Group 4 - Revision history

Cjcrowley at 14:00, 12 January 2012

2012-01-12T14:00:08Z

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	''Group Members: Jeff Tithof, Brad Taylor, Luis Jover, & Vlad Levenfeld''		''Group Members: Jeff Tithof, Brad Taylor, Luis Jover, & Vlad Levenfeld''

			[[Media: Group4_Metronomes_Talk.pdf \| Presentation]]. '''Papers:''' [[Media: Levenfeld_Vlad.pdf \| Vlad]], [[Media: Jover_finalpaper_metronomes_jover.pdf \| Luis]], [[Media: Taylor_Bradford.pdf \| Brad]], [[Media: Tithof_Metronomes.pdf \| Jeff]]

	The synchronization of coupled oscillations has proven to be a useful model for describing the dynamics of a large number of physical systems. Perhaps the simplest system which can capture such dynamics is that of two oscillators coupled through the motion of a free platform. In recent years, there has been a considerable amount of research on the case of $N=2$; however, little work has extended to larger numbers of oscillators. The goal of the current work is to quantify the time to synchronization for $N=2$ through $N=9$ metronomes on a movable platform, through experimental investigation and numerical simulation. Our results show a considerable level of agreement between the two, and discrepancies are explored. Possibilites for future work and potential improvements are noted.		The synchronization of coupled oscillations has proven to be a useful model for describing the dynamics of a large number of physical systems. Perhaps the simplest system which can capture such dynamics is that of two oscillators coupled through the motion of a free platform. In recent years, there has been a considerable amount of research on the case of $N=2$; however, little work has extended to larger numbers of oscillators. The goal of the current work is to quantify the time to synchronization for $N=2$ through $N=9$ metronomes on a movable platform, through experimental investigation and numerical simulation. Our results show a considerable level of agreement between the two, and discrepancies are explored. Possibilites for future work and potential improvements are noted.

Cjcrowley at 16:14, 11 January 2012

2012-01-11T16:14:51Z

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			'''Synchronization of metronomes'''

	''Group Members: Jeff Tithof, Brad Taylor, Luis Jover, & Vlad Levenfeld''		''Group Members: Jeff Tithof, Brad Taylor, Luis Jover, & Vlad Levenfeld''

Cjcrowley: Protected "Group 4" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

2012-01-09T16:51:40Z

Protected "Group 4" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

← Older revision	Revision as of 12:51, 9 January 2012
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Group4 at 14:44, 16 December 2011

2011-12-16T14:44:01Z

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	The phenomenon of synchronization is found throughout nature and in many applications		The phenomenon of synchronization is found throughout nature and in many applications
	of different fields of science<ref>Steven H. Strogatz, ~~{\it~~ Nonlinear Dynamics and Chaos}, Westview, Cambridge (1994).</ref>. Examples of synchronization include the flashing of firefly populations,		of different fields of science<ref>Steven H. Strogatz, Nonlinear Dynamics and Chaos, Westview, Cambridge (1994).</ref>. Examples of synchronization include the flashing of firefly populations,
	the firing of pacemaker cells in the heart, applications of Josephson junctions, coupled fiber		the firing of pacemaker cells in the heart, applications of Josephson junctions, coupled fiber
	laser arrays, and perhaps even epileptic seizures. Many instances of synchronization in nature and		laser arrays, and perhaps even epileptic seizures. Many instances of synchronization in nature and

Group4 at 14:42, 16 December 2011

2011-12-16T14:42:41Z

Show changes

Group4 at 13:49, 16 December 2011

2011-12-16T13:49:35Z

← Older revision		Revision as of 09:49, 16 December 2011
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	''Group Members: Luis Jover, Vlad Levenfeld~~, Brad Taylor, & Jeff Tithof~~''		''Group Members: Jeff Tithof, Brad Taylor, Luis Jover, & Vlad Levenfeld''

