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Experimental Approach to Nonlinear Dynamics and Chaos

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:
theoretical and applied mathematics (dynamical systems theory and perturbation theory),
theoretical physics (development of models for physical phenomena, application
of physical laws to explain the dynamics, and the topological characterization
of chaotic motions),
experimental physics (circuit diagrams and desktop experiments),
engineering (instabilities in mechanical, electrical, and optical systems), and
computer science (numerical algorithms in C and symbolic computations with
Mathematica).
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A major goal of this project is to show how to integrate tools from these di erent
disciplines when studying nonlinear systems.
Many sections of this book develop one speci c \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- eld, 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 con dence needed to begin their
own research into nonlinear systems.
Regardless of her eld of study, a student pursuing the material in this book
should have a rm grounding in Newtonian physics and a course in di erential
equations that introduces the qualitative theory of ordinary di erential 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 rst 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 e ects 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
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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.

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