Group 2 - Revision history

Cjcrowley at 13:59, 12 January 2012

2012-01-12T13:59:34Z

← Older revision		Revision as of 09:59, 12 January 2012
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	''Group Members: Chris Cordell, Andrew Hardin, Andrew Masse''		''Group Members: Chris Cordell, Andrew Hardin, Andrew Masse''

			[[Media: Plinko_Compressed.pdf\| Presentation]]. '''Papers:''' [[Media: Mass_Andrew.pdf \| Andrew]], [[Media: Hardin_Charles_-_Group_2_-_6268_Final_Paper.pdf \| Andrew]], [[Media: Cordell_Group2_Phys6268.pdf \| Chris]]

	Here we investigate the dynamics of a particle traveling through a periodic lattice of attractor/repellers. This problem is inspired from a pedagogical example of a chaotic system first described by Ed Lorenz, that of a free skier descending a hill with moguls (periodic bumps).		Here we investigate the dynamics of a particle traveling through a periodic lattice of attractor/repellers. This problem is inspired from a pedagogical example of a chaotic system first described by Ed Lorenz, that of a free skier descending a hill with moguls (periodic bumps).

Cjcrowley at 16:13, 11 January 2012

2012-01-11T16:13:07Z

← Older revision		Revision as of 12:13, 11 January 2012
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			'''Plinko dynamics'''

	''Group Members: Chris Cordell, Andrew Hardin, Andrew Masse''		''Group Members: Chris Cordell, Andrew Hardin, Andrew Masse''

			Here we investigate the dynamics of a particle traveling through a periodic lattice of attractor/repellers. This problem is inspired from a pedagogical example of a chaotic system first described by Ed Lorenz, that of a free skier descending a hill with moguls (periodic bumps).

	[[File:Plinkoseason37.jpeg \| thumb \| 300px \| Plinko game on The Price is Right ]]		[[File:Plinkoseason37.jpeg \| thumb \| 300px \| Plinko game on The Price is Right ]]

Cjcrowley: Protected "Group 2" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

2012-01-09T16:51:17Z

Protected "Group 2" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

← Older revision	Revision as of 12:51, 9 January 2012
(No difference)

Group2: /* Longer-Term Behavior */

2011-12-16T04:43:57Z

Longer-Term Behavior

← Older revision		Revision as of 00:43, 16 December 2011
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	===Longer-Term Behavior===		===Longer-Term Behavior===
	The transient region results, while interesting, only represent behavior of the system in keeping with the experiment. As the system evolves, the dynamics settle onto the final attractor and become more interesting, as shown in the image to the right. After 30 rows, a small group of similar starting positions shows that the runs remain on a remarkably similar diagonal trajectory, until they turn and begin a vertical descent. It is at this point that the attractor begins to pull these trajectories to different parts of the phase space, thus showing the sensitivity to initial conditions~~. This behavior is qualitatively similar to behavior demonstrated by the Lorenz normal-force ski slope seen at the right~~.		The transient region results, while interesting, only represent behavior of the system in keeping with the experiment. As the system evolves, the dynamics settle onto the final attractor and become more interesting, as shown in the image to the right. After 30 rows, a small group of similar starting positions shows that the runs remain on a remarkably similar diagonal trajectory, until they turn and begin a vertical descent. It is at this point that the attractor begins to pull these trajectories to different parts of the phase space, thus showing the sensitivity to initial conditions.

	~~[[File:Lorenztraj.png \| thumb \| 200px \| Long-term evolution of Lorenz ski slope trajectories [6]]]~~
	[[File:Subset_long_term.png \| thumb \| 200px \| Long-term evolution of magnetic ski slope trajectories]]		[[File:Subset_long_term.png \| thumb \| 200px \| Long-term evolution of magnetic ski slope trajectories]]