	The synchronization of coupled oscillations has proven to be a useful model for describing the dynamics of a large number of physical systems. Perhaps the simplest system which can capture such dynamics is that of two oscillators coupled through the motion of a free platform. In recent years, there has been a considerable amount of research on the case of $N=2$; however, little work has extended to larger numbers of oscillators. The goal of the current work is to quantify the time to synchronization for $N=2$ through $N=9$ metronomes on a movable platform, through experimental investigation and numerical simulation. Our results show a considerable level of agreement between the two, and discrepancies are explored. Possibilites for future work and potential improvements are noted.		The synchronization of coupled oscillations has proven to be a useful model for describing the dynamics of a large number of physical systems. Perhaps the simplest system which can capture such dynamics is that of two oscillators coupled through the motion of a free platform. In recent years, there has been a considerable amount of research on the case of $N=2$; however, little work has extended to larger numbers of oscillators. The goal of the current work is to quantify the time to synchronization for $N=2$ through $N=9$ metronomes on a movable platform, through experimental investigation and numerical simulation. Our results show a considerable level of agreement between the two, and discrepancies are explored. Possibilites for future work and potential improvements are noted.
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	for synchronization, the case of two identical, coupled oscillators is perhaps the simplest.		for synchronization, the case of two identical, coupled oscillators is perhaps the simplest.

	Research into the mutual ~~synchonization~~ of two identical coupled oscillators is believed to date back to the		Research into the mutual synchronization of two identical coupled oscillators is believed to date back to the
	work of Christiaan Huygens in the 1600's, in which he discovered and investigated the		work of Christiaan Huygens in the 1600's, in which he discovered and investigated the
	synchronization of two pendulum clocks hanging on a common support beam <ref>Huygens C. Œuvres complètes de Christiaan Huygens. vol. 17. The Hague: Martinus Nijhoff; 1932.</ref>. Motivated by the goal		synchronization of two pendulum clocks hanging on a common support beam <ref>Huygens C. Œuvres complètes de Christiaan Huygens. vol. 17. The Hague: Martinus Nijhoff; 1932.</ref>. Motivated by the goal
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	Significant progress has been made in explaining simple instances of synchronization. In 2002, Bennett		Significant progress has been made in explaining simple instances of synchronization. In 2002, Bennett
	et al. <ref>M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, ``Huygens's clocks," Proc. R. Soc. Lond. A 458, 563-579 (2002).</ref>		et al. <ref>M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, ``Huygens's clocks," Proc. R. Soc. Lond. A 458, 563-579 (2002).</ref>
	constructed an apparatus meant to recreate the setup ~~decribed~~ by Christiaan		constructed an apparatus meant to recreate the setup described by Christiaan
	Huygens. The experimental and numerical results agree qualitatively with records from Huygens's		Huygens. The experimental and numerical results agree qualitatively with records from Huygens's
	research. Concurrently, Pantaleone <ref>J. Pantaleone, ``Synchronization of metronomes," American Journal of Physics 70, 992-1000 (2002).</ref>		research. Concurrently, Pantaleone <ref>J. Pantaleone, ``Synchronization of metronomes," American Journal of Physics 70, 992-1000 (2002).</ref>
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	goal of the current work.		goal of the current work.

	== Experiment ==		= Experiment =
	~~[[File:ExperimentalSetupFigure.jpg‎\| thumb \| 400px \| alt = Experimental setup for the N=3 case. \| The experimental setup for the N=3 case. Note that this figure is not to scale.]]~~

			Phenomenologically, as expected, one observes more complicated dynamics as the number of coupled metronomes increases. Below are two videos, for the cases of $N=2$ and $N=8$, respectively, which demonstrate this. Our experimental goal is to quantify this behavior.

	~~Below is a video of synchronization for $N=2$ in our experiment.~~		<videoflash>AMwDIIeacvs</videoflash> <videoflash>h1wY13XspVI</videoflash>
	<videoflash>AMwDIIeacvs</videoflash>

	~~For $N=8$, one sees that full, in-phase synchronization is not achieved in the next video.~~
	~~<videoflash>h1wY13XspVI</videoflash>~~

			== Experimental Materials and Methods==

			[[File:ExperimentalSetupFigure.jpg‎\| thumb \| 400px \| alt = Experimental setup for the N=3 case. \| The experimental setup for the N=3 case. Note that this figure is not to scale.]]
	[~~http~~:~~//www~~.~~youtube~~.~~com/watch?v~~=~~h1wY13XspVI&feature=youtu~~.be]


	~~== Experimental Materials and Methods==~~

	The experimental set-up for this project is conveniently very simple, but care must be given to reduce external influences that can easily alter the dynamics of non-linear systems.		The experimental set-up for this project is conveniently very simple, but care must be given to reduce external influences that can easily alter the dynamics of non-linear systems.

