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36 Pages

### LizKenneyChaosProject

Course: ME 639, Fall 2009
School: Clarkson
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Word Count: 5467

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Project Elizabeth Chaos Kenney July 29, 2002 Abstract For this project, the Lorentz Model was studied. The equilibrium states and the stability of these equilibrium states were examined. Numerical experiments were performed and a periodic solution and a chaotic solution were found. For both solutions, various statistical quantities were evaluated. Next white noise was added to both the periodic and chaotic...

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Project Elizabeth Chaos Kenney July 29, 2002 Abstract For this project, the Lorentz Model was studied. The equilibrium states and the stability of these equilibrium states were examined. Numerical experiments were performed and a periodic solution and a chaotic solution were found. For both solutions, various statistical quantities were evaluated. Next white noise was added to both the periodic and chaotic solutions. Again, the same statistical quantities were evaluated. It was observed that when adding white noise to the periodic solution, the response statistics were greatly affected, which is quite evident in the time history plots. When adding white noise to the chaotic solution, there isnt a huge difference in the response statistics. The Lorentz Model was also expanded using the Karhunen-Loeve expansion. This illustrated that there was a high amount of energy surrounding the mode corresponding to one of the constants in the Lorentz Model. This energy played a role in the reconstruction of the x direction, while it did not signicantly affect the reconstruction in the y and z directions. 1 Equilibrium States and the Stability of the Equilibrium States The following equations are the nonlinear system known as the Lorentz model. dx = (y x) dt dy = rx y xz dt (1) (2) dz = b + xy (3) dt In these equations r, b, and are constants. To begin, the critical or equilibrium points were determined. To do this, dx , dy , and dz were set to zero and the equations 1,2, and 3 were dt dt dt solved. From this, the following critical points were determined. 1 x=y=z=0 for r > 1 x = r 1 y = r1 z =r1 (4) (5) (6) (7) To check the stability, the equations were linearized about the critical point x=y=z=0. By doing this, the characteristic determinate was found to be: r 1 From this determinate, the eigenvalues were determined to be: 1 1 2 2 + 1 + 4r (9) 2 2 2 For the system to be stable, all the eigenvalues must be negative. Therefore, for this to occur, the parameter r must be < 1. For the system to be unstable, at least one eigenvalue must be positive, therefore r > 1 will make the solution unstable. When r < 1, x=y=z=0 is the only stable solution. 1,2 = (8) 2 Numerical Experiments To perform the numerical experiments, equations 1, 2, and 3 were linearized and equations 10, 11, and 12 below were used. dx = (y x) dt dy =xy dt (10) (11) dz = bz (12) dt Using these equations, x(t), y(t), and z(t) were found using the Adams Bashforth step scheme. By solving for these equations, various plots of stochastic paratmeters were created. These plots include a 3-D plot, time history plots, poincare plots, cross correlation plots, autocorrelation plots, and power spectral density plots. The constants in equations 1, 2, and 3 were selected to create cases where both periodic and chaotic solutions existed. 2 2.1 Periodic Solution By selecting the following constants, a periodic solution was found = 10 r = 15 b = 8 3 (13) Figure 1 is a phase plot of the system. From this plot, it can be seen that when the constants above and the initial condition of (0,1,0) are both selected, the Lorentz system is periodic and it approaches the stable point of (-6.11, -6.11, 15). 3D Phase Plot of the periodic solution to the Lorentz model 30 25 20 15 10 5 0 20 10 0 10 5 y 20 10 x 0 10 5 15 Next, the time history responses were determined (Figure 2) as well as the Poincare plots (Figure 3). z Figure 1: 3D Phase Plots of Periodic Solution to the Lorentz Model 3 Time histories for the x, y, and z directions for the periodic solution 20 10 x 0 10 20 10 0 10 20 30 20 z 10 0 0 200 400 600 800 1000 1200 y 0 200 400 600 800 1000 1200 0 200 400 600 time (sec) 800 1000 1200 Figure 2: Time History Responses of Periodic Solution to the Lorentz Model Poincare maps for the periodic solution in the xy, xz, and yz planes 30 20 y 10 0 10 20 10 0 10 20 10 30 20 z 10 0 15 5 0 x 5 10 15 z 5 0 x 5 10 15 10 5 0 y 5 10 15 20 Figure 3: Poincare Plots of Periodic Solution to the Lorentz Model 4 The response statistics of the periodic solution were then analyzed. Table 1 contains the mean, mean square, and various moments for this system in each the x, y, and z directions. Mean Mean Square Moment (n=6) Moment (n=3) Moment (n=20) X -4.5164 35.2522 4.9164e4 0 3.4243e20 Y -4.5466 38.7074 8.8002e4 0 2.3845e21 Z 12.7164 183.2012 1.489e5 0 1.3775e22 Table 1: Response Statistics for Periodic Solution to the Lorentz System The next statistics that were investigated were the cross correlation and the autocorrelation. The cross correlation is a measure of the similarities or shared properties between two signals. In this case, the x-y correlation, y-z correlation, and x-z correlation were evaluated and are illustrated in Figure 4. This gure shows that there is a strong positive correlation between the x and y signals with the maximum value at a lag time of 1000 seconds. Conversely, there is a strong negative correlation between both the x and z signals and the y and z signals with the most negative point at a lag time of 1000 seconds. 