Group 3 2012 - Revision history

Cjcrowley: Protected "Group 3 2012" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

2014-10-29T02:52:10Z

Protected "Group 3 2012" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

← Older revision	Revision as of 22:52, 28 October 2014
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Group1-2012: /* Attractor Reconstruction */

2012-12-14T07:59:25Z

Attractor Reconstruction

← Older revision		Revision as of 03:59, 14 December 2012
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	Here, R is the radius of a circle in the plane, and it this parameter which is varied. The correlation dimension can then be calculated by plotting C as a function of R on a log-log plot. This plot is shown below the ~~and~~ dimension is given by the slope of the dotted line, 1.8.		Here, R is the radius of a circle in the plane, and it this parameter which is varied. The correlation dimension can then be calculated by plotting C as a function of R on a log-log plot. This plot is shown below and the dimension is given by the slope of the dotted line, 1.8.

	[[File:cor_dim_plot.png\|thumb\|center\|400px\|Figure 19: The correlation dimension.]]		[[File:cor_dim_plot.png\|thumb\|center\|400px\|Figure 19: The correlation dimension.]]

Group1-2012: /* Attractor Reconstruction */

2012-12-14T07:57:31Z

Attractor Reconstruction

← Older revision		Revision as of 03:57, 14 December 2012
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	==Attractor Reconstruction==		==Attractor Reconstruction==

	The idea behind attractor reconstruction is that the full 3-D phase space can be reconstructed from the measurement of only one variable. The method suggests that information about each independent variable is not self-contained ~~(ref = strogatz)~~. The method relies on a delay time and reconstructs the data according to the following equations:		The idea behind attractor reconstruction is that the full 3-D phase space can be reconstructed from the measurement of only one variable. The method suggests that information about each independent variable is not self-contained [7]. The method relies on a delay time and reconstructs the data according to the following equations:

Group1-2012: /* Simulation */

2012-12-14T07:56:53Z

Simulation

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	[[File:chua_bifurcation.png\|thumb\|400px\|alt=text\|Figure 1: A bifurcation diagram for Chua's circuit. The capacitance is varied.]]		[[File:chua_bifurcation.png\|thumb\|400px\|alt=text\|Figure 20: A bifurcation diagram for Chua's circuit. The capacitance is varied.]]

	These were solved using an adaptive Runge-Kutta fourth order method. The advantage of the simulation over experiment is that any of the three variable can determined to a high degree of accuracy. This bypassed the need of an attractor reconstruction, but also enabled a comparison of simulated data to calculations. Although we did not do a rigorous comparison of the two, the simulation did generate results that were comparable to those obtained in the results. Similar attrators were observed in 2D and the 3D image compared to the reconstructed scroll from the data.		These were solved using an adaptive Runge-Kutta fourth order method. The advantage of the simulation over experiment is that any of the three variable can determined to a high degree of accuracy. This bypassed the need of an attractor reconstruction, but also enabled a comparison of simulated data to calculations. Although we did not do a rigorous comparison of the two, the simulation did generate results that were comparable to those obtained in the results. Similar attrators were observed in 2D and the 3D image compared to the reconstructed scroll from the data.


	Another advantage of the simulation is the ability to run the experiment for any value of the resistance and tweak other values of the circuit components. This enabled the simulation to construct a bifurcation diagram for a circuit that meets the criteria for period doubling, a behavior unobservable in our experiment. The diagram is shown in figure ~~(12)~~ and plots the possible voltage values as a function of the capacitance.		Another advantage of the simulation is the ability to run the experiment for any value of the resistance and tweak other values of the circuit components. This enabled the simulation to construct a bifurcation diagram for a circuit that meets the criteria for period doubling, a behavior unobservable in our experiment. The diagram is shown in figure 20 and plots the possible voltage values as a function of the capacitance.