	Our system consists of between two to nine metronomes, a flat, thick poster-board platform, and two rollers (empty 12 fl. oz diet coke cans). The basic set-up is shown ~~here~~, with the metronomes always being placed along a 1-dimensional array.		Our system consists of between two to nine metronomes, a flat, thick poster-board platform, and two rollers (empty 12 fl. oz diet coke cans). The basic set-up is shown to the right, with the metronomes always being placed along a 1-dimensional array.
	[[File:SetupNeq8.jpg‎\| thumb \| 400px \| alt = ~~Experimental~~ setup for the N=8 case with the metronomes in motion\| ~~The~~ experimental setup for the N=8 case with the metronomes in motion.]]		[[File:SetupNeq8.jpg‎\| thumb \| 400px \| alt = A photo of the experimental setup for the N=8 case with the metronomes in motion\| A photo of the experimental setup for the N=8 case with the metronomes in motion.]]

	The metronomes are Wittner’s Super-Mini-Taktell (Series 880). We tested each metronomes capability of synchronizing in N=2 case. This resulted in us removing a metronome due to anomalous behavior. We tracked the data using a camera set for 42 frame per second that was connected to LabView on a PC. For data collection, we blacked out the pendulum surface and the reflective casing near the pendulum bob. This aided LabView in tracking a single white dot applied with white-out on each metronome bob. Similarly we spray-painted the poster-board platform black and applied a white dot on its side to track the motion. Finally we placed the entire system on a thin, smooth plastic block used to reduce frictional irregularities liable from the table surface.		The metronomes are Wittner’s Super-Mini-Taktell (Series 880). We tested each metronomes capability of synchronizing in N=2 case. This resulted in us removing a metronome due to anomalous behavior. We tracked the data using a camera set for 42 frame per second that was connected to LabView on a PC. For data collection, we blacked out the pendulum surface and the reflective casing near the pendulum bob. This aided LabView in tracking a single white dot applied with white-out on each metronome bob. Similarly we spray-painted the poster-board platform black and applied a white dot on its side to track the motion. Finally we placed the entire system on a thin, smooth plastic block used to reduce frictional irregularities liable from the table surface.
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	== ~~Theory~~ ==		= Theory & Simulation =

			== Equations of Motion ==
	The equations of motion for two coupled pendulums are available in Bennett et al.\cite{Bennett} and Wiesenfeld and Borrero-Echeverry\cite{Borrero}. These equations may easily be extended to $N$ metronomes:		The equations of motion for two coupled pendulums are available in Bennett et al.\cite{Bennett} and Wiesenfeld and Borrero-Echeverry\cite{Borrero}. These equations may easily be extended to $N$ metronomes:

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	\end{equation}		\end{equation}

	This nature of the escapement mechanism and the validity of this model is perhaps most visible for very low frequencies, as seen in the following video. Notice the behavior of the pendulum around $\phi=0$.		This nature of the escapement mechanism and the validity of this model is perhaps most visible for very low frequencies, as seen in the following video. Notice the "slow-down-then-kick" behavior of the pendulum around $\phi=0$.

	<videoflash>zBvBpVtrjA8</videoflash>		<videoflash>zBvBpVtrjA8</videoflash>
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	Incorporation of the equations and the parameters listed above into a computer model allows for a simple numerical investigation for the cases		Incorporation of the equations and the parameters listed above into a computer model allows for a simple numerical investigation for the cases
	of $N=2$ through $N=9$. For each case, we run 100 simulations with randomly selected		of $N=2$ through $N=9$. For each case, we run 100 simulations with randomly selected
	initial conditions (similar to the ``random" initial conditions of the experiment) of approximately constant total energy. The equations of motion are integrated in the simulation up to $\tau=2000$, and synchronization is determined from two criteria: (1) applying the same test from the experimental determination, but applied to the simulation $\~~phi~~$ variable, and (2) applying a ``strict sync test" in which all peak $\~~phi~~$ values must be within one time discretization. Runs which fail to satisfy the criteria are noted.		initial conditions (similar to the ``random" initial conditions of the experiment) of approximately constant total energy. The equations of motion are integrated in the simulation up to $\tau=2000$, and synchronization is determined from two criteria: (1) applying the same test from the experimental determination, but applied to the simulation $\phi_j$ variable, and (2) applying a ``strict sync test" in which all peak $\phi_j$ values must be within one time discretization. Runs which fail to satisfy the criteria are noted.