4 2 0 2 5 0 5 10 5 0 5 10 x 10 4 Cross correlations for the periodic solution cross correlation xy 0 4 x 10 500 1000 1500 2000 2500 cross correlation yz 0 4 x 10 500 1000 1500 2000 2500 cross correlation xz 0 500 1000 lag time (sec) 1500 2000 2500 Figure 4: Cross Correlations of the x-y, y-z, and x-z signals for the Periodic Solution to the Lorentz Model 5 The autocorrelations for the x, y, and z signals are found in Figure 5. An autocorrelation involves only one signal and provides information about the structure of the signal or its behavior in the time domain. For a periodic signal, it is expected that when the lag time is zero, the amplitude of the autocorrelation should be one and should decrease as time increases. Looking at the autocorrelation for the x, y, and z signals, this is exactly what occurs. Autocorrelation for the periodic solution 1 autocorrelation x 0.8 0.6 0.4 50 1 autocorrelation y 0.8 0.6 0.4 50 1 autocorrelation z 0.9 0.8 0.7 50 40 30 20 10 0 10 20 30 40 50 40 30 20 10 0 10 20 30 40 50 40 30 20 10 0 10 lag time (sec) 20 30 40 50 Figure 5: Autocorrelations of the x, y, and z signals for the Periodic Solution to the Lorentz Model The nal response statistic that was analyzed was the power spectral density. Figure 6 illustrates the estimated power spectral density for the x, y, and z signals. Power Spectral Density x Power Spectral Density for x, y, and z 50 0 50 100 150 50 0 50 100 150 50 0 50 100 150 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.6 0.7 0.8 0.9 1 Figure 6: Power Spectral Density of the x, y, and z signals for the Periodic Solution to the Lorentz Model Power Spectral Density z Power Spectral Density y 6 2.2 Chaotic Solution Figure 7 is a phase plot of the system when the following constants are selected. 8 (14) 3 From this plot it can be seen that when these constants and an initial condition of (0,1,0) are selected, the Lorentz system is chaotic. = 10 r = 50 b = 3D Phase Plot of the chaotic solution to the Lorentz model 120 100 80 60 40 20 0 60 40 20 0 20 40 y 60 30 10 20 x 0 20 10 30 40 Figure 7: 3-D Phase Plot for the Chaotic Solution to the Lorentz Model Next the time history responses were determined (Figure 8) as well as the Poincare plots (Figure 9). These plots also illustrate how by selecting the constants above, the Lorentz system becomes chaotic. Unlike Figure 2, the time history plots for the chaotic system (Figure 8) do not show a nice periodic signal. Also, the Poincare plots in Figure 3 show how the system nds its stable point, while Figure 9 illustrates how there are 2 unstable points that cause this system to be chaotic. z 7 Time histories for the x, y, and z directions for the chaotic solution 40 20 0 20 40 100 50 0 50 100 150 100 z 50 0 0 200 400 600 800 1000 1200 y 0 200 400 600 800 1000 1200 x 0 200 400 600 time (sec) 800 1000 1200 Figure 8: Time History Responses for the Chaotic Solution to the Lorentz Model Poincare maps for the chaotic solution in the xy, xz, and yz planes 150 100 y 50 0 30 100 50 0 50 100 30 150 100 z 50 0 60 20 10 0 x 10 20 30 40 z 20 10 0 x 10 20 30 40 40 20 0 y 20 40 60 Figure 9: Poincare Plots for the Chaotic Solution to the Lorentz Model 8 The response statistics of the chaotic solution were analyzed next. Table 2.2 contains the mean, mean square, and various moments for this system in each the x, y, and z directions. Mean Mean Square Moment (n=6) Moment (n=3) Moment (n=20) X -2.3072 137.4242 3.4579e7 0 1.0595e30 Y -2.4362 235.8749 1.8236e8 0 2.7052e32 Z 47.9134 2.4998e3 1.2757e8 0 8.2209e31 Table 2: Response Statistics for Chaotic Solution to the Lorentz System The next statistics that were investigated were the cross correlation, the autocorrelation, and the power spectral density of the signals. Figure 10 illustrates the cross correlation of the x and y signals, the x and z signals, and the y and z signals. These show that there is no real correlation between any of the signals. Again, similar to the periodic solution, there seems to be a negative correlation between the y and z signals and the x and z signals, although there sint point lag time where the correlation is stronger than the rest. Also, there once again is a positive correlation between the x and y signals, with the strongest correlation located at a lag time of 1000 seconds. However, at every other lag time, there appears to be almost no correlation. Figure 11 is the autocorrelation. Similar to the periodic case, at a lag time of 0, all the autocorrelations are at the maximum which is 1. Figure 12 is the power spectral density. This plots looks similar to the periodic case (Figure 6) with the same values at the frequencies 0 and 1. The main difference is in the slopes of the curves. 9 cross correlation xy 2 1 0 1 1 0 1 2 1 0 1 2 x 10 5 Cross correlations for the chaotic solution 0 5 x 10 500 1000 1500 2000 2500 cross correlation yz 0 5 x 10 500 1000 1500 2000 2500 cross correlation xz 0 500 1000 lag time (sec) 1500 2000 2500 Figure 10: Cross Correlations of the x-y, y-z, and x-z signals for the Chaotic Solution to the Lorentz Model Autocorrelation for the chaotic solution 1 autocorrelation x 0.5 0 50 1 autocorrelation y 40 30 20 10 0 10 20 30 40 50 0.5 0 50 1 autocorrelation z 0.95 0.9 0.85 50 40 30 20 10 0 10 20 30 40 50 40 30 20 10 0 10 lag time (sec) 20 30 40 50 Figure 11: Autocorrelations of the x, y, and z signals for the Chaotic Solution to the Lorentz Model 10 Power Spectral Density x Power Spectral Density for x, y, and z for chaotic solution 50 0 50 100 150 50 0 50 100 150 100 0 100 200 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.6 0.7 0.8 0.9 1 Power Spectral Density z Power Spectral Density y 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.6 0.7 0.8 0.9 1 Figure 12: Power