	==Conclusion==		==Conclusion==

Group1-2012: /* Attractor Reconstruction */

2012-12-14T07:56:18Z

Attractor Reconstruction

← Older revision		Revision as of 03:56, 14 December 2012
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	If the right delay time is chosen, then the attractor may be reconstructed. An optimal time choice is one that minimizes mutual information between the signal and its time-delayed signal ~~(ref = Fraser and Finney, 1986)~~.		If the right delay time is chosen, then the attractor may be reconstructed. An optimal time choice is one that minimizes mutual information between the signal and its time-delayed signal [13].

	[[File:mut_info.png\|thumb\|center\|400px]]		[[File:mut_info.png\|thumb\|center\|400px]]
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	Using this method, we were able to reconstruct the attractor in two and three dimensions. The reconstructed attractor in 2-D is shown in figure ~~(10)~~. We see that a double scroll has been created in phase space, and only the voltage across one of the capacitors was needed. A double scroll is reconstructed in 3-D as shown in figure ~~(11)~~.		Using this method, we were able to reconstruct the attractor in two and three dimensions. The reconstructed attractor in 2-D is shown in figure 17. We see that a double scroll has been created in phase space, and only the voltage across one of the capacitors was needed. A double scroll is reconstructed in 3-D as shown in figure 18.


	<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">		<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">
	{\|		{\|
	\| [[File:recon2.png\|400px\|thumb\|center\|Figure 1: The reconstructed attractor in 2D]]		\| [[File:recon2.png\|400px\|thumb\|center\|Figure 17: The reconstructed attractor in 2D]]
	\| [[File:recon3.png\|700px\|thumb\|center\|Figure 1: The reconstructed attractor in 3D]]		\| [[File:recon3.png\|700px\|thumb\|center\|Figure 18: The reconstructed attractor in 3D]]
	\|}		\|}
	</div>		</div>
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	Here, R is the radius of a circle in the plane, and it this parameter which is varied. The correlation dimension can then be calculated by plotting C as a function of R on a log-log plot. This plot is shown below the dimension is given by the slope of the dotted line, 1.8.		Here, R is the radius of a circle in the plane, and it this parameter which is varied. The correlation dimension can then be calculated by plotting C as a function of R on a log-log plot. This plot is shown below the and dimension is given by the slope of the dotted line, 1.8.

	[[File:cor_dim_plot.png\|thumb\|center\|400px\|Figure 1: The correlation dimension.]]		[[File:cor_dim_plot.png\|thumb\|center\|400px\|Figure 19: The correlation dimension.]]

	==Simulation==		==Simulation==

Group1-2012: /* Iterative Maps */

2012-12-14T07:54:45Z

Iterative Maps

← Older revision		Revision as of 03:54, 14 December 2012
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	<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">		<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">
	{\|		{\|
	\| [[File:map_screw.png\|400px\|thumb\|center\|Figure 1: Iterative map for screw attractor]]		\| [[File:map_screw.png\|400px\|thumb\|center\|Figure 14: Iterative map for screw attractor]]
	\| [[File:map_c1.png\|400px\|thumb\|center\|Figure 1: Iterative map for chaotic behavior across capacitor C1]]		\| [[File:map_c1.png\|400px\|thumb\|center\|Figure 15: Iterative map for chaotic behavior across capacitor C1]]
	\| [[File:map_c2.png\|400px\|thumb\|center\|Figure 1: Iterative map for chaotic behavior across capacitor C2]]		\| [[File:map_c2.png\|400px\|thumb\|center\|Figure 16: Iterative map for chaotic behavior across capacitor C2]]
	\|}		\|}
	</div>		</div>