			= Results & Conclusion =

	== Results ==		== Results ==

	[[File:TimeToSync_BestGuess.jpg‎\| thumb \| 300px \| alt = The time to synchronization for our best estimate of parameter values. \| The time to synchronization for experiment and simulation. Parameters are our ``best estimate," as indicated in Parameters section above, with $l=2.5cm$. Error bars indicate one standard deviation. The natural frequency of the simulation is about 3 times that of the experiment.]]		[[File:TimeToSync_BestGuess.jpg‎\| thumb \| 300px \| alt = The time to synchronization for our best estimate of parameter values. \| The time to synchronization for experiment and simulation. Parameters are our ``best estimate," as indicated in Parameters section above, with $l=2.5cm$. Error bars indicate one standard deviation. The natural frequency of the simulation is about 3 times that of the experiment.]]

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	Clearly, there is much to be learned before the phenomenon of synchronization is thoroughly understood, but a solid foundation is being constructed.		Clearly, there is much to be learned before the phenomenon of synchronization is thoroughly understood, but a solid foundation is being constructed.

	== References ==		= References =


	<references />		<references />

Group4 at 13:27, 16 December 2011

2011-12-16T13:27:54Z

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	Research into the mutual synchonization of two identical coupled oscillators is believed to date back to the		Research into the mutual synchonization of two identical coupled oscillators is believed to date back to the
	work of Christiaan Huygens in the 1600's, in which he discovered and investigated the		work of Christiaan Huygens in the 1600's, in which he discovered and investigated the
	synchronization of two pendulum clocks hanging on a common support beam. ~~Although his~~ goal		synchronization of two pendulum clocks hanging on a common support beam <ref>Huygens C. Œuvres complètes de Christiaan Huygens. vol. 17. The Hague: Martinus Nijhoff; 1932.</ref>. Motivated by the goal
	~~was to solve~~ the "Longitude Problem" ~~at the time, he did reach~~ some insightful conclusions about		of solving the ``Longitude Problem," Huygens reached some insightful conclusions about
	the nature of this system, eventually concluding that small movements of the support beam		the nature of this system, eventually concluding that small movements of the support beam
	with every oscillation were responsible for the antiphase synchronization that consistently occurred.		with every oscillation were responsible for the antiphase synchronization that consistently occurred.
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	of coupled mechanical metronomes," Chaos 19, 043120 (2009).</ref> reported agreement with Pantaleone's results in computer simulations.		of coupled mechanical metronomes," Chaos 19, 043120 (2009).</ref> reported agreement with Pantaleone's results in computer simulations.
	Additionally, these simulations were run for coupling of up to 100 metronomes and reportedly showed		Additionally, these simulations were run for coupling of up to 100 metronomes and reportedly showed
	chaotic and hyperchaotic behavior. A paper by Borrero-Echeverry ~~and Wiesenfeld~~ <ref>Daniel Borrero-Echeverry and Kurt Wiesenfeld, ``Huygens (and Others) Revisted,"		chaotic and hyperchaotic behavior. A paper by Wiesenfeld and Borrero-Echeverry <ref>Daniel Borrero-Echeverry and Kurt Wiesenfeld, ``Huygens (and Others) Revisted,"
	~~Publication~~ in ~~progress~~, Obtained through personal contact (2011).</ref>, currently in		Accepted for publication in Chaos, Obtained through personal contact (2011).</ref>, currently in
	the process of publication, ~~describes a theoretical model which encompasses~~ the ~~behavior of both types~~		the process of publication, investigates how varying the parameters of two coupled oscillators can affect the synchronization outcome.
	of oscillators ~~often studied, clocks and metronomes, which often contrast in the phase of synchronization~~
	~~encountered. This paper attributes~~ the ~~difference in~~ synchronization ~~phase to differences in parameters~~
	~~of the oscillators~~.

	An arguably strong theoretical understanding of the $N=2$ case has been developed in recent years;		An arguably strong theoretical understanding of the $N=2$ case has been developed in recent years;

Group4 at 08:03, 16 December 2011

2011-12-16T08:03:07Z

← Older revision		Revision as of 04:03, 16 December 2011
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	The experimental set-up for this project is conveniently very simple, but care must be given to reduce external influences that can easily alter the dynamics of non-linear systems.		The experimental set-up for this project is conveniently very simple, but care must be given to reduce external influences that can easily alter the dynamics of non-linear systems.