	In constructing the attractor that forms in the long-term evolution of the system, we hoped to see similar behavior to that of the long-term Lorenz attractor. This clear spiral was not found. Instead, the phase space is just showing the beginnings of a collapse into ordered chaos. A longer runtime would have, yet again, provided more insight. There are, however, a few notable features of the attractor. The diagonal and vertical attracting trajectories can be easily seen beginning to form globally. This clearly ordered system that is beginning to form demonstrates the oscillatory nature of nearby trajectory groups to repeatedly go between vertical and diagonal lines (the paths of least resistance in the Lorenz ski slope and the paths of highest magnetic attraction in the magnetic ski slope) and eventually diverging.		In constructing the attractor that forms in the long-term evolution of the system, we hoped to see similar behavior to that of the long-term Lorenz attractor (shown above). This clear spiral was not found. Instead, the phase space is just showing the beginnings of a collapse into ordered chaos. A longer runtime would have, yet again, provided more insight. There are, however, a few notable features of the attractor. The diagonal and vertical attracting trajectories can be easily seen beginning to form globally. This clearly ordered system that is beginning to form demonstrates the oscillatory nature of nearby trajectory groups to repeatedly go between vertical and diagonal lines (the paths of least resistance in the Lorenz ski slope and the paths of highest magnetic attraction in the magnetic ski slope) and eventually diverging.

	[[File:Long_term_row_30.png \| thumb \| 300px \| Long-term evolution of the attractor shows a contraction and the formation of clear structures]]		[[File:Long_term_row_30.png \| thumb \| 300px \| Long-term evolution of the attractor shows a contraction and the formation of clear structures]]

Group2: /* Longer-Term Behavior */

2011-12-16T04:42:37Z

Longer-Term Behavior

← Older revision		Revision as of 00:42, 16 December 2011
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	The transient region results, while interesting, only represent behavior of the system in keeping with the experiment. As the system evolves, the dynamics settle onto the final attractor and become more interesting, as shown in the image to the right. After 30 rows, a small group of similar starting positions shows that the runs remain on a remarkably similar diagonal trajectory, until they turn and begin a vertical descent. It is at this point that the attractor begins to pull these trajectories to different parts of the phase space, thus showing the sensitivity to initial conditions. This behavior is qualitatively similar to behavior demonstrated by the Lorenz normal-force ski slope seen at the right.		The transient region results, while interesting, only represent behavior of the system in keeping with the experiment. As the system evolves, the dynamics settle onto the final attractor and become more interesting, as shown in the image to the right. After 30 rows, a small group of similar starting positions shows that the runs remain on a remarkably similar diagonal trajectory, until they turn and begin a vertical descent. It is at this point that the attractor begins to pull these trajectories to different parts of the phase space, thus showing the sensitivity to initial conditions. This behavior is qualitatively similar to behavior demonstrated by the Lorenz normal-force ski slope seen at the right.

	[[File:Lorenztraj.png \| thumb \| ~~300px~~ \| ~~captions!~~]]		[[File:Lorenztraj.png \| thumb \| 200px \| Long-term evolution of Lorenz ski slope trajectories [6]]]
	[[File:Subset_long_term.png \| thumb \| ~~300px~~ \| ~~captions!~~]]		[[File:Subset_long_term.png \| thumb \| 200px \| Long-term evolution of magnetic ski slope trajectories]]

	In constructing the attractor that forms in the long-term evolution of the system, we hoped to see similar behavior to that of the long-term Lorenz attractor. This clear spiral was not found. Instead, the phase space is just showing the beginnings of a collapse into ordered chaos. A longer runtime would have, yet again, provided more insight. There are, however, a few notable features of the attractor. The diagonal and vertical attracting trajectories can be easily seen beginning to form globally. This clearly ordered system that is beginning to form demonstrates the oscillatory nature of nearby trajectory groups to repeatedly go between vertical and diagonal lines (the paths of least resistance in the Lorenz ski slope and the paths of highest magnetic attraction in the magnetic ski slope) and eventually diverging.		In constructing the attractor that forms in the long-term evolution of the system, we hoped to see similar behavior to that of the long-term Lorenz attractor. This clear spiral was not found. Instead, the phase space is just showing the beginnings of a collapse into ordered chaos. A longer runtime would have, yet again, provided more insight. There are, however, a few notable features of the attractor. The diagonal and vertical attracting trajectories can be easily seen beginning to form globally. This clearly ordered system that is beginning to form demonstrates the oscillatory nature of nearby trajectory groups to repeatedly go between vertical and diagonal lines (the paths of least resistance in the Lorenz ski slope and the paths of highest magnetic attraction in the magnetic ski slope) and eventually diverging.