Spectral Density of the x, y, and z signals for the Chaotic Solution to the Lorentz Model The periodic and chaotic solutions produced very different results. The time history response of the periodic solution eventually oscillated about a xed point, while the chaotic solutions time history response had no obvious pattern to it. The Poincare plots for both cases were also very different. There periodic case illustrated that there is a denite stable point, while the chaotic case appeared to have two unstable points. The cross correlations for the periodic case illustrated a strong positive or a strong negative correlation at a lag time of 1000, while in the chaotic case there were no strong correlations that could be determined for the y and x signals and the x and z signals, while there appeared to be a correlation at a lag time of 1000 seconds for the x and y signals. The autocorrelations for the periodic case started with an amplitude of 1 and slowly the amplitude of the wave began to decrease. In the chaotic case, the slope of the autocorrelation was slightly different than in the periodic case. The power spectral density for both the chaotic and periodic case appeared to be somewhat similar, although the chaotic power spectral density is not as smooth of a curve as the power spectral density for the periodic case. The values for the mean, mean square, and moments in the periodic case were much smaller than what was determined for the chaotic case. 11 3 Addition of White Noise 3.1 Periodic The following are the plots that correspond to the periodic solution to the Lorentz model with the addition of white noise. 3D Phase Plot of the periodic solution to the Lorentz model 30 25 20 15 10 5 0 15 10 12 5 0 y 5 2 0 x 6 4 10 8 14 Figure 13: 3D Phase Plots of Periodic Solution to the Lorentz Model with the Addition of White Noise Time histories for the x, y, and z directions for the periodic solution 15 10 x 5 0 15 10 5 0 5 30 20 z 10 0 0 100 200 300 400 500 600 700 800 900 1000 y 0 z 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 time (sec) 600 700 800 900 1000 Figure 14: Time History Responses of Periodic Solution to the Lorentz Model with the Addition of White Noise 12 Poincare maps for the chaotic solution in the xy, xz, and yz planes 30 20 10 0 10 2 15 10 5 0 5 2 30 20 10 0 10 4 2 0 2 4 6 y 8 10 12 14 16 z 0 2 4 6 x 8 10 12 14 z 0 2 4 6 x 8 10 12 14 y Figure 15: Poincare Plots of Periodic Solution to the Lorentz Model with the Addition of White Noise X 6.1910 40.4063 134.5193 0 9.8108e11 Y 3.7685 17.5954 586.4132 0 1.3277e14 Z 18.0536 332.9565 5.2001e3 0 1.9163e17 Mean Mean Square Moment (n=6) Moment (n=3) Moment (n=20) Table 3: Response Statistics for Periodic Solution to the Lorentz System with the Addition of White Noise The addition of white noise to the periodic case is obvious when looking at Figures 13, 14, and 15. The main structure of the signal is still there, but it has obviously been disturbed by the white noise added. The response statistics (Table 3.1) are also greatly affected by the addition of white noise. The cross correlations are now all positive, while originally there were negative correlations. The autocorrelations are somewhat similar, although there is a difference between them. The power spectral density is also different. Not only are the values at the frequencies 0 and 1 different, there is also more noise in the signal. 13 cross correlation xy 3 2 1 0 8 6 4 2 0 x 10 4 Cross correlations for the periodic solution 0 4 x 10 200 400 600 800 1000 1200 1400 1600 1800 2000 cross correlation yz cross correlation xz 15 10 5 0 0 4 x 10 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 lag time (sec) 1400 1600 1800 2000 Figure 16: Cross Correlations of the x-y, y-z, and x-z signals for the Periodic Solution to the Lorentz Model with the Addition of White Noise Autocorrelation for the periodic solution 1 autocorrelation x 0.95 0.9 0.85 50 1 autocorrelation y 0.9 0.8 0.7 0.6 50 1 autocorrelation z 0.98 0.96 0.94 0.92 50 40 30 20 10 0 10 lag time (sec) 20 30 40 50 40 30 20 10 0 10 20 30 40 50 40 30 20 10 0 10 20 30 40 50 Figure 17: Autocorrelations of the x, y, and z signals for the Periodic Solution to the Lorentz Model with the Addition of White Noise 14 Power Spectral Density x Power Spectral Density for x, y, and z for periodic solution 40 20 0 20 40 40 20 0 20 40 50 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.6 0.7 0.8 0.9 1 Power Spectral Density z Power Spectral Density y 0 50 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.6 0.7 0.8 0.9 1 Figure 18: Power Spectral Density of the x, y, and z signals for the Periodic Solution to the Lorentz Model with the Addition of White Noise 3.2 Chaotic The following are the plots that correspond to the chaotic solution to the Lorentz model with the addition of white noise. 3D Phase Plot of the chaotic solution to the Lorentz model 120 100 80 60 40 20 0 60 40 20 0 20 40 y 60 30 10 20 x 0 20 10 30 40 Figure 19: 3D Phase Plots of Chaotic Solution to the Lorentz Model with the Addition of White Noise z 15 Time histories for the x, y, and z directions for the chaotic solution 40 20 0 20 40 100 50 0 50 100 150 100 z 50 0 0 100 200 300 400 500 600 700 800 900 1000 y 0 100 200 300 400 500 600 700 800 900 1000 x 0 100 200 300 400 500 time (sec) 600 700 800 900 1000 Figure 20: Time History Responses of Chaotic Solution to the Lorentz Model with the Addition of White Noise Poincare maps for the chaotic solution in the xy, xz, and yz planes 150 100 y 50 0 30 100 50 0 50 100 30 150 100 z 50 0 60 20 10 0 x 10 20 30 40 z 20 10 0 x 10 20 30 40 40 20 0 y 20 40 60 Figure 21: Poincare Plots of Chaotic Solution to the Lorentz Model with the Addition of White Noise 16 Unlike the periodic case, where