	Our system consists of between two to nine metronomes, a flat, thick poster-board platform, and two rollers (empty 12 fl. oz diet coke cans). The basic set-up is shown here, with the metronomes always being placed along a 1-dimensional array. The metronomes are Wittner’s Super-Mini-Taktell (Series 880). We tested each metronomes capability of synchronizing in N=2 case. This resulted in us removing a metronome due to anomalous behavior. We tracked the data using a camera set for 42 frame per second that was connected to LabView on a PC. For data collection, we blacked out the pendulum surface and the reflective casing near the pendulum bob. This aided LabView in tracking a single white dot applied with white-out on each metronome bob. Similarly we spray-painted the poster-board platform black and applied a white dot on its side to track the motion. Finally we placed the entire system on a thin, smooth plastic block used to reduce frictional irregularities liable from the table surface.		Our system consists of between two to nine metronomes, a flat, thick poster-board platform, and two rollers (empty 12 fl. oz diet coke cans). The basic set-up is shown here, with the metronomes always being placed along a 1-dimensional array.
			[[File:SetupNeq8.jpg‎\| thumb \| 400px \| alt = Experimental setup for the N=8 case with the metronomes in motion\| The experimental setup for the N=8 case with the metronomes in motion.]]

			The metronomes are Wittner’s Super-Mini-Taktell (Series 880). We tested each metronomes capability of synchronizing in N=2 case. This resulted in us removing a metronome due to anomalous behavior. We tracked the data using a camera set for 42 frame per second that was connected to LabView on a PC. For data collection, we blacked out the pendulum surface and the reflective casing near the pendulum bob. This aided LabView in tracking a single white dot applied with white-out on each metronome bob. Similarly we spray-painted the poster-board platform black and applied a white dot on its side to track the motion. Finally we placed the entire system on a thin, smooth plastic block used to reduce frictional irregularities liable from the table surface.

Group4: /* Experimental Materials and Methods */

2011-12-16T07:55:43Z

Experimental Materials and Methods

← Older revision		Revision as of 03:55, 16 December 2011
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	The experimental set-up for this project is conveniently very simple, but care must be given to reduce external influences that can easily alter the dynamics of non-linear systems.		The experimental set-up for this project is conveniently very simple, but care must be given to reduce external influences that can easily alter the dynamics of non-linear systems.

	Our system consists of between two to nine metronomes, a flat, thick poster-board platform, and two rollers (empty 12 fl. oz diet coke cans). The basic set-up is shown ~~in figure 1~~, with the metronomes always being placed along a 1-dimensional array. The metronomes are Wittner’s Super-Mini-Taktell (Series 880). We tested each metronomes capability of synchronizing in N=2 case. This resulted in us removing a metronome due to anomalous behavior. We tracked the data using a camera set for 42 frame per second that was connected to LabView on a PC. For data collection, we blacked out the pendulum surface and the reflective casing near the pendulum bob. This aided LabView in tracking a single white dot applied with white-out on each metronome bob. Similarly we spray-painted the poster-board platform black and applied a white dot on its side to track the motion. Finally we placed the entire system on a thin, smooth plastic block used to reduce frictional irregularities liable from the table surface.		Our system consists of between two to nine metronomes, a flat, thick poster-board platform, and two rollers (empty 12 fl. oz diet coke cans). The basic set-up is shown here, with the metronomes always being placed along a 1-dimensional array. The metronomes are Wittner’s Super-Mini-Taktell (Series 880). We tested each metronomes capability of synchronizing in N=2 case. This resulted in us removing a metronome due to anomalous behavior. We tracked the data using a camera set for 42 frame per second that was connected to LabView on a PC. For data collection, we blacked out the pendulum surface and the reflective casing near the pendulum bob. This aided LabView in tracking a single white dot applied with white-out on each metronome bob. Similarly we spray-painted the poster-board platform black and applied a white dot on its side to track the motion. Finally we placed the entire system on a thin, smooth plastic block used to reduce frictional irregularities liable from the table surface.

