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Measuring Chaos in a Double Pendulum

Measuring Chaos in a Double Pendulum. Vasha Dutell, Patrick Freeman, Duncan Luiten, and Professor Eric Torrence UO Undergraduate Research Symposium. Motivation & Background. Simple system, chaotic motion Chaos: small Δ initial ➔ large Δ final

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Measuring Chaos in a Double Pendulum

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  1. Measuring Chaos in a Double Pendulum Vasha Dutell, Patrick Freeman, Duncan Luiten, and Professor Eric Torrence UO Undergraduate Research Symposium UO Undergraduate Research Symposium

  2. Motivation & Background • Simple system, chaotic motion • Chaos: small Δinitial➔ large Δfinal • Exponential separation characterized by Lyapunov Exponent UO Undergraduate Research Symposium

  3. Two Modes of Attack Simulation Physical Pendulum • MATLAB generated • Runge-Kutta Method • No Friction • Double-bar pendulum • Released from high angle • Circular dots for tracking • Casio EX-F1 at ~5 feet • 600 fps analyzed in MATLAB UO Undergraduate Research Symposium

  4. Circle Detection & Tracking • Circle Detection – Circular Hough • Accumulation Array • Circle Differentiation • Angles Extracted UO Undergraduate Research Symposium

  5. Phase Space Plots • 4 parameters: θ,Φ, δθ, δΦ • Angle 1 vs. its Angular Velocity • Chaotic ➔ Periodic UO Undergraduate Research Symposium

  6. Lyapunov Exponent • 4 parameters: θ,Φ, δθ, δΦ • |δZ(t)|≈|δZ(t0)|eλt • At least 1 positive to show chaotic behavior • Rosenstein Method • Nearest Neighbors • Temporal Separation UO Undergraduate Research Symposium

  7. Challenges/Problems • Camera • Lighting • Energy Function • Tracking & Interpolation • Circle Sizes • Could Switch Direction

  8. Future Analysis • Fractal Dimension of attractor • Angle 2 vs. its velocity • Use Box-counting method

  9. Special Thanks to • Professor Eric Torrence • Bryan Boggs • Isaac HastingsHauss • Professor Richard Taylor • Alexander Elrich (Simulation) • UO Machine Shop Personnel • Ian Pilgrim (Box Counting Analysis) UO Undergraduate Research Symposium

  10. Questions?

  11. Lyapunov Exponent Calculation

  12. Attractor Plot

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