the addition of white noise was very evident, Figure 19, 20, and 21 illustrate that the chaotic solution with white noise doesnt look much different than the original chaotic solution. Mean Mean Square Moment (n=6) Moment (n=3) Moment (n=20) X -0.2058 108.6045 1.9192e7 0 1.4888e29 Y -2.5672 220.1332 1.4606e8 0 1.2909e32 Z 46.8334 2.3967e3 1.2617e8 0 7.9233e31 Table 4: Response Statistics for Chaotic Solution to the Lorentz System with the Addition of White Noise 4 cross correlation xy 15 10 5 0 5 1 0 1 2 3 5 0 5 x 10 Cross correlations for the chaotic solution 0 5 x 10 200 400 600 800 1000 1200 1400 1600 1800 2000 cross correlation yz 0 4 x 10 200 400 600 800 1000 1200 1400 1600 1800 2000 cross correlation xz 10 0 200 400 600 800 1000 1200 lag time (sec) 1400 1600 1800 2000 Figure 22: Cross Correlations of the x-y, y-z, and x-z signals for the Chaotic Solution to the Lorentz Model with the Addition of White Noise The response statistics for the chaotic solution with the addition of white noise are more to similar the original chaotic solution than what was observed in the periodic cases. The parameters are not exactly the same, but they are much closer to one another than what was observed in the periodic cases. The same is true for the cross correlations and the autocorrelations. The power spectral density plots are not quite the same and appear to be similar to the power spectral density plots for the periodic case with white noise. 17 Autocorrelation for the chaotic solution 1 autocorrelation x 0.5 0 0.5 50 1 autocorrelation y 0.5 0 0.5 50 1 autocorrelation z 0.95 0.9 0.85 0.8 50 40 30 20 10 0 10 lag time (sec) 20 30 40 50 40 30 20 10 0 10 20 30 40 50 40 30 20 10 0 10 20 30 40 50 Figure 23: Autocorrelations of the x, y, and z signals for the Chaotic Solution to the Lorentz Model with the Addition of White Noise Power Spectral Density x Power Spectral Density for x, y, and z for chaotic solution 40 20 0 20 40 40 20 0 20 40 100 50 0 50 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.6 0.7 0.8 0.9 1 Power Spectral Density z Power Spectral Density y 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.6 0.7 0.8 0.9 1 Figure 24: Power Spectral Density of the x, y, and z signals for the Chaotic Solution to the Lorentz Model with the Addition of White Noise 18 By comparing the cases when white noise is present and when it is absent, it can be seen that the white noise disrupts the solution to the Lorentz model signicantly. Figure 25 illustrates the x signal for the periodic solution with and without white noise, while Figure 26 shows the chaotic case. Not only are the phase plots, time history responses, and poincare plots altered, but all the response statistics are also affected. It is interesting to note that for both the chaotic and periodic solutions, the xy cross correlation has a similar spike at a lag time of 1000 for both the case with white noise and without white noise. There appears to be a similarity in the autocorrelation of the z signal as well. Periodic solution with and without white noise 15 periodic with white noise periodic 10 x 5 0 0 200 400 600 time(sec) 800 1000 1200 Figure 25: Periodic Solutions with and without White Noise Chaotic solution with and without white noise 40 chaotic chaotic with white noise 30 20 10 0 x 10 20 30 40 0 200 400 600 time (sec) 800 1000 1200 Figure 26: Chaotic Solutions with and without White Noise 19 4 Karhunen-Loeve Expansion For the chaotic model with white noise, an orthogonal basis set for the Karhunen-Loeve expansions was analyzed. With the system of equations for the Lorentz model being dened as the vector (r), the integral eigenvalue problem can be written as follows: Rij (t, t )j (t )dt = i (t) Rij is the ensemble averaged two-point correlation tensor and is dened as Rij (t, t ) = ri (t)rj (t ) (16) (15) This kernel Rij will always be symmetric. Due to this property, the following properties hold: There is a discrete set of solution to equation 15 Rij (t, t )n (t )dt = n n (t) j i (17) D The solution to equation 15 can be selected such that n are orthonormal: p (t)q (t)dt = pq i i (18) D The innite series of solutions, n , can be used to reconstruct the original uctuating quantity, r(t): ri (t) = n=1 an n (t) i (19) where the coefcients are dened as: an = D ui (t)i (n) (t)dt (20) The eigenvalues and eigenvectors of the two-point correlation tensor were then determined. Using the Karhunen-Loeve Expansion Matlab Script provided at the end of this report, reconstruction of the Lorentz model was performed. The kernel was constructed from 3000 realizations of the model, each excited with white noise. The white noise algorithm was seeded with the time to ensure independence between the blocks. Figure 27 illustrates the original x, y, and z values. The rst parameter to be evaluated was the percent mean square energy that each eigenvalue carries. From Figure 28 it can be seen that most of the energy is located in the rst few modes, with little energy is the higher modes. Next, the rst three eigenfunctions were examined (Figure 29). There are a total of 300 eigenfunctions. Using the rst set of eigenfunctions, reconstruction was done. This can be seen in Figure 30 for x, y, and z. Reconstruction was done for up to modes 5, 10, 45, 50, 55, 100, and 300 (Figures 31 - 37). At mode 300, all modes are used, and therefore, the reconstruction is exactly the same as the original data set. 20 original data 120 x y z 100 80 60 40 x, y, z 20 0 20 40 60 0 10 20 30 40 50 time 60 70 80 90 100 Figure 27: Original Data Set Percent of Mean Square Energy for the Eigenvalues 100 90 80 70 energy contribution, % 60 50 40 30 20 10 0 0 10 10 1 10 mode 2 Figure 28: Percent of Mean Square Energy for the Eigenvalues 21 First