adam Hilger, 1987.

This set of reprints can be considered an extension of Cvitanovic's compilations; sort of a "Volume 2" dealing with Hamiltonian systems.

Edited by Kim, Jong Hyun, and Stringer, John, Applied Chaos, John Wiley & Sons, 1992.

This is a nice collection of articles about applying the ideas of chaos to real physical systems. Topics covered range from applications in engineering and physical sciences to physiology. There is also an appendix which is a transcript of the discussions about many of papers in the book by the participants of the meeting.

Edited by Ott, E., Sauer T., and Yorke, J. A., Coping with Chaos, John Wiley & Sons, 1994.

A more contemporary set of reprints. Very good in practical applications of chaos.

Edited by Weigend, A., and Gershenfeld, N.A., Time Series Prediction: Forecasting the Future and Understanding the Past, Addison-Wesley, 1994.

These proceedings grew out of a "competition" to analyze various data sets that was held at the Sante Fe Institue in 1992. The purpose was to compare and contrast the various methods of time series analysis which would hopefully lead to new understanding and insights about studying nonlinear time series. This is an excellent collection of articles that surveys the latest techniques in this growing field.

Edited by S. Vohra, M. Spano, M. Shlesinger, L. Pecora and W. Ditto, Proceedings of the First Experimental Chaos Conference, World Scientific, 1992.

Edited by W. Ditto, L. Pecora, M. Shlesinger, M. Spano and S. Vohra, Proceedings of the Second Experimental Chaos Conference, World Scientific, 1995.

Edited by R. Harrison, W. Liu, L. Pecora, M. Spano and S. Vohra, Proceedings of the Third Experimental Chaos Conference, World Scientific, (in press).

These proceedings contain a broad range of papers reporting on real experimental results and applications of chaotic and other nonlinear phenomena. Examples of fields covered: electronics, condensed matter, biology, fluid dynamics, spatio-temporal systems, meteorology, communications, control systems, optics, and chemistry.

Abraham, R. H., and Shaw, C. D., Dynamics: the geometry of behavior, Addison-Wesley, 1992.

A visual mathematics book that uses no formulas and only colorful pictures to explain concepts in nonlinear dynamics. Many of the pictures do a nice job of clarifying the concepts in dynamical systems. However, it is probably more helpful as a supplementary text after the reader has read one of the introductory texts mentioned above.

Field, Michael, and Golubitsky, Martin, Symmetry in Chaos, Oxford University Press, 1992.

A visually delightful book!! Not only is this book filled with eye-catching colorful pictures, it also does a wonderful job of explaining how they were generated. They also draw many interesting parallels between the computer-generated pictures and the structures that occur in nature or are man-made. Check out the Symmetric Chaos site!

Gleick, James, Chaos : Making a New Science, Viking, 1987.

A book that catapulted Chaos into the public's eyes. Written by a journalist, thus it tends to be a little melodramatic, but it does give a flavor of the nature of chaos. It also gives interesting sketches of many of the key personalities in the development of chaos. Especially interesting and relevant is the part where it describes how James Yorke "had discovered Lorenz and given the science of chaos its name" (pg. 65).

Lorenz, Edward N, The Essence of Chaos, University of Washington Press, 1993.

This book grew out of a set of lectures that Lorenz gave to a general audience at the University of Washington, so the math is kept to a minimal (actually, no formulas appear until Appendix 2). It's always nice to look at the science of chaos from the perspective of one of the pioneers in the field. Also included is the text of a talk that he gave in Washington, D.C. in 1972 which gave rise to the popular term, "the Butterfly Effect".

Stewart, Ian, Does God Play Dice? The Mathematics of Chaos, B. Blackwell, 1989.

Another popular science book that tries to give the public a flavor of chaos. This one assumes a more mathematically literate lay reader.

Haake, Fritz, Quantum signatures of chaos, Springer-Verlag, 1991. Reviewed by A. Ortiz-Tapia.

How does the transition occur from quantum mechanical properties to classical properties showing deterministic chaos? ``Quantum signatures of Chaos'' includes quantum aspects of nonlinear dynamics, antiunitary symmetries (generalized time reversal) and the quantum mechanics of dissipative systems. There is a set of problems at the end of each chapter. Quantum observables can display effectively irreversible behaviour when they are coupled to an appropiate environmental system containing many degrees of freedom. Even in closed quantum systems with relatively few degrees of freedom, behaviour resembling damping is possible, provided the system displays chaotic motion in the classical limit.

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