Group4 at 07:54, 16 December 2011

2011-12-16T07:54:08Z

← Older revision		Revision as of 03:54, 16 December 2011
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	[http://www.youtube.com/watch?v=h1wY13XspVI&feature=youtu.be]		[http://www.youtube.com/watch?v=h1wY13XspVI&feature=youtu.be]


			== Experimental Materials and Methods==

			The experimental set-up for this project is conveniently very simple, but care must be given to reduce external influences that can easily alter the dynamics of non-linear systems.

			Our system consists of between two to nine metronomes, a flat, thick poster-board platform, and two rollers (empty 12 fl. oz diet coke cans). The basic set-up is shown in figure 1, with the metronomes always being placed along a 1-dimensional array. The metronomes are Wittner’s Super-Mini-Taktell (Series 880). We tested each metronomes capability of synchronizing in N=2 case. This resulted in us removing a metronome due to anomalous behavior. We tracked the data using a camera set for 42 frame per second that was connected to LabView on a PC. For data collection, we blacked out the pendulum surface and the reflective casing near the pendulum bob. This aided LabView in tracking a single white dot applied with white-out on each metronome bob. Similarly we spray-painted the poster-board platform black and applied a white dot on its side to track the motion. Finally we placed the entire system on a thin, smooth plastic block used to reduce frictional irregularities liable from the table surface.


			During our week in the lab we ran 20 usable data-runs for $N={2,3,5,6,7,9}$, 19 data-runs for $N=4$, and 12 runs for $N=8$. The metronomes were set for 200 beats per minute, which corresponds to 100 periods per minute. For each run we set up LabView to track the platform and the metronome bobs. Once the camera was tracking we stabilized the platform by hand as we set the metronomes one-by-one in motion. As the last metronome was set in motion we let go of the platform. We considered this the beginning of the run. The system was allowed to evolve for 90 seconds or longer depending on whether synchronization seemed to occur. For some of the higher N cases synchronization did not occur despite allowing the metronomes to wind completely down.

			== Experimental Analysis ==

			Our goal was to determine the dependence of time to synchronization on the number of metronomes. We used the raw data output of times and x-positions (from the camera lens’ point of view in pixel units). We considered a strict definition of synchrony where metronomes are in-phase.

			Due to camera distortions and small tracking errors, we chose to consider the turning points of the data as opposed to the entire time series. By using the findpeaks function in Matlab our time-series data was reduced to a vector of peak times for each metronome. We chose to look at peaks occurring at only one turning point ensuring our analysis would not falsely detect anti-phase phase-locking as an occurrence of our stricter synchronization.

			We developed an algorithm for determining synchrony for use with the experimental data and the simulation data. The peak times will occur at the same time for systems in inphase synchrony. However for the real data, this is not expected even when synchronization has phenomenologically occurred because of slight noise in the system, errors in tracking, and discretization of time. As a result we cannot discount synchronization from being achieved even if peaks occur at slightly different times for the metronomes. This motivates a time-$\epsilon$ definition of synchronization. We introduce the model parameter $\epsilon_{time}$ where if all metronome peaks occur within it we consider that time an instant of synchrony. Full synchronization occurs when so many instances of synchrony occur in a row (defined by another model parameter, $r$).

			After trimming the data of all time before the platform is destabilized and determining the peaks, the output data is reduced to a vector of peak times for each metronome. The size of each vector is not necessarily the same, however any instance of synchrony must contain any elements from each vector so we look for instances of synchrony by arbitrarily choosing one vector and considering its time elements in succession, $v_{arb}$. In order to reduce the arbitrariness we used an algorithm that updated the reference time for placing the $\epsilon_{time}$ ball. Initially we look around the arbitrary metronome peak time. Any peak times from other metronomes within the $\epsilon_{time}$ are placed in a set with the original reference peak time and the average time becomes the new reference for the $\epsilon_{time}$. This procedure is continued until no further peak times are picked up. At this point, if the number of elements is equal to $N$ we consider it a syncing instance and place the reference time in a set, $\mathcal{S}$, and we move to the next peak time in $v_{arb}$. If an instance of syncing does not occur $\mathcal{S}$ becomes the empty set and we move to the next peak time in $v_{arb}$. When the cardinality of $\mathcal{S}=r$ we take the $min({\mathcal{S}})$ as the time of synchronization, $t_{sync}$ for the run.


	== Theory ==		== Theory ==