• Chaos: Making a New Science, by J. Gleick. (A journalist's lucid description of the main ideas and contributors to 'chaos theory'.)
• Chance and Chaos, by D. Ruelle. (26 short essays covering topics from physics and mathematics, probability, games, historical remarks, economics, information, algorithmic complexity, intelligence and ... the true meaning of sex, to name a few. Ruelle is one of the top mathematical physicists.)
• Does God Play Dice?, by I. Stewart. (A mathematician's entertaining survey of the historical development of the scientific ideas that lead to 'chaos theory', along with accessible explanations of the mathematics involved.)
• Fractals: Endlessly Repeated Geometric Figures, by H. Lauwerier. (A nice little pocketbook full of interesting examples and clear explanations.)
• What is Random?, by E. Beltrami. (Discussion of what is meant by a 'random' process including generating such processes and relations to information.)

• Exploring Chaos, by B. Davies.
• A First Course in Chaotic Dynamical Systems (second edition), by R. Devaney
• Encounters with Chaos, by Denny Gulick
• A First Course in Discrete Dynamical Systems (second edition), by R.A. Holmgren
• Chaos and Fractals: New Frontiers of Science, by H.-O. Peitgen, H. Jurgens, and D. Saupe. 1000 pages covering fractals, dynamics, Julia sets and the Mandelbrot set, plus discussions of specialized topics such as renormalization, Brownian motion, fractal image compression, etc. Lots of diagrams, pictures and color plates. Mostly discussion, but the relevant mathematical formulae are also presented.
• Invitation to Dynamical Systems, by E.R. Scheinerman
• An Eye for Fractals: A Graphic and Photographic Essay, by M. McGuire. (Beautiful photographs of fractals in nature, as well as clear explanations.)

• Ergodic Problems of Classical Mechanics, by V.I. Arnold and A.A. Avez (4th year - graduate level)
• An Introduction to Dynamical Systems, by D.K. Arrowsmith and C.M. Place (4th year level)
• The Science of Fractal Images, by M.F. Barnsley et. al. (Fractals, random fractals, algorithms, fractal landscapes, Julia and the Mandelbrot set. Most articles can be read as an introduction, but they also contain advanced material.)
• Complex Dynamics, by L. Carleson and T.W. Gamelin (3rd - 4th year level; an introductory book on the dynamics of iterated complex functions (Julia sets, etc), requires some knowledge of complex numbers and complex functions)
• Iterated Maps on the Interval as Dynamical Systems, by P. Collet and J-P Eckmann (graduate level)
• Fractal Geometry: Mathematical foundations and applications, by Kenneth Falconer (3rd - 4th year level. A very complete treatment of fractals from a mathematical perspective. Very readable. Certainly worthy as a textbook for a 'serious' course on fractals and related topics)
• Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, by J. Guckenheimer and P. Holmes (4th year - graduate level)
• Chaos in Classical and Quantum Mechanics, by M.C. Gutzwiller (4th year - graduate level)
• Differential Equations, Dynamical Systems, and Linear Algebra, by M.W. Hirsch and S. Smale (3rd year level)
• Regular and Stochastic Motion, by A.J. Lichtenberg and M.A. Lieberman (4th year - graduate level)
• The Fractal Geometry of Nature, by B. Mandelbrot. (Fractals in mathematics and nature, with applications, history, anecdotes, etc., written by the man who invented the 'fractal'.)
• One-Dimensional Dynamics, by W. de Melo and S. van Strien. (Mathematically complete and rigorous discussion of the dynamics of functions from the interval [0,1] to [0,1], such as the logistic function. Graduate level.)
• Dynamics in One Complex Variable, by J. Milnor (graduate level). This book describes the current state of the art of the mathematical theory of Julia sets of rational complex functions.
• Holomorphic Dynamics, by S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda (graduate level)
• The Beauty of Fractal Images by H.-O. Peitgen and P.H. Richter. Beautifully illustrated with many high quality colour pictures of Julia sets and the Mandelbrot set, along with extensive discussion.
• Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, by Clark Robinson. (4th year to graduate level).
• Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, by S.H. Strogatz (3rd-4th year level)
• Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists", by J.M.T. Thompson and H.B. Stewart (4th year level)
• Fractals Everywhere, by M.F. Barnsley (4th year level)
• One-Dimensional Dynamics, by W. de Melo and S. van Strien (4th year-graduate level).

• Fractal Image Compression:
  • Fractal Image Compression by Y. Fischer, in Appendix A of Chaos and Fractals: New Frontiers of Science H.-O Peitgen, H. Jurgens, D. Saupe, Springer-Verlag, 1992.
  • Fractal Image Compression. Theory and Application, Yuval Fisher, editor. Springer-Verlag, 1995.
  • Waterloo Fractal Compression Project at the University of Waterloo (http://links.uwaterloo.ca).
  • Fractal and Wavelet Image Compression Techniques, Stephen Welstead, SPIE Optical Engineering Press, 1999.
• Multifractal Measures, C.J.G. Evertsz, B.B. Mandelbrot, in Appendix B of Chaos and Fractals: New Frontiers of Science H.-O Peitgen, H. Jurgens, D. Saupe, Springer-Verlag, 1992.
• The Science of Fractal Images, by M.F. Barnsley, R.L. Devaney, B.B. Mandelbrot, H.-O. Peitgen, D. Saupe, R.F. Voss. (Articles on generating fractal landscapes, random fractals, modelling real world images with fractals, etc)
• Fractal Physiology, by J.B. Bassingthwaighte, L.S. Liebovitch, and B.J. West
• Fractals, by Jens Feder (applications to physics)
• The Theory of Evolution and Dynamical Systems, by J. Hofbauer and K. Sigmund
• A Wavelet Tour of Signal Processing, by S. Mallat (see section 6.4 on Multifractal Signals)
• Fractals in Chemistry, by W.G. Rothschild (1998).

