Group 1 2012 - Revision history

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Protected "Group 1 2012" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

← Older revision	Revision as of 22:50, 28 October 2014
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Group4 2012 at 14:35, 14 December 2012

2012-12-14T14:35:58Z

← Older revision		Revision as of 10:35, 14 December 2012
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	''Group members: Ross Granowski, Aemen Lodhi, Andrew Champion, Suchithra Ravi''		''Group members: Ross Granowski, Aemen Lodhi, Andrew Champion, Suchithra Ravi''

	[[Media: duffingpreproposal.pdf \| Pre-proposal presentation]]		* [[Media: duffingpreproposal.pdf \| Pre-proposal presentation]]
	[[Media: duffingfinal.pdf \| Final presentation]]		* [[Media: duffingfinal.pdf \| Final presentation]]
			* Reports: [[Media: duffingChampion.pdf \| Andrew]], [[Media: duffingGranowski.pdf \| Ross]]

	[[File: Tw_duffing.png \| frame \| Phase portrait of a duffing oscillator [http://en.wikipedia.org/wiki/File:Tw_duffing.png]]]		[[File: Tw_duffing.png \| frame \| Phase portrait of a duffing oscillator [http://en.wikipedia.org/wiki/File:Tw_duffing.png]]]

Group4 2012: /* Attractor Reconstruction. Estimating the Largest Lyapunov Exponent. */

2012-12-14T14:08:32Z

Attractor Reconstruction. Estimating the Largest Lyapunov Exponent.

← Older revision		Revision as of 10:08, 14 December 2012
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	\|-		\|-
	\| [[Image:lyapunovExponent.png\|400px\|thumb\|Plot of <math>\frac{1}{\Delta t}\langle \ln d_{j}(i) \rangle</math> (on the ''y''-axis) versus <math>i</math> (on the ''x''-axis). The slope of the least-squares fit line gives us an estimate of the largest Lyapunov exponent.\|alt=image]]		\| [[Image:lyapunovExponent.png\|400px\|thumb\|Plot of <math>\frac{1}{\Delta t}\langle \ln d_{j}(i) \rangle</math> (on the ''y''-axis) versus <math>i</math> (on the ''x''-axis). The slope of the least-squares fit line gives us an estimate of the largest Lyapunov exponent.\|alt=image]]
	\|		\| [[Image:2dReconstructedAttractor.png\|400px\|thumb\|FIG 10b: The 2D-embedded reconstructed attractor. The units for each axis are Volts.\|alt=image]]
	\|}		\|}
	</div>		</div>

Group4 2012: /* Conclusion */

2012-12-14T14:02:50Z

Conclusion

← Older revision		Revision as of 10:02, 14 December 2012
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	We must acknowledge that the forced Duffing oscillator is now something of a classical example of a chaotic dynamical system, and it has been thoroughly studied in theory and experiment. Although we did not necessarily do anything new, we found it to be a very deep and complicated system. It was a great chance to apply the theory which was presented in the class. In the course of our experiment and analysis we learned about a number of interesting and fruitful techniques for the study of chaotic dynamics.		We must acknowledge that the forced Duffing oscillator is now something of a classical example of a chaotic dynamical system, and it has been thoroughly studied in theory and experiment. Although we did not necessarily do anything new, we found it to be a very deep and complicated system. It was a great chance to apply the theory which was presented in the class. In the course of our experiment and analysis we learned about a number of interesting and fruitful techniques for the study of chaotic dynamics.

