RESOURCE LETTER Roger H. Stuewer, Editor School of Physics and Astronomy, 116 Church Street University of Minnesota, Minneapolis, Minnesota 55455 This is one of a series of Resource Letters on different topics intended to guide college physicists, astronomers, and other scientists to some of the literature and other teaching aids that may help improve course content in specified fields. @The letter E after an item indicates elementary level or material of general interest to persons becoming informed in the field. The letter I, for intermediate level, indicates material of somewhat more specialized nature; and the letter A, indicates rather specialized or advanced material.# No Resource letter is meant to be exhaustive and complete; in time there may be more than one letter on some of the main subjects of interest. Comments on these materials as well as suggestions for future topics will be welcomed. Please send such communications to Professor Roger H. Stuewer, Editor, AAPT Resource Letters, School of Physics and Astronomy, 116 Church Street SE, University of Minnesota, Minneapolis, MN 55455. Resource Letter: ND-1: Nonlinear Dynamics Robert C. Hilborn Department of Physics, Amherst College, Amherst, Massachusetts 01002-5000 Nicholas B. Tufillaro Measurement Systems Department, Hewlett–Packard Company, 1501 Page Mill Road, MS4AD, Palo Alto, California 94304-1126 ~Received 10 July 1996; accepted 13 March 1997! This Resource Letter provides an introductory guide to the literature on nonlinear dynamics. Journal articles and books are cited for the following topics: general aspects of nonlinear dynamics and applications of nonlinear dynamics to various fields of physics, other sciences, and a few areas outside the sciences. Software and Internet resources are given also. © 1997 American Association of Physics Teachers. I. INTRODUCTION There are two important themes in nonlinear dynamics. Nonlinear dynamics and its subdiscipline ‘‘chaos theory’’ One emphasizes the temporal behavior of systems, most with have swept over the landscape of science, mathematics, and no significant spatial variation. The other emphasizes spatial engineering in the past two decades. The subject has ben- structures and the formation of spatial patterns. Most begin- efited from some good public relations ~Ref. 75!, but its stay- ning courses in nonlinear dynamics focus on the temporal ing power is rooted in the universality of its methods and behavior of ‘‘low-dimensional’’ systems, that is, those gov- results: Although each nonlinear system may be quite dis- erned by a few degrees of freedom, for which there now tinct in its detailed behavior, most nonlinear systems fall into exists a fairly complete body of theoretical results and ex- broad classes with common qualitative and quantitative fea- perimental techniques ~Sec. IV B and Ref. 90!. Methods of tures. Many of the theoretical and experimental advances in nonlinear time-series analysis ~Sec. V D and Ref. 53!,in nonlinear science, moreover, can be explored with only a which a sequence of values of a single dynamical variable is rudimentary background in mathematics and a pocket calcu- used to determine the qualitative and quantitative measures lator or with simple computer programs ~Refs. 42, 67, and of the temporal dynamical behavior of the entire system, Sec. VII!. Experiments, such as the forced vibrations of a have dominated both the theoretical and experimental meth- simple pendulum ~Ref. 12!, a bouncing ball ~Ref. 27!,ora odologies and are already finding their way into many prac- semiconductor diode circuit ~Sec. VI B!, are straightforward tical applications. and inexpensive. Because nonlinear effects are observed in The extension of these notions to higher-dimensional sys- everyday phenomena—such as a dripping faucet ~Sec. VI I! tems, to problems of ‘‘spatiotemporal’’ chaos, and to the or avalanches in sandpiles ~Sec. VI F!—they capture the question of turbulence is still a subject of active debate and imagination of students at all levels. inquiry ~Sec. VI F!. So too is the question of the correspon- Nonlinear dynamics can appear at several points in the dence between chaos in classical and quantum-mechanical undergraduate or graduate curriculum. It can be included as a systems ~Sec. VI J!. Both issues, though, have motivated special topic in a standard course in classical mechanics some exciting new experimental and theoretical work. ~Ref. 79! or differential equations ~Ref. 48!. Courses focus- The crucial theoretical construct in nonlinear dynamics is ing on nonlinear dynamics at the junior, senior, or beginning phase space ~also called state space!. Each of the ~indepen- graduate level are also becoming common ~Sec. IV B!. Sev- dent! dynamical variables is used as a coordinate to