Eigenfunction 0.15 mode shape, 1(t) 0.1 0.05 0 0.05 0.2 mode shape, 2(t) i 0.1 0 2(t) x 2(t) y 2 z (t) 0 0.1 0.2 0.3 0.4 0.5 0.6 Second Eigenfunction 0.7 0.8 0.9 1 1(t) x 1(t) y 1 z (t) i 0.1 0.2 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 Third Eigenfunction 0.7 0.8 0.9 3(t) x 3(t) y 3(t) z 1 mode shape, 3(t) i 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 time 0.6 0.7 0.8 0.9 0.2 1 Figure 29: First Three Eigenfunctions for the x, y, and z directions 40 reconstructed x original x 30 60 reconstructed y original y 120 reconstructed z original z 40 100 20 20 10 0 60 x y z 0 50 time 100 80 0 20 10 40 20 40 20 30 0 50 time 100 60 0 0 50 time 100 Figure 30: Reconstruction up to Mode 1 22 40 reconstructed x original x 30 60 reconstructed y original y 120 reconstructed z original z 100 40 20 20 10 0 x y z 80 60 0 20 10 40 20 20 40 0 30 0 50 time 100 60 0 50 time 100 20 0 50 time 100 Figure 31: Reconstruction up to Mode 5 40 reconstructed x original x 30 60 reconstructed y original y 120 reconstructed z original z 100 40 20 20 10 0 x y z 80 60 0 20 10 40 20 20 40 0 30 0 50 time 100 60 0 50 time 100 20 0 50 time 100 Figure 32: Reconstruction up to Mode 10 23 40 reconstructed x original x 30 60 reconstructed y original y 120 reconstructed z original z 100 40 20 20 10 0 x y z 80 60 0 20 10 40 20 20 40 0 30 0 50 time 100 60 0 50 time 100 20 0 50 time 100 Figure 33: Reconstruction up to Mode 45 40 reconstructed x original x 30 60 reconstructed y original y 120 reconstructed z original z 100 40 20 20 10 0 x y z 80 60 0 20 10 40 20 20 40 0 30 0 50 time 100 60 0 50 time 100 20 0 50 time 100 Figure 34: Reconstruction up to Mode 50 24 40 reconstructed x original x 30 60 reconstructed y original y 120 reconstructed z original z 100 40 20 20 10 0 x y z 80 60 0 20 10 40 20 20 40 0 30 0 50 time 100 60 0 50 time 100 20 0 50 time 100 Figure 35: Reconstruction up to Mode 55 40 reconstructed x original x 30 60 reconstructed y original y 120 reconstructed z original z 40 100 20 20 10 0 60 x y z 0 50 time 100 80 0 20 10 40 20 40 20 30 0 50 time 100 60 0 0 50 time 100 Figure 36: Reconstruction up to Mode 100 25 30 reconstructed x original x 50 reconstructed y original y 40 100 reconstructed z original z 20 30 80 20 10 10 60 0 0 40 x y 10 10 20 20 30 20 40 0 30 0 50 time 100 50 z 0 50 time 100 20 0 50 time 100 Figure 37: Reconstruction up to Mode 300 An interesting phenomenon that occurred was that while the y and z directions were reconstructed fairly accurately at lower mode numbers, the x direction wasnt. It wasnt until mode 50 that the x direction was reconstructed in a manner that was close to the original data. Similarly, 50 is also the value selected for the constant r in the Lorentz model. This is the constant which determines if the solution is periodic or chaotic. It was then changed to 40 to determine if the reconstruction in any way depended on this value. Figures 38 - 40 show that before reaching mode 40, the x direction is not reconstructed well, while after mode 40, it is reconstructed closer to the original value. Other values were selected for r and the same phenomenon occurred. Therefore, it was concluded that the constant r plays a role in the reconstruction of the Lorentz model. The values of the coefcients were then plotted against the mode number and it showed that there was energy located at these higher modes. Figures 41 and 42 show the coefcients for the case with r = 40 and r = 50 respectively. 26 30 reconstructed x original x 25 40 reconstructed y original y 30 90 reconstructed z original z 80 20 20 15 70 60 10 10 50 5 0 x y z 40 0 10 30 5 20 10 30 20 10 0 50 time 100 0 0 15 20 0 50 time 100 40 50 time 100 Figure 38: Reconstruction up to Mode 30 with r = 40 30 reconstructed x original x 25 50 reconstructed y original y 40 90 reconstructed z original z 80 20 30 70 15 20 10 10 5 60 50 40 x y 0 0 10 5 20 20 30 10 z 30 40 0 50 time 100 10 15 0 20 0 50 time 100 10 0 50 time 100 Figure 39: Reconstruction up to Mode 40 with r = 40 27 30 reconstructed x original x 25 50 reconstructed y original y 40 90 reconstructed z original z 80 20 30 70 15 20 10 10 5 60 50 40 x y 0 0 10 5 20 20 30 10 z 30 40 0 50 time 100 10 15 0 20 0 50 time 100 10 0 50 time 100 Figure 40: Reconstruction up to Mode 50 with r = 40 8 x 10 4 7 6 5 4 3 2 1 0 0 10 20 30 40 50 60 70 80 90 100 Figure 41: Coefcients for r = 40 28 1.2 x 10 3 1 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 70 80 90 100 Figure 42: Coefcients for r = 50 What this reconstruction illustrated was that there is a large amount of energy surrounding the mode number that is equal to the chaotic parameter r. This large amount of energy affects the reconstruction in the x direction signicantly more than the y and z directions. 29 5 Matlab Script This is the Matlab script that was used to create the gures. % x = -ax + ay % y = bx - y - xz % z = -cz + xy close all clear clc step=0.01; t=1000; x=[0,1,0]; a=[10,15,8/3]; %a for chaotic [10,50,8/3], a for periodic [10,15,8/3] %without white noise %r=zeros(3,t+1); %r(:,1) = x; %for i=1:t %r(:,i+1) = r(:,i) + step.*[-a(1).*r(1,i) + a(1).*r(2,i); a(2).*r(1,i) - (r(2,i) + r(1,i).*r(3,i));(-a(3))*r(3,i) + r(1,i).*r(2,i)]; %end; %with white noise % initialize rand(state,sum(100*clock)); x = 10*(rand(size(t))-0.5); r = zeros(3,t); r(:,1) = x; % Solve for r for i = 1:t-1, r(:,i+1) = r(:,i) + 0.5*rand(size(t)) + step.*[-a(1).*r(1,i) + a(1).*r(2,i); a(2).*r(1,i) - (r(2,i) + r(1,i).*r(3,i)); (-a(3))*r(3,i) + r(1,i).*r(2,i)]; end figure plot3(r(1,:),r(2,:),r(3,:)) title(3D Phase Plot of the periodic solution to the Lorentz model) xlabel(x) ylabel(y) zlabel(z) grid on figure subplot(3,1,1), plot(r(1,:)) title(Time histories for the x, y, and z directions for the periodic s...