• Simple Mathematical Models With Very Complicated Dynamics, by R.M. May. Nature, Vol. 261, June 10, 1976, pp459-467. (One of the first articles to recognize chaotic behavior in simple systems. Very readable.)
• Is the Solar System Stable?, by J. Moser. Mathematical Intelligencer, Vol. 1, No. 65, 1978. This article describes the famous (and still unsolved) problem about the stability of the solar system (i.e., Will the planets continue to move in regular orbits or will one of them some day collide with another and be ejected from the solar system?). This was the motivating problem that led to the development of modern dynamical systems theory (as first laid down by Poincare a hundred years ago). Moser has made many important contributions to the theory of dynamical systems and celestial mechanics.
• White and Brown Music, Fractal Curves and One-Over-f Fluctuations, by Martin Gardner in the Mathematical Games column of Scientific American , April, 1978, pp16-32.
• Strange Attractors, by D. Ruelle. Mathematical Intellingencer , Vol 2, No. 3, 1980, pp126-137
• Fractals and Self Similarity, by J. E. Hutchinson. Indiana University Mathematics Journal, Vol. 30, No. 5 , 1981). This article lays the mathematical foundations of Iterated Function Systems, along with the 'geometric measure theory' of fractals.
• Roads to Chaos, by L.P. Kadanoff. Physics Today, December, 1983, pp46-53
• Special Issue on Fractals, Scientific American, August 1985.
• Where Can One Hope To Profitably Apply The Ideas Of Chaos?, by D. Ruelle. Physics Today, July 1994, pp24-30
• Complex Analytic Dynamics on the Riemann Sphere, by Paul Blanchard, Bulletin of the American Mathematical Society, Vol. 11, Number 1, July 1984, pp 85-141. This review article discusses the state-of-the-art of complex dynamics (Julia sets, Mandelbrot set, etc).
• Similarity Between the Mandelbrot Set and Julia Sets, by Tan Lei, Communications in Mathematical Physics, 134 (1990), pp 587-617.
• Developments in Chaotic Dynamics, by L-S Young. Notices of the American Mathematical Society , November 1998, pp1318-1328.
• Analysis on Fractals, by R. Strichartz. Notices of the American Mathematical Society , November 1999, pp1199-1208.
• The Quadratic Family as a Qualitatively Solvable Model of Chaos, by M. Lyubich. Notices of the American Mathematical Society, October 2000, pp1042-1052.
• Periodicity versus Chaos in One-Dimensional Dynamics, by Hans Thunberg. SIAM Review, Vol. 43, No. 1 (March 2001), pp.3-30. This is a review article about the dynamics of one-dimensional maps f:R -> R such as the logistic function. Topics covered are; types of orbits (periodic vs chaotic), families of maps and their bifurcations, invariant densities, universality.

• Fractals: An Animated Discussion. (Interviews with Mandelbrot and Lorenz, as well as explanations of fractals and chaos, and Mandelbrot and Julia sets.)
• The Beauty and Complexity of the Mandelbrot Set. (A tour and discussion of the Mandelbrot set by one of the mathematicians who have made fundamental discoveries about it; J. Hubbard of Cornell.)
• Chaos, Science and the Unexpected (An episode of David Suzuki's The Nature of Things).
• Chaos, Fractals and Dynamics: Computer Experiments (R. Devaney explains some basic examples).
• Chaos (Documentary).
• Chaos: filmed and narrated by Rudy Rucker.

• The Wikipedia entry on Fractals
• University of Texas at Austin, Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems. Introduction to chaos, fractals, and more.
• University of Maryland Chaos group. Pictures, history, and many references.
• Hypertextbook on Chaos. Book on chaos and fractals, and LOTS of links (see Appendix A.3).
• MathSoft's Feigenbaum web page. A discussion of the bifurcation diagram of the logistic function, plus additional references.
• John Milnor's web page at Stony Brook. His recent and ongoing work on dynamics of one complex variable. (Advanced material.)
• History of Mathematics web page at the University of St Andrews Scotland. (Also has an extensive collection of information on many topics in mathematics.)
• Dmitry Brant's VB programs for drawing Julia sets, the Mandelbrot set, and some chaotic systems. (You don't need to have VB; you can download the executable files. However, you may need to download a VB runtime program too if you get a missing DLL file error.)
• XAOS: Downloadable program for exploring the Mandelbrot sets and their associated Julia sets

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