Measuring Chaos in a Double Pendulum
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Measuring Chaos in a Double Pendulum
Vasha Dutell, Patrick Freeman, Duncan Luiten, and Professor Eric Torrence
UO Undergraduate Research Symposium
UO Undergraduate Research Symposium
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Motivation & Background
• Simple system, chaotic motion• Chaos: small Δinitial large Δ➔ final
• Exponential separation characterized by Lyapunov Exponent
UO Undergraduate Research Symposium
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Two Modes of Attack
Simulation
• MATLAB generated• Runge-Kutta Method• No Friction
Physical Pendulum
• 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
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Circle Detection & Tracking
• Circle Detection – Circular Hough
• Accumulation Array• Circle Differentiation• Angles Extracted
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Phase Space Plots
UO Undergraduate Research Symposium
• 4 parameters: θ,Φ, δθ, δΦ
• Angle 1 vs. its Angular Velocity
• Chaotic Periodic ➔
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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
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Challenges/Problems• Camera
o Lighting
• Energy Function
• Tracking & Interpolationo Circle Sizeso Could Switch Direction
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Future Analysis
• Fractal Dimension of attractor• Angle 2 vs. its velocity• Use Box-counting method
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Special Thanks to
• Professor Eric Torrence• Bryan Boggs• Isaac Hastings Hauss• Professor Richard Taylor• Alexander Elrich (Simulation)• UO Machine Shop Personnel• Ian Pilgrim (Box Counting Analysis)
UO Undergraduate Research Symposium
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Questions?
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Lyapunov Exponent Calculation
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Attractor Plot