Measuring Chaos in a Double Pendulum Vasha Dutell, Patrick Freeman, Duncan Luiten, and Professor...

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Transcript of Measuring Chaos in a Double Pendulum Vasha Dutell, Patrick Freeman, Duncan Luiten, and Professor...

Measuring Chaos in a Double Pendulum

Measuring Chaos in a Double PendulumVasha Dutell, Patrick Freeman, Duncan Luiten, and Professor Eric TorrenceUO Undergraduate Research Symposium UO Undergraduate Research Symposium Motivation & BackgroundSimple system, chaotic motionChaos: small initial large finalExponential separation characterized by Lyapunov Exponent

UO Undergraduate Research Symposium Two Modes of AttackSimulationMATLAB generatedRunge-Kutta MethodNo FrictionPhysical PendulumDouble-bar pendulumReleased from high angleCircular dots for trackingCasio EX-F1 at ~5 feet600 fps analyzed in MATLAB UO Undergraduate Research Symposium Circle Detection & TrackingCircle Detection Circular Hough Accumulation ArrayCircle DifferentiationAngles Extracted

UO Undergraduate Research Symposium Phase Space Plots UO Undergraduate Research Symposium 4 parameters: ,, , Angle 1 vs. its Angular VelocityChaotic Periodic Lyapunov Exponent4 parameters: ,, , |Z(t)||Z(t0)|etAt least 1 positive to show chaotic behavior

Rosenstein MethodNearest NeighborsTemporal Separation

UO Undergraduate Research Symposium Challenges/ProblemsCameraLightingEnergy FunctionTracking & InterpolationCircle SizesCould Switch Direction

Future AnalysisFractal Dimension of attractorAngle 2 vs. its velocityUse Box-counting method

Special Thanks toProfessor Eric TorrenceBryan BoggsIsaac Hastings HaussProfessor Richard TaylorAlexander Elrich (Simulation)UO Machine Shop PersonnelIan Pilgrim (Box Counting Analysis)

UO Undergraduate Research Symposium Questions?Lyapunov Exponent CalculationAttractor Plot

Biochemistry: A Short Course Second Edition Tymoczko Berg Stryer © 2013 W. H. Freeman and Company CHAPTER 30 Amino Acid Degradation and the Urea Cycle
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This book by Linton Freeman provides an excellent history of the
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Homework Homework Assignment #13 Review Section 6.2 Page 389, Exercises: 25 – 33(EOO), 35 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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