	We still have number of questions regarding the Duffing oscillator and our experiment which we find very interesting. We would like to explore several different directions in the future. In particular, we would like to analyze forced oscillators with asymmetric potential wells, which can be modeled by setting <math>V(x)= \frac{\beta }{4}x^{4}+\frac{\epsilon_{1}}{3}x^{3}-\frac{\alpha}{2} x^{2}+\epsilon_{2}x</math>. We would also like to study the case of nonsinusoidal periodic forcing<ref name="VariousPeriodicForces"/>, where we replace <math>\cos</math> with some other function <math>\rho(\omega t)</math> which is possibly piecewise smooth (or merely piecewise <math>C^{\alpha}(\mathbb{R})</math>). Exploring either of these questions will require further refining the experimental setup, continued study of numerical simulations, and some new mathematical/theoretical techniques.		We still have number of questions regarding the Duffing oscillator and our experiment which we find very interesting. We would like to explore several different directions in the future. In particular, we would like to analyze forced oscillators with asymmetric potential wells<ref name="Asymmetric"/>, which can be modeled by setting <math>V(x)= \frac{\beta }{4}x^{4}+\frac{\epsilon_{1}}{3}x^{3}-\frac{\alpha}{2} x^{2}+\epsilon_{2}x</math>. We would also like to study the case of nonsinusoidal periodic forcing<ref name="VariousPeriodicForces"/>, where we replace <math>\cos</math> with some other function <math>\rho(\omega t)</math> which is possibly piecewise smooth (or merely piecewise <math>C^{\alpha}(\mathbb{R})</math>). Exploring either of these questions will require further refining the experimental setup, continued study of numerical simulations, and some new mathematical/theoretical techniques.

	= References =		= References =

Group4 2012: /* References */

2012-12-14T14:02:15Z

References

← Older revision		Revision as of 10:02, 14 December 2012
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	<ref name="BergerNunes">Berger, J. and G. Nunes Jr. (1997). A mechanical duffing oscillator for the undergraduate laboratory. ''Am. J. Phys.''. 65, 275-296.</ref>		<ref name="BergerNunes">Berger, J. and G. Nunes Jr. (1997). A mechanical duffing oscillator for the undergraduate laboratory. ''Am. J. Phys.''. 65, 275-296.</ref>
	<ref name="FraserSwinney">Fraser, A. and H. Swinney (1986). Independent coordinates for strange attractors from mutual information. ''Phys. Rev. A,'' 33, 1134-1140.</ref>		<ref name="FraserSwinney">Fraser, A. and H. Swinney (1986). Independent coordinates for strange attractors from mutual information. ''Phys. Rev. A,'' 33, 1134-1140.</ref>
	~~<ref name="GuckenheimerHolmes">Guckenheimer, J. and P. Holmes (1983). ''Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields''. Springer-Verlag.</ref>~~
	<ref name="Holmes">Holmes, P. (1979). A nonlinear oscillator with a strange attractor. ''Phil. Trans. R. Soc. Lond''. A 292, 419–448.</ref>		<ref name="Holmes">Holmes, P. (1979). A nonlinear oscillator with a strange attractor. ''Phil. Trans. R. Soc. Lond''. A 292, 419–448.</ref>
	<ref name="MoonHolmes">Moon, F. and P. Holmes (1979). A magnetoelastic strange attractor. ''Journal of Sound and Vibration''. 65, 841-846.</ref>		<ref name="MoonHolmes">Moon, F. and P. Holmes (1979). A magnetoelastic strange attractor. ''Journal of Sound and Vibration''. 65, 841-846.</ref>
	~~<ref name="Morozov">Morozov, A. (1973). Approach to a complete qualitative study of duffing’s equation. ''USSR Comput. Math.~~
	~~math. Phys. 13''(5), 45–66.</ref>~~
	<ref name="VariousPeriodicForces">Ravichandran, V., V. Chinnathambi, and S. Rajasekar (2006). Effect of various periodic forces on duffing oscillator. ''Pramana 67''(2), 351–356.</ref>		<ref name="VariousPeriodicForces">Ravichandran, V., V. Chinnathambi, and S. Rajasekar (2006). Effect of various periodic forces on duffing oscillator. ''Pramana 67''(2), 351–356.</ref>
	<ref name="Rosenstein">Rosenstein, M.T. and Collins, J.J. and De Luca, C.J. (1993). A practical method for calculating largest Lyapunov exponents from small data sets. "Physica D: Nonlinear Phenomena". 65, 117-134.</ref>		<ref name="Rosenstein">Rosenstein, M.T. and Collins, J.J. and De Luca, C.J. (1993). A practical method for calculating largest Lyapunov exponents from small data sets. "Physica D: Nonlinear Phenomena". 65, 117-134.</ref>
			<ref name="Asymmetric">S. Jeyakumari, V. Chinnathambi, S. Rajasekar, M.A.F. Sanjuan (2011). Vibrational resonance in an asymmetric duffing oscillator. "I. J. Bifurcation and Chaos". 21, 275-286.</ref>
	</references>		</references>