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ME 639SIMULATION OF CHAOS, PROJECT-1Spring 2008Select a nonlinear deterministic dynamical system for detail analysis. 1. Study the equilibrium states and periodic orbit solutions. (Analyze the stability of the equilibrium states.) 2. Perform numerical
Clarkson - ME - 326
ME 326G. Ahmadi Compressible Flow Regimes Thermodynamics Mach Number Speed of Sound Isentropic Flows with Area Change Variations with Mach number Shock Waves Nozzle and Diffusers Flows with Friction Flows with Heat TransferME 326G. AhmadiFlows with F
Clarkson - ME - 326
ME 326G. Ahmadi Compressible Flow Regimes Thermodynamics Mach Number Speed of Sound Isentropic Flows with Area Change Variations with Mach number Shock Waves Nozzle and Diffusers Flows with Friction Flows with Heat TransferME 326G. AhmadiME 326G. Ah
Clarkson - ME - 326
ME 326G. Ahmadi Compressible Flow Regimes Thermodynamics Mach Number Speed of Sound Isentropic Flows with Area Change Variations with Mach number Shock Waves Nozzle and Diffusers Flows with Friction Flows with Heat TransferME 326G. AhmadiEnergy Equat
Clarkson - ME - 326
ME 326G. Ahmadi Compressible Flow Regimes Thermodynamics Mach Number Speed of Sound Compressible Flows with Area Change Variations with Mach number Shock Waves Nozzle and Diffusers Flows with Friction Flows with Heat TransferME 326G. AhmadiMach Numbe
Clarkson - ME - 326
ME 326G. AhmadiOutline ME 326Stream Function Vorticity Velocity Potential Irrotational Flows Flow NetG. AhmadiIncompressible Fluid Cartesian Coordinates Define Stream Functionr u v V = + =0 x y u= y v= xG. AhmadiME 326 2 u v 2 + = 0 x y x y x
Clarkson - ME - 326
ME 326G. AhmadiME 326Outline Irrotational Flows Simple Flows Source/Sink Flows Vortex Flows Doublet Flows Flow Superposition Flow over a CylinderG. AhmadiEuler Equation ContinuitydV = g p cfw_ cfw_ dt Body Force Pr essure ForceGiven a Irrotational
Clarkson - ME - 326
ME 326G. AhmadiOutline ME 326Flows Past Immersed Bodies Boundary Layer Flows (laminar) Blasius Solution Momentum Integral Method Turbulent Boundary Layer FlowsG. AhmadiUoUol / l &lt; 1G. AhmadiLaminar Boundary LayerME 326Boundary Layer Thicknes
Clarkson - ME - 326
ME 326G. AhmadiME 326G. AhmadiOutline ME 326Forces and Toques Drag of Spheres Drag of Cylinders Drag Coefficient for 2D Objects Drag Coefficient for 3D Objects Lift Force for an AirfoilG. AhmadiME 326G. AhmadiDrag Force CD = 1 2 V A 2C D = C D
Clarkson - ME - 326
ME 326G. AhmadiOutline Cartesian Coordinates Cylindrical Axial Flows Cylindrical Rotating FlowsME 326G. AhmadiIncompressible Fluidu u u u p 2u 2u 2u ( + u +v + w ) = g x - + ( 2 + 2 + 2 ) t x y z x x y z 2v 2v 2v p v v v v + ( 2 + 2 + 2 ) + w ) = g
Clarkson - ME - 326
ME 326G. AhmadiOutline Conservation of Mass Balance of Momentum Navier-Stokes EquationME 326G. AhmadiIncompressible Fluid Cartesian Coordinatesr u v w + = V = 0 + x y zPolar Coordinatesr 1 ( rv r ) v V = [ + ]=0 r r ME 326G. Ahmadiy yy + yy y
Clarkson - ME - 326
ME 326G. Ahmadi4Constant Coefficient Equations 4Euler Equation 4Total Differentials 4Separable EquationsME 326G. AhmadiConstant Coefficient Equationsd y dy + 2 - 8y = 0 2 dx dxSolution2Boundary Conditionsx = 0, y = 1 x = , y = 0y = AeME 326mx
Clarkson - ME - 326
ME 326-Fluid MechanicsG. AhmadiME 326-Fluid MechanicsG. AhmadiME 326-Fluid MechanicsG. AhmadiME 326-Fluid MechanicsG. AhmadiME 326-Fluid MechanicsG. AhmadiPumps Compressors Wind TurbinesME 326-Fluid MechanicsG. AhmadiME 326-Fluid MechanicsG.