construct eral books aim to introduce nonlinear dynamics to students the state space for the system. For a deterministic system, the who are not necessarily science majors ~Sec. IV A and Ref. future behavior is determined by the current state of the sys- 10!. tem, represented as a point in state space. As a system 822 Am. J. Phys. 65 ~9!, September 1997 © 1997 American Association of Physics Teachers 822 evolves in time, its state-space representation maps out a Europhysics Letters trajectory in that space. Sets of trajectories form a phase Physics Letters A portrait. These phase portraits often have interesting geomet- Physical Review Letters ric properties. For example, chaotic systems can have phase Other journals that often have articles on nonlinear dy- portraits with a fractal geometric structure. namics: The important and distinctive features of nonlinear behav- American Journal of Physics ior are as follows: Computers in Physics European Journal of Physics I Symmetry-breaking, either temporal or spatial. The tem- International Journal of Nonlinear Mechanics poral response of a system need not be the same as that of Journal of Statistical Physics the ‘‘driving force.’’ Even autonomous systems ~those with no explicit time-dependent forcing! can spontane- ously develop complex temporal behavior. The most dra- III. CONFERENCE PROCEEDINGS matic of these broken symmetries is chaotic behavior, Several conferences on nonlinear dynamics are held regu- which is bounded, completely aperiodic behavior. For larly. nonlinear systems with significant spatial variation, the 1. SIAM Conference on Applications of Dynamical Systems ~every other spatial patterns may be independent of the boundary con- year!. ditions. 2. Proceedings of the First Experimental Chaos Conference, edited by I Dramatic changes in behavior, called bifurcations, which S. Vohra et al. ~World Scientific, Singapore, 1992!. occur over extremely small parameter ranges. There are 3. Proceedings of the Second Experimental Chaos Conference, edited several common classes of bifurcation events—including by W. Ditto et al. ~World Scientific, Singapore, 1995!. period doubling, intermittency, and crises—that organize 4. Measures of Spatio-Temporal Dynamics, edited by A. M. Albano, P. E. Rapp, N. B. Abraham, and A. Passamante ~North-Holland, Amster- the transitions between regular ~periodic! behavior and dam, 1996!. chaotic behavior. I Sensitive dependence on initial conditions ~the so-called Butterfly Effect!~Refs. 5 and 9!: Small changes in initial IV. TEXTBOOKS AND EXPOSITIONS conditions may lead to qualitatively and quantitatively dif- ferent long-term behavior. Such sensitivity leads to the A. Popularizations loss of long-term predictability even if the system is com- 5. Chaos, Making a New Science, J. Gleick ~Viking, New York, 1987!. pletely deterministic. A best-seller that chronicles the development of the scientific study of chaos. Includes good biographical sketches of many of the major play- I Universal scaling laws for the transitions between chaotic ers in the development of chaos. ~E! and regular behavior. 6. Does God Play Dice? The Mathematics of Chaos, I. Stewart ~Black- well, New York, 1989!. For the scientifically literate reader. Empha- Much of the analysis of the temporal behavior of nonlinear sizes the mathematical approach to chaos, but does pay some attention systems is carried out using time-series data from a single to experiments. ~E! dynamical variable. The series may be generated by using a 7. Order Out of Chaos, I. Prigogine and I. Stengers ~Bantam, New York, ‘‘stroboscopic’’ technique: Every time the state-space trajec- 1984!. A wide-ranging look at the new paradigm: how order can de- tory intersects a plane in the state space, forming a so-called velop from randomness in nonequilibrium systems. ~E! Poincare´ section, a data point is recorded. From such a se- 8. Newton’s Clock: Chaos in the Solar System, I. Peterson ~Freeman, ries, the topology and dynamics of the entire attractor can be New York, 1993!. An excellent introduction to the history of nonlinear dynamics. ~E! reconstructed. See the references in Sec. V C and the texts in 9. The Essence of Chaos, E. N. Lorenz ~University of Washington Press, Sec. IV B. Seattle, 1993!. The text of popular lectures given by Lorenz, one of the We have selected references that emphasize the breadth pioneers of chaos. ~E! and connectedness of nonlinear dynamics at a level appro- priate for someone new to the field, although we have in- cluded advanced-level citations for historically important pa- B. Introductory texts for science students pers or those that have played a significant role in the The following are listed more or less in order of increasing modern development of nonlinear dynamics. demands