Group4 2012: /* Attractor Reconstruction. Estimating the Largest Lyapunov Exponent. */

2012-12-14T14:00:35Z

Attractor Reconstruction. Estimating the Largest Lyapunov Exponent.

← Older revision		Revision as of 10:00, 14 December 2012
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	We followed the method outlined in Fraser and Swinney<ref name="FraserSwinney"/> for reconstructing attractors from time series. We use the time series <math>X(t)</math> from our chaotic trial whose graph is shown in FIG. 6 and we plot <math>t\mapsto (X(t), X(t+\tau), X(t+2\tau))</math>, where <math>\tau</math> is chosen to be the first local minimum of the mutual information (see FIG. 9) for <math>(X(t), X(t+\tau))</math>. We do not have the space here to explain this concept in detail, but, heuristically, we want <math>X(t)</math> and <math>X(t+\tau)</math> to be as statistically independent as possible. The 3-dimensional embedding of the reconstructed attractor which results is shown in FIG. 10.		We followed the method outlined in Fraser and Swinney<ref name="FraserSwinney"/> for reconstructing attractors from time series. We use the time series <math>X(t)</math> from our chaotic trial whose graph is shown in FIG. 6 and we plot <math>t\mapsto (X(t), X(t+\tau), X(t+2\tau))</math>, where <math>\tau</math> is chosen to be the first local minimum of the mutual information (see FIG. 9) for <math>(X(t), X(t+\tau))</math>. We do not have the space here to explain this concept in detail, but, heuristically, we want <math>X(t)</math> and <math>X(t+\tau)</math> to be as statistically independent as possible. The 3-dimensional embedding of the reconstructed attractor which results is shown in FIG. 10.

	Using the reconstructed attractor plot, we then followed the method in Rosenstein et. al. for estimating the largest Lyapunov exponent. We use an algorithm to find the nearest neighbor of each point on the reconstructed attractor, breaking our time series into a set of closest pairs of points. Then letting <math>d_{j}(i)</math> denote the distance between the <math>j</math>th pair of nearest neighbors at time <math>i\Delta t</math>, we plot the line <math>y(i)=\frac{1}{\Delta t}\langle \ln d_{j}\rangle</math>, where <math>\langle\cdot \rangle</math> is the average over ''j''. The largest Lyapunov exponent is then given by the slope of the least squares fit to this line.		Using the reconstructed attractor plot, we then followed the method in Rosenstein et. al.<ref name="Rosenstein"/> for estimating the largest Lyapunov exponent. We use an algorithm to find the nearest neighbor of each point on the reconstructed attractor, breaking our time series into a set of closest pairs of points. Then letting <math>d_{j}(i)</math> denote the distance between the <math>j</math>th pair of nearest neighbors at time <math>i\Delta t</math>, we plot the line <math>y(i)=\frac{1}{\Delta t}\langle \ln d_{j}\rangle</math>, where <math>\langle\cdot \rangle</math> is the average over ''j''. The largest Lyapunov exponent is then given by the slope of the least squares fit to this line.

	With this method, we found the largest Lyapunov exponent to be .0196 - which is positive, as we would expect for a system exhibiting chaotic behavior (see FIG. 11).		With this method, we found the largest Lyapunov exponent to be .0196 - which is positive, as we would expect for a system exhibiting chaotic behavior (see FIG. 11).