			[[File:Long_term_row_30.png \| thumb \| 300px \| Long-term evolution of the attractor shows a contraction and the formation of clear structures]]

	The dissimilarities between the Lorenz system and the magnetic system are likely due to three main contributions. The first is obvious: the Lorenz system is a passive force system whereas our magnetic system produces active attracting force on the puck at all times and from all pegs. That said, the forces of pegs not near the puck are nearly zero (thus why the finite span of magnets is an acceptable configuration and also likely the cause of the 1:1 trajectory behavior) and aid in keeping the overall nature of our system similar to the Lorenz system. The second significant difference is the set of simplifying assumptions. In the Lorenz study, the downhill speed of the puck is fixed; in this system, it is not fixed and depends on gravity and magnetic attraction which speeds and slows the puck. The Lorenz system is therefore semi-conservative as energy lost or gained from the slope is matched by the puck speeding up or slowing. The final difference is that the Lorenz system was integrated over a much longer slope, allowing the system to fully evolve. Due to computational and schedule limitations, this was not possible for our system.		The dissimilarities between the Lorenz system and the magnetic system are likely due to three main contributions. The first is obvious: the Lorenz system is a passive force system whereas our magnetic system produces active attracting force on the puck at all times and from all pegs. That said, the forces of pegs not near the puck are nearly zero (thus why the finite span of magnets is an acceptable configuration and also likely the cause of the 1:1 trajectory behavior) and aid in keeping the overall nature of our system similar to the Lorenz system. The second significant difference is the set of simplifying assumptions. In the Lorenz study, the downhill speed of the puck is fixed; in this system, it is not fixed and depends on gravity and magnetic attraction which speeds and slows the puck. The Lorenz system is therefore semi-conservative as energy lost or gained from the slope is matched by the puck speeding up or slowing. The final difference is that the Lorenz system was integrated over a much longer slope, allowing the system to fully evolve. Due to computational and schedule limitations, this was not possible for our system.

Group2: /* Attractor Structure */

2011-12-16T04:39:57Z

Attractor Structure

← Older revision		Revision as of 00:39, 16 December 2011
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	The transient region results, while interesting, only represent behavior of the system in keeping with the experiment. As the system evolves, the dynamics settle onto the final attractor and become more interesting, as shown in the image to the right. After 30 rows, a small group of similar starting positions shows that the runs remain on a remarkably similar diagonal trajectory, until they turn and begin a vertical descent. It is at this point that the attractor begins to pull these trajectories to different parts of the phase space, thus showing the sensitivity to initial conditions. This behavior is qualitatively similar to behavior demonstrated by the Lorenz normal-force ski slope seen at the right.		The transient region results, while interesting, only represent behavior of the system in keeping with the experiment. As the system evolves, the dynamics settle onto the final attractor and become more interesting, as shown in the image to the right. After 30 rows, a small group of similar starting positions shows that the runs remain on a remarkably similar diagonal trajectory, until they turn and begin a vertical descent. It is at this point that the attractor begins to pull these trajectories to different parts of the phase space, thus showing the sensitivity to initial conditions. This behavior is qualitatively similar to behavior demonstrated by the Lorenz normal-force ski slope seen at the right.

	[[Lorenztraj.png \| thumb \| 300px \| captions!]]		[[File:Lorenztraj.png \| thumb \| 300px \| captions!]]
	[[~~Subset long term~~.png \| thumb \| 300px \| captions!]]		[[File:Subset_long_term.png \| thumb \| 300px \| captions!]]

