![Page 1: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/1.jpg)
Fractal Basins in a Magnetic Pendulum System
Galen Craven, Chris Martin, Fredrik Ekström, and Jung Byun
![Page 2: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/2.jpg)
“Designed to illustrate the random forces that
effect us all. Can you make order
out of this chaos? If so, you could
probably write a bestseller”
-ThinkGeek.com
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- www.bugman123.com
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Background
• Toy system that has complex dynamics and fractal formation
• Basins of attraction / fractal boundaries
• Interesting Patterns
![Page 5: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/5.jpg)
• Gravitational force
• Magnetic force
• Damping
![Page 6: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/6.jpg)
Final State Sensitivity
• The magnets are stable fixed points
• Simulation of trajectories from almost same initial conditions
• Example videos of a few runs (from almost same initial conditions)
![Page 7: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/7.jpg)
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![Page 9: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/9.jpg)
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Goals
• Recreate fractal using experimental data
• Investigate the dynamics of the system
• Optimize parameters for simulations
![Page 11: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/11.jpg)
Experimental Setup
• Features
– Pendulum
– Free 2D pivot
– Magnet on Rod
– Base magnets
– Correct Polarity “base attracting pendulum”
![Page 12: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/12.jpg)
Measurements to Take
• Initial Conditions
– Measured with Picture of initial state
• Final Position
– Final position of pendulum
– (1,2,3) or (r,g,b)
![Page 13: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/13.jpg)
Things to Control
• Need initial Velocity to be zero
Vo = 0
• Stabilize on aluminum ring
• Better than trying to hold steady
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Special Considerations
• Magnet Strength
• Length of Pendulum
• Mass of Pendulum
• Separation of Magnets
• Orientation of magnets
• Distance of Base magnets from Pendulum
• Friction in pivot joint
![Page 15: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/15.jpg)
Physical Experiment
![Page 16: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/16.jpg)
Data Acquisition
• Static Pictures
• Whiteout on base to help
• Final Position tagged onto photo
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Difficulties
• zero velocity initial condition
• Light intensity variations
![Page 18: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/18.jpg)
Image processing
• Need to find position of white dot
• Tried using a threshold and compute centroid, but did not work
• Marked position manually instead
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![Page 21: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/21.jpg)
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Model
• Force from magnet:
yy
xx
dyyxxAmF
i
i
ii
i 2/3222 )()(
1
H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science
(Springer, 2004)
'
'
y
xRmFd
• Damping force:
y
xCmFg
• Gravitational force:
![Page 23: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/23.jpg)
Model
'
)()(''
')()(
''
2/3222
2/3222
Rydyyxx
yyACyy
Rxdyyxx
xxACxx
ii
i
ii
i
![Page 24: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/24.jpg)
Parameter estimation
![Page 25: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/25.jpg)
Parameter estimation
• Equation:
• Solution:
• Gives R and C
• d and A trickier
CxRxx '''
)sin()exp( ttx
![Page 26: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/26.jpg)
Parameter estimation
• Minimize
• Improved agreement a bit but not much
misses # ),,,( dRCAN
![Page 27: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/27.jpg)
d ≈ 6 mm
Points ≈ 2300
![Page 28: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/28.jpg)
d ≈ 4 mm
Points ≈ 1200
![Page 29: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/29.jpg)
d ≈ 6 mm
![Page 30: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/30.jpg)
d ≈ 4 mm
![Page 31: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/31.jpg)
• C = 25 N/m
• R = 0.41 N/m·s
• A = 0.0003 N/m^2
d = 0.005 m
d = 0.006 m
d = 0.004 m
Simulations Varying d
![Page 32: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/32.jpg)
• C = 25 N/m
• R = 0.41 N/m·s
• d = 0.004 m
A = 0.00038 N/m^2
A = 0.00018 N/m^2 A = 0.00028 N/m^2
Simulations Varying A
![Page 33: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/33.jpg)
• C = 25 N/m
• R = 0.41 N/m·s
• d = 0.005 m
• A = 0.0001 N/m^2
Time Dependence of Final State
(How long it takes to reach final state)
![Page 34: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/34.jpg)
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Conclusions
• Varying Opinions – Opinion 1: Experiment is so flawed that the
measured points must be rejected. Simulation accepted
– Opinion 2: Data is good. Reduction of dimensionality in simulation model is a poor approximation
• Agreement – Chaotic systems are difficult to probe
experimentally
![Page 36: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/36.jpg)
Improvements
• Automate: More points and less variance in initial velocities
• Working parameter optimization algorithm
• Automatic centroid tracking
![Page 37: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/37.jpg)
Contributions
• Fredrick – Experimental Design
– Data Acquisition
– Data Analysis
– Simulation Code
• Chris – Experimental Design
– Data Acquisition
– Data Analysis
• Jung – Experimental Design
– Data Acquisition
– Data Analysis
• Galen – Experimental Design
– Data Acquisition
– Data Analysis
– Simulation Code
![Page 38: Fractal Basins in a Magnetic Pendulum Systemnldlab.gatech.edu/w/images/2/28/2010_MagnetChaos.pdf · H. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science](https://reader033.vdocuments.us/reader033/viewer/2022050307/5f6ff657e2c45a47c762bd2f/html5/thumbnails/38.jpg)
Acknowledgments
• Andrei
• Nick
• Dr. Goldman
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Thank You