Group1-2012: /* Lyapunov Exponent */

2012-12-14T07:54:20Z

Lyapunov Exponent

← Older revision		Revision as of 03:54, 14 December 2012
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	<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">		<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">
	{\|		{\|
	\| [[File:traj_short_time.png\|500px\|thumb\|center\|Figure 1: Development of two nearby trajectories over a short time]]		\| [[File:traj_short_time.png\|500px\|thumb\|center\|Figure 10: Development of two nearby trajectories over a short time]]
	\| [[File:traj_long_time.png\|500px\|thumb\|center\|Figure 1: Development of two nearby trajectories over a long time]]		\| [[File:traj_long_time.png\|500px\|thumb\|center\|Figure 11: Development of two nearby trajectories over a long time]]
	\|}		\|}
	</div>		</div>

	Two nearby points were chosen in phase space by scanning the data and finding any two points that were separated by no more than 0.0075 V across each capacitor. These two points were of course separated in time by quite a bit. Finding points any closer would prove difficult or impossible despite having data collected over about ten seconds (an incredible amount of time for the dynamics of the circuit, which can change rapidly on the order of a millisecond). Two nearby trajectories in phase space are shown in figure ~~(6)~~ over a short time. In such a time, these trajectories are nearly locked on and only separate by a small amount in the end. Figure ~~(7)~~ shows the two trajectories over a long time. By this time, the trajectories have completely different behavior and no longer resemble each other. The system is chaotic, for a small separation in initial conditions has grown to something enormous. Figure ~~(8)~~ emphasizes the divergence by showing how the two trajectories compare on the voltage-time plot. They begin locked on, but after a few milliseconds they take on completely different behavior.		Two nearby points were chosen in phase space by scanning the data and finding any two points that were separated by no more than 0.0075 V across each capacitor. These two points were of course separated in time by quite a bit. Finding points any closer would prove difficult or impossible despite having data collected over about ten seconds (an incredible amount of time for the dynamics of the circuit, which can change rapidly on the order of a millisecond). Two nearby trajectories in phase space are shown in figure 10 over a short time. In such a time, these trajectories are nearly locked on and only separate by a small amount in the end. Figure 11 shows the two trajectories over a long time. By this time, the trajectories have completely different behavior and no longer resemble each other. The system is chaotic, for a small separation in initial conditions has grown to something enormous. Figure 12 emphasizes the divergence by showing how the two trajectories compare on the voltage-time plot. They begin locked on, but after a few milliseconds they take on completely different behavior.

	[[File:traj_time.png\|thumb\|400px\|alt=text\|Figure 1: Two nearby trajectories across capacitor C1]]		[[File:traj_time.png\|thumb\|400px\|alt=text\|Figure 12: Two nearby trajectories across capacitor C1]]


	A true description of two nearby trajectories requires n Lyapunov exponents in n-dimensional space. Each independent variable contributes to the total chaos of the system. This was not the method used here. Instead, the maximal Lyapunov exponent was measured. The maximal Lyapunov exponent is a measure of how the distance between two nearby trajectories in phase space diverges in time ~~(ref = strogatz)~~.		A true description of two nearby trajectories requires n Lyapunov exponents in n-dimensional space. Each independent variable contributes to the total chaos of the system. This was not the method used here. Instead, the maximal Lyapunov exponent was measured. The maximal Lyapunov exponent is a measure of how the distance between two nearby trajectories in phase space diverges in time [7].


	Only a single value for the Lyapunov exponent is obtained using this method. However, the Lyapunov exponent depends on the initial conditions ~~(ref = strogatz)~~ and therefore depends on which attractor the trajectories start in, since the two attractors are not identical. So two Lyapunov exponents are expected, one for each attractor. In determining the Lyapunov exponent for each attractor, it was important to ensure that these trajectories did not leave the attractor they started in too quickly. Otherwise, the calculated value would be influenced by both attractors. Keeping in mind that the Lyapunov exponent varies in phase space, an average Lyapunov exponent can be defined for the whole system.		Only a single value for the Lyapunov exponent is obtained using this method. However, the Lyapunov exponent depends on the initial conditions and therefore depends on which attractor the trajectories start in, since the two attractors are not identical [7]. So two Lyapunov exponents are expected, one for each attractor. In determining the Lyapunov exponent for each attractor, it was important to ensure that these trajectories did not leave the attractor they started in too quickly. Otherwise, the calculated value would be influenced by both attractors. Keeping in mind that the Lyapunov exponent varies in phase space, an average Lyapunov exponent can be defined for the whole system.