	In constructing the attractor that forms in the long-term evolution of the system, we hoped to see similar behavior to that of the long-term Lorenz attractor. This clear spiral was not found. Instead, the phase space is just showing the beginnings of a collapse into ordered chaos. A longer runtime would have, yet again, provided more insight. There are, however, a few notable features of the attractor. The diagonal and vertical attracting trajectories can be easily seen beginning to form globally. This clearly ordered system that is beginning to form demonstrates the oscillatory nature of nearby trajectory groups to repeatedly go between vertical and diagonal lines (the paths of least resistance in the Lorenz ski slope and the paths of highest magnetic attraction in the magnetic ski slope) and eventually diverging.		In constructing the attractor that forms in the long-term evolution of the system, we hoped to see similar behavior to that of the long-term Lorenz attractor. This clear spiral was not found. Instead, the phase space is just showing the beginnings of a collapse into ordered chaos. A longer runtime would have, yet again, provided more insight. There are, however, a few notable features of the attractor. The diagonal and vertical attracting trajectories can be easily seen beginning to form globally. This clearly ordered system that is beginning to form demonstrates the oscillatory nature of nearby trajectory groups to repeatedly go between vertical and diagonal lines (the paths of least resistance in the Lorenz ski slope and the paths of highest magnetic attraction in the magnetic ski slope) and eventually diverging.

Group2: /* Longer-Term Behavior */

2011-12-16T04:39:16Z

Longer-Term Behavior

← Older revision		Revision as of 00:39, 16 December 2011
Line 125:		Line 125:
	===Longer-Term Behavior===		===Longer-Term Behavior===
	The transient region results, while interesting, only represent behavior of the system in keeping with the experiment. As the system evolves, the dynamics settle onto the final attractor and become more interesting, as shown in the image to the right. After 30 rows, a small group of similar starting positions shows that the runs remain on a remarkably similar diagonal trajectory, until they turn and begin a vertical descent. It is at this point that the attractor begins to pull these trajectories to different parts of the phase space, thus showing the sensitivity to initial conditions. This behavior is qualitatively similar to behavior demonstrated by the Lorenz normal-force ski slope seen at the right.		The transient region results, while interesting, only represent behavior of the system in keeping with the experiment. As the system evolves, the dynamics settle onto the final attractor and become more interesting, as shown in the image to the right. After 30 rows, a small group of similar starting positions shows that the runs remain on a remarkably similar diagonal trajectory, until they turn and begin a vertical descent. It is at this point that the attractor begins to pull these trajectories to different parts of the phase space, thus showing the sensitivity to initial conditions. This behavior is qualitatively similar to behavior demonstrated by the Lorenz normal-force ski slope seen at the right.

			[[Lorenztraj.png \| thumb \| 300px \| captions!]]
			[[Subset long term.png \| thumb \| 300px \| captions!]]

	In constructing the attractor that forms in the long-term evolution of the system, we hoped to see similar behavior to that of the long-term Lorenz attractor. This clear spiral was not found. Instead, the phase space is just showing the beginnings of a collapse into ordered chaos. A longer runtime would have, yet again, provided more insight. There are, however, a few notable features of the attractor. The diagonal and vertical attracting trajectories can be easily seen beginning to form globally. This clearly ordered system that is beginning to form demonstrates the oscillatory nature of nearby trajectory groups to repeatedly go between vertical and diagonal lines (the paths of least resistance in the Lorenz ski slope and the paths of highest magnetic attraction in the magnetic ski slope) and eventually diverging.		In constructing the attractor that forms in the long-term evolution of the system, we hoped to see similar behavior to that of the long-term Lorenz attractor. This clear spiral was not found. Instead, the phase space is just showing the beginnings of a collapse into ordered chaos. A longer runtime would have, yet again, provided more insight. There are, however, a few notable features of the attractor. The diagonal and vertical attracting trajectories can be easily seen beginning to form globally. This clearly ordered system that is beginning to form demonstrates the oscillatory nature of nearby trajectory groups to repeatedly go between vertical and diagonal lines (the paths of least resistance in the Lorenz ski slope and the paths of highest magnetic attraction in the magnetic ski slope) and eventually diverging.