Clarkson - ME - 437
ME 437/537-ParticleG. AhmadiAerosols are suspension of solid or liquid particles in a gas. Dust, smoke, mists, fog, haze, and smog are common aerosols. Aerosol particles are found in different shapes.ME 437/537-ParticleG. AhmadiEquivalent area diamet
Clarkson - ME - 437
ME 437/ME 537PARTICLE TRANSPORT, DEPOSITION AND REMOVALGoodarz Ahmadi Department of Mechanical and Aeronautical Engineering Clarkson University Potsdam, NY 13699-57251Air pollution and smog.Particle trajectories in a hot gas filtration vessel.2Samp
Clarkson - ME - 326
Clarkson - ME - 326
ME 326FLOW SIMULATION, PROJECT-1Spring 2004Part 1: Duct Flow Develop a grid and analyze the developing flow in a channel, which is 2 cm wide and 11 cm long. a) For an air velocity of 0.1 m/s, compare the velocity profile with the exact laminar flow sol
Clarkson - ME - 326
ME326 - INTERMEDIATE FLUID MECHANICS Spring 2004 Textbook: Fluid Mechanics, F.M. White, McGraw-Hill Book Co., Special Edition. (Part of 5th ) Instructors: G. Ahmadi (CAMP 267, 268-2322) Office Hours: MTW 2:20 - 4:00 p.m. Course Site: http:/www.clarkson.ed
UCCS - ECE - 3610
Set #1Due Monday February 2, 2004Make note of the following:ECE 3610 Homework Problems Papers are due at the start of class Write only on one side of the paper Please try if possible to start each new problem on a clean sheet of paper Use engineering
UCCS - ECE - 3610
ECE 3610 Engineering Probability and StatisticsSpring Semester 2004Instructor:Mark Wickert Office: EB-226 wickert@eas.uccs.edu http:/eceweb.uccs.edu/wickert/ece3610/Phone: 262-3500 Fax: 262-3589Office Hrs: Required Texts: Study Guides Optional Softwa
University of Scranton - CS - 341
Project Name Use Case Specification: Create AccountName Brief Description Actor(s) Flow of Events Basic Flow This use case starts when the User accesses the sign in feature of the system. 1. The system prompts the User for his/her username and password.
UVA - FR - 245
Timber Harvesting - 5Logging System Planning The successful implementation of any specialized logging system is dependent upon successful planning. With a specialized logging system, it is possible to do a more efficient job under particular conditions.
Northeastern University - COM - 1355
Copyright 2000 William D ClingerModified March 2002 by William D Clinger*Outline of lecture. pitfalls of benchmarking importance of code improvement (optimization) examining the code generated by a compiler disassembly delayed branch instructions
George Mason - M - 322
Homework # 7Hand-in (Due Wednesday, April 22 - 3pm): Sec. 6.1 - 9a, 11, 18 Sec. 6.2 - 6 Additional Problems: Sec. 6.1 - 1, 2, 5, 8, 10 Sec. 6.2 - 1, 7, 9, 14
George Mason - M - 322
Homework # 6Hand-in (Due Wednesday, April 1 - 3pm): Sec. 5.1 - 14 Sec. 5.2 - 18a Sec. 5.4 - 17 (Hint: Consider the charactersitic polynomial of A), 18a, b Additional Problems: Sec. 5.1 - 1, 2b,c, 3a,b, 8, 11, 17 Sec. 5.2 - 1, 2a,b,d, 3a,
George Mason - M - 322
Homework # 4Hand-in (Due Friday, March 6 - 3pm): Let T be the map from R^2 to itself, given by perpindicular projection onto the line y = mx. Find the matrix of T with respect to the standard basis. (Hint: Do this by first finding a basis of R^2 such
George Mason - M - 322
Homework # 5Hand-in (Due Monday, March 23 - 3pm): Sec. 4.2 - 8, 25 Sec. 4.3 - 12, 14 Additional Problems: Sec. 4.2 - 1, 5, 6, 26, 27 Sec. 4.3 - 1, 10, 15, 20
George Mason - M - 322
Homework # 3 There is no hand-in homework. Additional Problems: Sec. 2.4 - 1,3,6,14,16,22 Sec. 2.5 - 1,2,5,6a,b,c,9, 10
George Mason - M - 322
Homework # 2Hand-in (Due Monday, Feb. 16 at 3pm): Sec. 1.6 - 15, 34a Sec. 2.1 - 14a, 15 Additional Problems: Sec. 1.6 - 1, 2,3,5, 6, 10a, 11, 12, 13,14,17 Sec. 2.1 - 1,2,3,5,9,10,16,17,18
George Mason - M - 322
Homework # 1Hand-in (Due Monday, Feb. 2 at 3pm): Sec. 1.2 - 18 Sec. 1.3 - 12 Sec. 1.4 - 10 Sec. 1.5 - 14Additional Problems: Sec. 1.2 - 1, 11, 12, 13,14,17, 21 Sec. 1.3 - 1, 2, 5, 8, 9,19, 20 Sec. 1.4 - 1,2,7,8 Sec. 1.5 - 1, 2,4,6,8, 11,
Bellarmine - CS - 699810
Venti: a new approach to archival storageSean Quinlan and Sean Dorward Bell Labs, Lucent Technologies AbstractThis paper describes a network storage system, called Venti, intended for archival data. In this system, a unique hash of a blocks contents act
Grinnell College - CS - 151
Today in 151: Symbols and ListsOverview:* What is a symbol?* What is a list?* Lab* ReflectionAdministrivia:* Eschew obfuscation* Any thoughts on the Eboards?* Do you mind visitors from other classes? (Other 151, 153, 195)* Homework 1 due + Th
Langston - CS - 699810
NFS Version 3 Design and ImplementationBrian Pawlowski Chet Juszczak Peter Staubach Carl Smith Diane Lebel David Hitz AbstractThis paper describes a new version of the Network File System (NFS) that supports access to files larger than 4GB and increases
UNL - PSYCH - 941
Power, Effect Size &amp; Sample Size* r? ? power .20 .30 .40 .50 .60 .70 .80 .90 .10 124 208 .15 32 93 .20 21 53 74 95 143 167 191 255 .25 15 34 47 60 90 105 120 160 .30 14 24 33 42 62 72 82 109 .35 13 18 24 30 45 52 59 78 .40 11 14 19 23 34 39 44 58 .45 9 11
CSB-SJU - BIOL - 106
Grading Rubric for Class Leadername_ date _General _ _ _ _ _ _ _ _ Arrives five minutes before class (1 pt) Music selected &amp; started (1 pt) Greeting of the day selected on overhead (1 pt) Ready as time-keeper; i.e., has gong (1 pt) Checks that the atten
CSB-SJU - BIOL - 106
Plants and Human Affairs Biology 106name _Plant Portrait1. Scientific Name: _ 2. Common Name: _ 3. Family name (technical):_ 4. Family Name (common): _ 5. Growth habit: tree 6. Part(s) of plant used: 7. Description of economic uses: shrub herb vine8.
