An Experimental Approach to Nonlinear Dynamics and Chaos is a textbook and a reference work designed for advanced undergraduate and beginning graduate students. This book provides an elementary introduction to the basic theoretical and experimental tools necessary to begin research into the nonlinear behavior of mechanical, electrical, optical, and other systems. A focus of the text is the description of several desktop experiments, such as the nonlinear vibrations of a current-carrying wire placed between the poles of an electromagnet and the chaotic patterns of a ball bouncing on a vibrating table. Each of these experiments is ideally suited for the small-scale environment of an undergraduate science laboratory.
In addition, the book includes software that simulates several systems described in this text. The software provides the student with the opportunity to immediately explore nonlinear phenomena outside of the laboratory. The feedback of the interactive computer simulations enhances the learning process by promoting the formation and testing of experimental hypotheses. Taken together, the text and associated software provide a hands-on introduction to recent theoretical and experimental discoveries in nonlinear dynamics.
Studies of nonlinear systems are truly interdisciplinary, ranging from experimental analyses of the rhythms of the human heart and brain to attempts at weather prediction. Similarly, the tools needed to analyze nonlinear systems are also interdisciplinary and include techniques and methodologies from all the sciences. The tools presented in the text include those of:
Many sections of this book develop one specific ``tool'' needed in the analysis of a nonlinear system. Some of these tools are mathematical, such as the application of symbolic dynamics to nonlinear equations; some are experimental, such as the necessary circuit elements required to construct an experimental surface of section; and some are computational, such as the algorithms needed for calculating fractal dimensions from an experimental time series. We encourage students to try out these tools on a system or experiment of their own design. To help with this, Appendix I provides an overview of possible projects suitable for research by an advanced undergraduate. Some of these projects are in acoustics (oscillations in gas columns), hydrodynamics (convective loop--Lorenz equations, Hele-Shaw cell, surface waves), mechanics (oscillations of beams, stability of bicycles, forced pendulum, compass needle in oscillating B-field, impact-oscillators, chaotic art mobiles, ball in a swinging track), optics (semiconductor laser instabilities, laser rate equations), and other systems showing complex behavior in both space and time (video-feedback, ferrohydrodynamics, capillary ripples).
This book can be used as a primary or reference text for both experimental and theoretical courses. For instance, it can be used in a junior level mathematics course that covers dynamical systems or as a reference or lab manual for junior and senior level physics labs. In addition, it can serve as a reference manual for demonstrations and, perhaps more importantly, as a source book for undergraduate research projects. Finally, it could also be the basis for a new interdisciplinary course in nonlinear dynamics. This new course would contain an equal mixture of mathematics, physics, computing, and laboratory work. The primary goal of this new course is to give students the desire, skills, and confidence needed to begin their own research into nonlinear systems.
Regardless of her field of study, a student pursuing the material in this book should have a firm grounding in Newtonian physics and a course in differential equations that introduces the qualitative theory of ordinary differential equations. For the latter chapters, a good dose of mathematical maturity is also helpful.
To assist with this new course we are currently designing labs and software, including complementary descriptions of the theory, for the bouncing ball system, the double scroll LRC circuit, and a nonlinear string vibrations apparatus. The bouncing ball package has been completed and consists of a mechanical apparatus (a loudspeaker driven by a function generator and a ball bearing), the Bouncing Ball simulation system for the Macintosh computer, and a lab manual. This package has been used in the Bryn Mawr College Physics Laboratory since 1986.
This text is the first step in our attempt to integrate nonlinear theory with easily accessible experiments and software. It makes use of numerical algorithms, symbolic packages, and simple experiments in showing how to attack and unravel nonlinear problems. Because nonlinear effects are commonly observed in everyday phenomena (avalanches in sandpiles, a dripping faucet, frost on a window pane), they easily capture the imagination and, more importantly, fall within the research capabilities of a young scientist. Many experiments in nonlinear dynamics are individual or small group projects in which it is common for a student to follow an experiment from conception to completion in an academic year.
In our opinion nonlinear dynamics research illustrates the finest aspects of small science. It can be the effort of a few individuals, requiring modest funding, and often deals with ``homemade'' experiments which are intriguing and accessible to students at all levels. We hope that this book helps its readers in making the transition from studying science to doing science.
We thank Neal Abraham, Al Albano, and Paul Melvin for providing detailed comments on an early version of this manuscript. We also thank the Department of Physics at Bryn Mawr College for encouraging and supporting our efforts in this direction over several years. We would also like to thank the text and software reviewers who gave us detailed comments and suggestions. Their corrections and questions guided our revisions and the text and software are better for their scrutiny.
One of the best parts about writing this book is getting the chance to thank all our co-workers in nonlinear dynamics. We thank Neal Abraham, Kit Adams, Al Albano, Greg Alman, Ditza Auerbach, Remo Badii, Richard Bagley, Paul Blanchard, Reggie Brown, Paul Bryant, Gregory Buck, Josefina Casasayas, Lee Casperson, R. Crandall, Predrag Cvitanovic, Josh Degani, Bob Devaney, Andy Dougherty, Bonnie Duncan, Brian Fenny, Neil Gershenfeld, Bob Gilmore, Bob Gioggia, Jerry Gollub, David Griffiths, G. Gunaratne, Dick Hall, Kath Hartnett, Doug Hayden, Gina Luca and Lois Hoffer-Lippi, Phil Holmes, Reto Holzner, Xin-Jun Hou, Tony Hughes, Bob Jantzen, Raymond Kapral, Kelly and Jimmy Kenison-Falkner, Tim Kerwin, Greg King, Eric Kostelich, Pat Langhorne, D. Lathrop, Wentian Li, Barbara Litt, Mark Levi, Pat Locke, Amy Lorentz, Takashi Matsumoto, Bruce McNamara, Tina Mello, Paul Melvin, Gabriel Mindlin, Tim Molteno, Ana Nunes, Oliver O'Reilly, Norman Packard, R. Ramshankar, Peter Rosenthal, Graham Ross, Miguel Rubio, Melora and Roger Samelson, Wes Sandle, Peter Saunders, Frank Selker, John Selker, Bill Sharpf, Tom Shieber, Francesco Simonelli, Lenny Smith, Hernán Solari, Tom Solomon, Vernon Squire, John Stehle, Rich Superfine, M. Tarroja, Mark Taylor, J. R. Tredicce, Hans Troger, Jim Valerio, Don Warrington, Kurt Wiesenfeld, Stephen Wolfram, and Kornelija Zgonc.
We would like to express a special word of thanks to Bob Gilmore, Gabriel Mindlin, Hernán Solari, and the Drexel nonlinear dynamics group for freely sharing and explaining their ideas about the topological characterization of strange sets. We also thank Tina Mello for experimental expertise with the bouncing ball system, Tim Molteno for the design and construction of the string apparatus, and Amy Lorentz for programing expertise with Mathematica.
Nick would like to acknowledge agencies supporting his research in nonlinear dynamics, which have included Sigma Xi, the Fulbright Foundation, Bryn Mawr College, Otago University Research Council, the Beverly Fund, and the National Science Foundation.
Nick would like to offer a special thanks to Mom and Dad for lots of home cooked meals during the writing of this book, and to Mary Ellen and Tracy for encouraging him with his chaotic endeavors from an early age.
Tyler would like to thank his own family and the Duncan and Dill families for their support during the writing of the book.
Jeremy and Tyler would like to thank the exciting teachers in our lives. Jeremy and Tyler dedicate this book to all inspiring teachers.
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