Group2: /* Attractor Structure */

2011-12-16T04:36:42Z

Attractor Structure

← Older revision		Revision as of 00:36, 16 December 2011
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	When looking at the simulation attractor structure as the puck passes row 3, it appears to be qualitatively similar to that seen in the Lorenz simulation. There is a certain folding structure visible as each row of magnets is passed, which evolves in a similar manner to the Lorenz image shown previously. This is not an exact comparison since the underlying physics being modeled are different between the two simulations, but the similarities show that the problems represent similar ski-slope dynamical systems. One key difference is that Lorenz notes that a sufficient downslope distance is required to establish the attractor structure, which is probably not being achieved in the magnetic Plinko simulation. A longer magnet sheet would provide information in this regard.		When looking at the simulation attractor structure as the puck passes row 3, it appears to be qualitatively similar to that seen in the Lorenz simulation. There is a certain folding structure visible as each row of magnets is passed, which evolves in a similar manner to the Lorenz image shown previously. This is not an exact comparison since the underlying physics being modeled are different between the two simulations, but the similarities show that the problems represent similar ski-slope dynamical systems. One key difference is that Lorenz notes that a sufficient downslope distance is required to establish the attractor structure, which is probably not being achieved in the magnetic Plinko simulation. A longer magnet sheet would provide information in this regard.

			===Longer-Term Behavior===
			The transient region results, while interesting, only represent behavior of the system in keeping with the experiment. As the system evolves, the dynamics settle onto the final attractor and become more interesting, as shown in the image to the right. After 30 rows, a small group of similar starting positions shows that the runs remain on a remarkably similar diagonal trajectory, until they turn and begin a vertical descent. It is at this point that the attractor begins to pull these trajectories to different parts of the phase space, thus showing the sensitivity to initial conditions. This behavior is qualitatively similar to behavior demonstrated by the Lorenz normal-force ski slope seen at the right.

			In constructing the attractor that forms in the long-term evolution of the system, we hoped to see similar behavior to that of the long-term Lorenz attractor. This clear spiral was not found. Instead, the phase space is just showing the beginnings of a collapse into ordered chaos. A longer runtime would have, yet again, provided more insight. There are, however, a few notable features of the attractor. The diagonal and vertical attracting trajectories can be easily seen beginning to form globally. This clearly ordered system that is beginning to form demonstrates the oscillatory nature of nearby trajectory groups to repeatedly go between vertical and diagonal lines (the paths of least resistance in the Lorenz ski slope and the paths of highest magnetic attraction in the magnetic ski slope) and eventually diverging.

			The dissimilarities between the Lorenz system and the magnetic system are likely due to three main contributions. The first is obvious: the Lorenz system is a passive force system whereas our magnetic system produces active attracting force on the puck at all times and from all pegs. That said, the forces of pegs not near the puck are nearly zero (thus why the finite span of magnets is an acceptable configuration and also likely the cause of the 1:1 trajectory behavior) and aid in keeping the overall nature of our system similar to the Lorenz system. The second significant difference is the set of simplifying assumptions. In the Lorenz study, the downhill speed of the puck is fixed; in this system, it is not fixed and depends on gravity and magnetic attraction which speeds and slows the puck. The Lorenz system is therefore semi-conservative as energy lost or gained from the slope is matched by the puck speeding up or slowing. The final difference is that the Lorenz system was integrated over a much longer slope, allowing the system to fully evolve. Due to computational and schedule limitations, this was not possible for our system.

	= Conclusions =		= Conclusions =

Group2 at 00:31, 16 December 2011

2011-12-16T00:31:59Z

← Older revision		Revision as of 20:31, 15 December 2011
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	== Ski-Slope Dynamics ==		== Ski-Slope Dynamics ==
	[[File:skislope.png \| thumb \| 250px \| Mogul field on a ski-slope ]]		[[File:skislope.png \| thumb \| 250px \| Mogul field on a ski-slope [6] ]]
	[[File:Lorenz_attractor.PNG \| thumb \| 250px \| Progression of attractor structure descending down the ski-slope ]]		[[File:Lorenz_attractor.PNG \| thumb \| 250px \| Progression of attractor structure descending down the ski-slope [6] ]]
	While Plinko as shown on the Price is Right is a random process, there are similar types of scenarios wherein an object descends through a regular field of obstructions that alter its trajectory as it descends. One such scenario is ski-slope dynamics [6]. In this scenario, a series of moguls are located on a slope, and a sliding object is allowed to descend under the force of gravity. As the sliding object interacts with the moguls, the trajectory is altered.		While Plinko as shown on the Price is Right is a random process, there are similar types of scenarios wherein an object descends through a regular field of obstructions that alter its trajectory as it descends. One such scenario is ski-slope dynamics [6]. In this scenario, a series of moguls are located on a slope, and a sliding object is allowed to descend under the force of gravity. As the sliding object interacts with the moguls, the trajectory is altered.