Group4 2012: /* References */

2012-12-14T13:59:55Z

References

← Older revision		Revision as of 09:59, 14 December 2012
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	math. Phys. 13''(5), 45–66.</ref>		math. Phys. 13''(5), 45–66.</ref>
	<ref name="VariousPeriodicForces">Ravichandran, V., V. Chinnathambi, and S. Rajasekar (2006). Effect of various periodic forces on duffing oscillator. ''Pramana 67''(2), 351–356.</ref>		<ref name="VariousPeriodicForces">Ravichandran, V., V. Chinnathambi, and S. Rajasekar (2006). Effect of various periodic forces on duffing oscillator. ''Pramana 67''(2), 351–356.</ref>
			<ref name="Rosenstein">Rosenstein, M.T. and Collins, J.J. and De Luca, C.J. (1993). A practical method for calculating largest Lyapunov exponents from small data sets. "Physica D: Nonlinear Phenomena". 65, 117-134.</ref>
	</references>		</references>

Group4 2012: /* Conclusion */

2012-12-14T13:58:02Z

Conclusion

← Older revision		Revision as of 09:58, 14 December 2012
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	We must acknowledge that the forced Duffing oscillator is now something of a classical example of a chaotic dynamical system, and it has been thoroughly studied in theory and experiment. Although we did not necessarily do anything new, we found it to be a very deep and complicated system. It was a great chance to apply the theory which was presented in the class. In the course of our experiment and analysis we learned about a number of interesting and fruitful techniques for the study of chaotic dynamics.		We must acknowledge that the forced Duffing oscillator is now something of a classical example of a chaotic dynamical system, and it has been thoroughly studied in theory and experiment. Although we did not necessarily do anything new, we found it to be a very deep and complicated system. It was a great chance to apply the theory which was presented in the class. In the course of our experiment and analysis we learned about a number of interesting and fruitful techniques for the study of chaotic dynamics.

	We still have number of questions regarding the Duffing oscillator and our experiment which we find very interesting. We would like to explore several different directions in the future. In particular, we would like to analyze forced oscillators with asymmetric potential wells, which can be modeled by setting <math>V(x)= \frac{\beta }{4}x^{4}+\frac{\epsilon_{1}}{3}x^{3}-\frac{\alpha}{2} x^{2}+\epsilon_{2}x</math>. We would also like to study the case of nonsinusoidal periodic forcing, where we replace <math>\cos</math> with some other function <math>\rho(\omega t)</math> which is possibly piecewise smooth (or merely piecewise <math>C^{\alpha}(\mathbb{R})</math>). Exploring either of these questions will require further refining the experimental setup, continued study of numerical simulations, and some new mathematical/theoretical techniques.		We still have number of questions regarding the Duffing oscillator and our experiment which we find very interesting. We would like to explore several different directions in the future. In particular, we would like to analyze forced oscillators with asymmetric potential wells, which can be modeled by setting <math>V(x)= \frac{\beta }{4}x^{4}+\frac{\epsilon_{1}}{3}x^{3}-\frac{\alpha}{2} x^{2}+\epsilon_{2}x</math>. We would also like to study the case of nonsinusoidal periodic forcing<ref name="VariousPeriodicForces"/>, where we replace <math>\cos</math> with some other function <math>\rho(\omega t)</math> which is possibly piecewise smooth (or merely piecewise <math>C^{\alpha}(\mathbb{R})</math>). Exploring either of these questions will require further refining the experimental setup, continued study of numerical simulations, and some new mathematical/theoretical techniques.

	= References =		= References =

Group4 2012: /* Results */

2012-12-14T13:56:29Z

Results

← Older revision		Revision as of 09:56, 14 December 2012
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	= Results =		= Results =

			===Early Apparatus Video===
			<videoflash>9Lt8kfEg5FA</videoflash>
			===Beam Becoming Stuck in a Well===
			<videoflash>7gS2Z92mufg</videoflash>
			===Final Apparatus Video===
			<videoflash>NCeBxAi-D6E</videoflash>

	== Parameter Estimation ==		== Parameter Estimation ==

Group4 2012: /* Attractor Reconstruction. Estimating the Largest Lyapunov Exponent. */

2012-12-14T13:51:20Z

Attractor Reconstruction. Estimating the Largest Lyapunov Exponent.

