# 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 dierent

disciplines when studying nonlinear systems.

Many sections of this book develop one specic \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 condence 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 dierential

equations that introduces the qualitative theory of ordinary dierential 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 eects 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.