	[[File:lyapunov.png\|thumb\|400px\|left\|alt=text\|Figure 1: Calculating the Lyapunov exponent for two nearby trajectories]]		[[File:lyapunov.png\|thumb\|400px\|left\|alt=text\|Figure 13: Calculating the Lyapunov exponent for two nearby trajectories]]

	The Lyapunov exponent is determined by plotting the log of the distance between the two trajectories as a function of time. As a result, linear behavior is expected. One of these plots is shown in Figure ~~(9)~~ and a line is fitted to the first part. There is a saturation point where the linear behavior disappears because the trajectories are confined to the attractor.		The Lyapunov exponent is determined by plotting the log of the distance between the two trajectories as a function of time. As a result, linear behavior is expected. One of these plots is shown in Figure 13 and a line is fitted to the first part. There is a saturation point where the linear behavior disappears because the trajectories are confined to the attractor.


	This procedure was done for a number of initial conditions (each of which do not leave the attractor they start in too quickly) and an average was calculated for each attractor. The left attractor and right attractor have an average Lyapunov exponent of 1.0 and 1.7, respectively. These averages have a standard deviation of around 0.3, although the dependence of the Lyapunov exponent on position in phase space implies the distribution isn’t normal anyways; there is some natural variance in the system. The variance obtained is due to this, but also due to the inexact measurement of the Lyapunov exponent, for there is no exact method for fitting a line to the curve.		This procedure was done for a number of initial conditions (each of which do not leave the attractor they start in too quickly) and an average was calculated for each attractor. The left attractor and right attractor have an average Lyapunov exponent of 1.0 and 1.7, respectively. These averages have a standard deviation of around 0.3, although the dependence of the Lyapunov exponent on position in phase space implies the distribution isn’t normal anyways; there is some natural variance in the system. The variance obtained is due to this, but also due to the inexact measurement of the Lyapunov exponent, for there is no exact method for fitting a line to the curve.
	In addition, a t_horizon was defined from the average Lyapunov exponent of the system, 1.33.		In addition, a t_horizon was defined from the average Lyapunov exponent of the system, 1.33 [7].

Group1-2012: /* Experimental Results */

2012-12-14T07:51:33Z

Experimental Results

← Older revision		Revision as of 03:51, 14 December 2012
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	<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">		<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">
	{\|		{\|
	\| [[File:time_periodic.png\|400px\|thumb\|center\|Figure 1: Periodic signal across a capacitor]]		\| [[File:time_periodic.png\|400px\|thumb\|center\|Figure 7: Periodic signal across a capacitor]]
	\| [[File:time_screw.png\|400px\|thumb\|center\|Figure 1: Semi-periodic signal across a capacitor]]		\| [[File:time_screw.png\|400px\|thumb\|center\|Figure 8: Semi-periodic signal across a capacitor]]
	\| [[File:time_chaos.png\|400px\|thumb\|center\|Figure 1: Chaotic signal across a capacitor]]		\| [[File:time_chaos.png\|400px\|thumb\|center\|Figure 9: Chaotic signal across a capacitor]]
	\|}		\|}
	</div>		</div>

Group1-2012: /* Experimental Results */

2012-12-14T07:51:03Z

Experimental Results

← Older revision		Revision as of 03:51, 14 December 2012
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	==Experimental Results==		==Experimental Results==