← Older revision		Revision as of 09:51, 14 December 2012
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	== Attractor Reconstruction. Estimating the Largest Lyapunov Exponent. ==		== Attractor Reconstruction. Estimating the Largest Lyapunov Exponent. ==

	We followed the method outlined in Fraser and Swinney for reconstructing attractors from time series. We use the time series <math>X(t)</math> from our chaotic trial whose graph is shown in FIG. 6 and we plot <math>t\mapsto (X(t), X(t+\tau), X(t+2\tau))</math>, where <math>\tau</math> is chosen to be the first local minimum of the mutual information (see FIG. 9) for <math>(X(t), X(t+\tau))</math>. We do not have the space here to explain this concept in detail, but, heuristically, we want <math>X(t)</math> and <math>X(t+\tau)</math> to be as statistically independent as possible. The 3-dimensional embedding of the reconstructed attractor which results is shown in FIG. 10.		We followed the method outlined in Fraser and Swinney<ref name="FraserSwinney"/> for reconstructing attractors from time series. We use the time series <math>X(t)</math> from our chaotic trial whose graph is shown in FIG. 6 and we plot <math>t\mapsto (X(t), X(t+\tau), X(t+2\tau))</math>, where <math>\tau</math> is chosen to be the first local minimum of the mutual information (see FIG. 9) for <math>(X(t), X(t+\tau))</math>. We do not have the space here to explain this concept in detail, but, heuristically, we want <math>X(t)</math> and <math>X(t+\tau)</math> to be as statistically independent as possible. The 3-dimensional embedding of the reconstructed attractor which results is shown in FIG. 10.

	Using the reconstructed attractor plot, we then followed the method in Rosenstein et. al. for estimating the largest Lyapunov exponent. We use an algorithm to find the nearest neighbor of each point on the reconstructed attractor, breaking our time series into a set of closest pairs of points. Then letting <math>d_{j}(i)</math> denote the distance between the <math>j</math>th pair of nearest neighbors at time <math>i\Delta t</math>, we plot the line <math>y(i)=\frac{1}{\Delta t}\langle \ln d_{j}\rangle</math>, where <math>\langle\cdot \rangle</math> is the average over ''j''. The largest Lyapunov exponent is then given by the slope of the least squares fit to this line.		Using the reconstructed attractor plot, we then followed the method in Rosenstein et. al. for estimating the largest Lyapunov exponent. We use an algorithm to find the nearest neighbor of each point on the reconstructed attractor, breaking our time series into a set of closest pairs of points. Then letting <math>d_{j}(i)</math> denote the distance between the <math>j</math>th pair of nearest neighbors at time <math>i\Delta t</math>, we plot the line <math>y(i)=\frac{1}{\Delta t}\langle \ln d_{j}\rangle</math>, where <math>\langle\cdot \rangle</math> is the average over ''j''. The largest Lyapunov exponent is then given by the slope of the least squares fit to this line.

← Older revision		Revision as of 10:35, 14 December 2012
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	''Group members: Ross Granowski, Aemen Lodhi, Andrew Champion, Suchithra Ravi''		''Group members: Ross Granowski, Aemen Lodhi, Andrew Champion, Suchithra Ravi''

	[[Media: duffingpreproposal.pdf \| Pre-proposal presentation]]		* [[Media: duffingpreproposal.pdf \| Pre-proposal presentation]]
	[[Media: duffingfinal.pdf \| Final presentation]]		* [[Media: duffingfinal.pdf \| Final presentation]]
			* Reports: [[Media: duffingChampion.pdf \| Andrew]], [[Media: duffingGranowski.pdf \| Ross]]

	[[File: Tw_duffing.png \| frame \| Phase portrait of a duffing oscillator [http://en.wikipedia.org/wiki/File:Tw_duffing.png]]]		[[File: Tw_duffing.png \| frame \| Phase portrait of a duffing oscillator [http://en.wikipedia.org/wiki/File:Tw_duffing.png]]]