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An Experimental Approach to Nonlinear Dynamics and Chaos Is a Textb O Ok and A An Exp erimental Approach to Nonlinear Dynamics and Chaos by Nicholas B Tullaro Tyler Abb ott and Jeremiah PReilly INTERNET ADDRESSES nbtreededu c COPYRIGHT TUFILLARO ABBOTT AND REILLY Published by AddisonWesley v b Novemb er i ii iii iv Contents Preface xi Intro duction What is nonlinear dynamics What is in this b o ok Some Terminology Maps Flows and Fractals References and Notes Bouncing Ball Intro duction Mo del Stationary Table Impact Relation for the Oscillating Table The Equations of Motion Phase and Velo city Maps Parameters High Bounce Approximation Qualitative Description of Motions Trapping Region Equilibrium Solutions Sticking Solutions Perio d One Orbits and Perio d Doubling Chaotic Motions Attractors Bifurcation Diagrams References and Notes Problems Quadratic Map Intro duction Iteration and Dierentiation Graphical Metho d Fixed Points v vi CONTENTS Perio dic Orbits Graphical Metho d Perio d One Orbits Perio d Two Orbit Stability Diagram Bifurcation Diagram Lo cal Bifurcation Theory Saddleno de Perio d Doubling Transcritical Perio d Doubling Ad Innitum Sarkovskiis Theorem Sensitive Dep endence Fully Develop ed Chaos Hyp erb olic InvariantSets Symb olic Dynamics Top ological Conjugacy Symb olic Co ordinates Whats in a name Lo cation Alternating Binary Tree Top ological Entropy Usage of Mathematica References and Notes Problems String Intro duction Exp erimental Apparatus SingleMo de Mo del Planar Vibrations Dung Equation Equilibrium States Unforced Phase Plane Extended Phase Space Global Cross Section Resonance and Hysteresis Linear Resonance Nonlinear Resonance Resp onse Curve Hysteresis Basins of Attraction Homo clinic Tangles Nonplanar Motions Free Whirling Resp onse Curve CONTENTS vii Torus Attractor Circle Map Torus Doubling Exp erimental Techniques Exp erimental Cross Section Emb edding Power Sp ectrum Attractor Identication Correlation Dimension References and Notes Problems Dynamical Systems Theory Intro duction Flows and Maps Flows Poincare Map Susp ension of a Map Creed and Quest Asymptotic Behavior and Recurrence InvariantSets Limit Sets and Nonwandering Chain Recurrence Expansions and Contractions DerivativeofaMap Jacobian of a Map Divergence of a Vector Field Dissipative and Conservative Equation of First Variation Fixed Points Stability Linearization Hyp erb olic Fixed Points Saddles Sources and Sinks Invariant Manifolds Center Manifold Theorem Homo clinic and Hetero clinic Points Example Laser Equations Steady States Eigenvalues of a Matrix Eigenvectors Stable Focus Smale Horsesho e From Tangles to Horsesho es viii CONTENTS Horsesho e Map Symb olic Dynamics From Horsesho es to Tangles Hyp erb olicity Lyapunov Characteristic Exp onent References and Notes Problems Knots and Templates Intro duction Perio dic Orbit Extraction Algorithm Lo cal Torsion Example Dung Equation Knot Theory Crossing Convention Reidemeister Moves Invariants and Linking Numb ers Braid Group Framed Braids Relative Rotation Rates Templates Motivation and Geometric Description Algebraic Description Lo cation of Knots Calculation of Relative Rotation Rates Intertwining Matrices Horsesho e Lorenz Dung Template References and Notes Problems App endix A Bouncing Ball Co de App endix B Exact Solutions for a Cubic Oscillator App endix C Ode Overview App endix D Discrete Fourier Transform App endix E Henons Trick App endix F Perio dic Orbit Extraction Co de CONTENTS ix App endix G Relative Rotation Rate Package App endix H Historical Comments App endix I Pro jects x CONTENTS Preface An Experimental Approach to Nonlinear Dynamics and Chaos is a textb o ok and a reference work designed for advanced undergraduate and b eginning graduate stu dents This b o ok provides an elementary intro duction to the basic theoretical and exp erimental to ols necessary to b egin researchinto the nonlinear b ehavior of me chanical electrical optical and other systems A fo cus of the text is the description of several desktop exp eriments such as the nonlinear vibrations of a currentcarrying wire placed b etween the p oles of an electromagnet and the chaotic patterns of a ball b ouncing on a vibrating table Each of these exp eriments is ideally suited for the smallscale environment of an undergraduate science lab oratory In addition the b o ok includes software that simulates several systems describ ed in this text The software provides the student with the opp ortunity to immedi ately explore nonlinear phenomena outside of the lab oratory The feedbackofthe interactive computer simulations enhances the learning pro cess by promoting the formation and testing of exp erimental hyp otheses Taken together the text and asso ciated software provide a handson intro duction to recent theoretical and ex p erimental discoveries in nonlinear dynamics Studies of nonlinear systems are truly interdisciplinary ranging from exp erimen tal analyses of the rhythms of the human heart and brain to attempts at weather prediction Similarly the to ols needed to analyze nonlinear systems are also inter disciplinary and include techniques and metho dologies from all the sciences The to ols presented in the text include those of theoretical and applied mathematics dynamical systems theory and p erturba tion theory theoretical physics development of mo dels for physical phenomena application of physical laws to explain the dynamics and the top ological characterization of chaotic motions exp erimental physics circuit diagrams and desktop exp eriments engineering instabilities in mechanical electrical and optical systems and computer science numerical algorithms in C and symb olic computations with Mathematica xi xii A ma jor goal of this pro ject is to showhowtointegrate to ols from these dierent disciplines when studying nonlinear systems Many sections of this b o ok develop one sp ecic to ol needed in the analysis of a nonlinear system Some of these to ols are mathematical such as the appli cation of symb olic dynamics to nonlinear equations some are exp erimental such as the necessary circuit elements required to construct an exp erimental surface of section and some are computational such as the algorithms needed for calculat ing fractal dimensions from an exp erimental time series We encourage students to try out these to ols on a system or exp erimentoftheirown design To help with this App endix I provides an overview of p ossible pro jects suitable for researchby an advanced undergraduate Some of these pro jects are in acoustics oscillations in gas columns hydro dynamics convective lo opLorenz equations HeleShaw cell surface waves mechanics oscillations of b eams stability of bicycles forced p endulum compass needle in oscillating Beld impactoscillators chaotic art mo biles ball in a swinging track optics semiconductor laser instabilities laser rate equations and other systems showing complex b ehavior in b oth space and time videofeedback ferrohydro dynamics capillary ripples This b o ok can b e used as a primary or reference text for b oth exp erimental and theoretical courses For instance it can b e 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 p erhaps more imp ortantly as a source b o ok for undergraduate research pro jects Finally it could also b e the basis for a new interdisciplinary course in nonlinear dynamics This new course would contain an equal mixture of mathematics physics computing and lab oratory work The primary goal of this new course is to give students the desire skills and condence needed to b egin their own researchinto nonlinear systems Regardless of her eld of study a student pursuing the material in this b o ok should have a rm grounding in Newtonian physics and a course in dierential equations that intro duces the qualitative theory of ordinary dierential equations For the latter chapters a go o d 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
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