	There are several different routes to chaos in Chua’s circuit. In the experiment, the route observed was periodic behavior, followed by a “screw attractor”, and finally chaotic behavior, referred to as a “double scroll” in Chua’s circuit (11). This contrasts to period doubling, whose requirements of the function g(V) did not meet the specifications of our circuit (12). The meaning of a “screw attractor” is most evident by looking at the phase space diagram for the two voltages in figure 5. Out of the periodic behavior, a single chaotic attractor was born as the resistance increased. This may be referred to as “semi-periodic” behavior, meaning that it is almost periodic, but has some chaotic aspects.		There are several different routes to chaos in Chua’s circuit. In the experiment, the route observed was periodic behavior, followed by a “screw attractor”, and finally chaotic behavior, referred to as a “double scroll” in Chua’s circuit [11]. This contrasts to period doubling, whose requirements of the function g(V) did not meet the specifications of our circuit [12]. The meaning of a “screw attractor” is most evident by looking at the phase space diagram for the two voltages in figure 5. Out of the periodic behavior, a single chaotic attractor was born as the resistance increased. This may be referred to as “semi-periodic” behavior, meaning that it is almost periodic, but has some chaotic aspects.

Group1-2012: /* Experimental Results */

2012-12-14T07:50:37Z

Experimental Results

← Older revision		Revision as of 03:50, 14 December 2012
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	==Experimental Results==		==Experimental Results==

	There are several different routes to chaos in Chua’s circuit. In the experiment, the route observed was periodic behavior, followed by a “screw attractor”, and finally chaotic behavior, referred to as a “double scroll” in Chua’s circuit (~~Ref. Harada 1996~~). This contrasts to period doubling, whose requirements of the function g(V) did not meet the specifications of our circuit (~~Chua 1993~~). The meaning of a “screw attractor” is most evident by looking at the phase space diagram for the two voltages in figure ~~(4)~~. Out of the periodic behavior, a single chaotic attractor was born as the resistance increased. This may be referred to as “semi-periodic” behavior, meaning that it is almost periodic, but has some chaotic aspects.		There are several different routes to chaos in Chua’s circuit. In the experiment, the route observed was periodic behavior, followed by a “screw attractor”, and finally chaotic behavior, referred to as a “double scroll” in Chua’s circuit (11). This contrasts to period doubling, whose requirements of the function g(V) did not meet the specifications of our circuit (12). The meaning of a “screw attractor” is most evident by looking at the phase space diagram for the two voltages in figure 5. Out of the periodic behavior, a single chaotic attractor was born as the resistance increased. This may be referred to as “semi-periodic” behavior, meaning that it is almost periodic, but has some chaotic aspects.


	When the resistance was increased further, another attractor was born, forming a double scroll. The phase portrait for this behavior is shown in Figure ~~(5)~~. Trajectories in phase space are confined to this double scroll and orbit around these two attractors in a complicated way. It is this complicated behavior that we wish to analyze.		When the resistance was increased further, another attractor was born, forming a double scroll. The phase portrait for this behavior is shown in Figure 6. Trajectories in phase space are confined to this double scroll and orbit around these two attractors in a complicated way. It is this complicated behavior that we wish to analyze.

	<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">		<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">
	{\|		{\|
	\| [[File:phase_periodic.png\|400px\|thumb\|center\|Figure 1: Phase space diagram for periodic behavior]]		\| [[File:phase_periodic.png\|400px\|thumb\|center\|Figure 4: Phase space diagram for periodic behavior]]
	\| [[File:phase_screw.png\|400px\|thumb\|center\|Figure 1: Phase space diagram for semi-periodic behavior]]		\| [[File:phase_screw.png\|400px\|thumb\|center\|Figure 5: Phase space diagram for semi-periodic behavior]]
	\| [[File:phase_chaos.png\|400px\|thumb\|center\|Figure 1: Phase space diagram for chaotic behavior]]		\| [[File:phase_chaos.png\|400px\|thumb\|center\|Figure 6: Phase space diagram for chaotic behavior]]
	\|}		\|}
	</div>		</div>