Group2: /* Conclusions */

2011-12-16T00:24:55Z

Conclusions

← Older revision		Revision as of 20:24, 15 December 2011
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	= Conclusions =		= Conclusions =
	A magnetic Plinko board exhibits characteristics which are consistent with results seen for a ski-slope dynamics problem. Clear basins of attraction form for a magnetic puck dropping through a sheet of magnets, particularly when the puck initially passes close to one of the magnets. Mapping cross-slope velocity against cross-slope location as the puck travels down the Plinko board shows the formation of a folded structure similar to that seen in Lorenz's ski-slope dynamics simulation. Experimental results and simulation results of the magnetic Plinko board are in qualitative agreement under the assumptions made for the values of modeling parameters in the simulation. Inconsistencies arise due to physical errors and uncertainties in the Plinko board, location of magnets, and determination of relevant parameters that limits direct quanitative comparison. Limitations in the experimental setup due to the puck ~~faling~~ off the side of the Plinko board prevented acquisition of data for initial conditions located far off center, which also limited the ability to validate the simulation results with the experiment. What qualitative data was taken shows an attractor structure that is similar in shape between the simulation and experiment, indicating that the simulation is a good model for the puck traveling through the magnet sheet.		A magnetic Plinko board exhibits characteristics which are consistent with results seen for a ski-slope dynamics problem. Clear basins of attraction form for a magnetic puck dropping through a sheet of magnets, particularly when the puck initially passes close to one of the magnets. Mapping cross-slope velocity against cross-slope location as the puck travels down the Plinko board shows the formation of a folded structure similar to that seen in Lorenz's ski-slope dynamics simulation. Experimental results and simulation results of the magnetic Plinko board are in qualitative agreement under the assumptions made for the values of modeling parameters in the simulation. Inconsistencies arise due to physical errors and uncertainties in the Plinko board, location of magnets, and determination of relevant parameters that limits direct quanitative comparison. Limitations in the experimental setup due to the puck falling off the side of the Plinko board prevented acquisition of data for initial conditions located far off center, which also limited the ability to validate the simulation results with the experiment. What qualitative data was taken shows an attractor structure that is similar in shape between the simulation and experiment, indicating that the simulation is a good model for the puck traveling through the magnet sheet.

	To further investigate the chaotic structure of magnetic Plinko, more experimental data points would need to be taken. Determining a method to allow for a wider range of initial conditions would allow for a more thorough construction of the attractor as the puck passes each row of magnets. Within the simulation, running a finer resolution sweep of initial conditions would better fill in the structure of the attractor for magnet rows further down the board. Improving the models for the magnetic force would also potentially affect the attractor shape. A series of rare earth magnets will create a complicated magnetic field that is not being modeled in the current simulation. Some of these underlying magnetic field interactions may better explain the tendency of the puck to travel along certain paths as it descends through the magnet sheet. A longer magnetic Plinko board allows for further tracking of the formation of the attractor, which may show better agreement with the Lorenz simulation that tracked trajectories for a large downslope distance.		To further investigate the chaotic structure of magnetic Plinko, more experimental data points would need to be taken. Determining a method to allow for a wider range of initial conditions would allow for a more thorough construction of the attractor as the puck passes each row of magnets. Within the simulation, running a finer resolution sweep of initial conditions would better fill in the structure of the attractor for magnet rows further down the board. Improving the models for the magnetic force would also potentially affect the attractor shape. A series of rare earth magnets will create a complicated magnetic field that is not being modeled in the current simulation. Some of these underlying magnetic field interactions may better explain the tendency of the puck to travel along certain paths as it descends through the magnet sheet. A longer magnetic Plinko board allows for further tracking of the formation of the attractor, which may show better agreement with the Lorenz simulation that tracked trajectories for a large downslope distance.