Syracuse - CSE - 681
Architecture of the Ultimate Extensible Distributed SystemJim Fawcett CSE681 Software Modeling and Analysis Fall 2006Your Assignment Your supervisor just handed you a spec forimplementation of: Distributed system with universal connectability using
Salisbury - ASTRO - 108
Stellar DeathLecture ElevenFate of StarsSalisbury UniversityThe Fate of StarsIf you recall from our discussion about the birth of modern astronomy, Tycho observed a bright new star in the heavens that lasted for Luminosity a few weeks in 1572 A.D. th
Harvey Mudd College - PHYSICS - 516
Genetic Algorithms and the Traveling Salesman ProblembyKylie Bryant Arthur Benjamin, AdvisorAdvisor:Second Reader: (Lisette de Pillis)December 2000 Department of MathematicsAbstractGenetic Algorithms and the Traveling Salesman Problem by Kylie Brya
Salisbury - ASTRO - 108
Stellar FormationThe Fate of the SunThe Main SequenceAt the core, main-sequence stars are all very much alike. All main-sequence stars convert hydrogen into helium by the nuclear fusion processes of PP Chain (lower mass stars) or the CNO Cycle. This is
Salisbury - ASTRO - 108
Stellar FormationLecture NineThe Stellar BirthSalisbury UniversityThe Formation of StarsStars are born in regions of high-density interstellar clouds. (I.S.M.)I.S.M. (Inter-Stellar Medium)Two phases: Gaseous 99% of ISM 1. Largely Hydrogen (~73%) 2.
Salisbury - ASTRO - 108
The StarsLecture EightThe Heavenly StarsSalisbury UniversityThe Properties of StarsEven with the naked-eye, the night sky is strung with bright pinpoints of light we call &quot;stars.&quot; Along with a dark night sky the unaided eye can easily pick out a few
Salisbury - ASTRO - 108
The SunLecture SevenThe SunSalisbury UniversityThe SunThe PhotosphereThe visible surface of the sun. Thin layer of gas (less than 500km deep) from which we receive the majority of the Suns light. Average surface temperature ~ 6000K The photosphere i
Salisbury - ASTRO - 108
Atoms &amp; StarlightLecture SixAtoms &amp; StarlightSalisbury UniversityAtoms and StarlightWe have already concluded that stars are very much distant objects. The great gulf of distance between the stars and our own solar system makes the study of stars, th
Salisbury - ASTRO - 108
Light &amp; TelescopesLecture FiveLight &amp; TelescopesSalisbury UniversityThe most exciting phrase to hear in science, the one that heralds the most discoveries, is not &quot;Eureka!,&quot; but &quot;That's funny.&quot; - Isaac AsimovThe Electromagnetic SpectrumIn the 1860s,
Salisbury - ASTRO - 108
Ancient AstronomyLecture FourAncient AstronomySalisbury University&quot;Eppur Si Muove&quot; -&quot;(And, yet it moves!&quot;)Some Early Cosmologies: &quot;Models of the Universe&quot; - A History BriefPlato &amp; Aristotle: (around 350 B.C.)Plato remains the best known of all the
Salisbury - ASTRO - 108
Patterns in the SkyLecture TwoPatterns in the SkySalisbury UniversityThe Sky: .Naked Eye Stars: &quot;naked eye&quot; = without a telescope ~ 6000 stars ~ 88 Constellations: - mesopotamia &gt;1 1 - latin &amp; modern &lt; 2 2In the past these were taken very seriously
Salisbury - ASTRO - 108
Tides &amp; EclipsesLecture ThreeTides &amp; EclipsesSalisbury UniversityTides &amp; EclipsesTIDES: Stated simply, tides are the vertical movement of water, specifically the alternate rise (flood) and fall (ebb) of water in the ocean. The word tide derives from
Salisbury - ASTRO - 108
IntroductionLecture OneThe Beckoning SkySalisbury UniversityIntroductionFirst Quiz: Are there more stars in the visible universe than grains of sand on all the beaches of Earth? Heavy elements in your body formed long ago deep inside stars? Is the u
Southern Illinois University Edwardsville - ECE - 482
FINALCOML: H-5/7/10/15/25, Q-10/15/25 IND: H-10/15/25, Q-20/25PALCE16V8 FamilyEE CMOS 20-Pin Universal Programmable Array LogicDISTINCTIVE CHARACTERISTICSs Pin and function compatible with all 20-pin GAL devices s Electrically erasable CMOS technolog
USC - A - 100
ASTRONOMY 100 Dr. Werner Dppen 2 pm, 12 December 1997 FINAL EXAM _ Exam Number Name (Please Print)1.One of the principal lessons of this course has been that _ . a) b) c) d) e) we are at the center of the Universe our Sun is a very special star the plan
East Los Angeles College - COURSE - 200
Unsupervised LearningZoubin GhahramaniGatsby Computational Neuroscience Unit University College London, UK zoubin@gatsby.ucl.ac.uk http:/www.gatsby.ucl.ac.uk/~zoubinSeptember 16, 2004Abstract We give a tutorial and overview of the eld of unsupervised
University of Florida - STAT - 6127
Multivariate Relationships Goal: Show a causal relationship between two variables (X Y) Elements of a cause-and-effect relationship: Association between variables (based on methods weve covered this semester) Correct time order (X occurs before Y) Elimi
University of Toronto - CS - 236
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University of Toronto - CS - 236
%!PS-Adobe-2.0 %Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %Title: midterm.dvi %Pages: 7 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips midterm %DVIPSParameters: dpi=300, comments removed %DVIPSSour
University of Toronto - CS - 236
%!PS-Adobe-2.0 %Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %Title: midterm.dvi %Pages: 9 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips midterm -o %DVIPSParameters: dpi=300, comments removed %DVIPSS
Wayne State University - CHM - 6440
SyllabusThe course will consist of lectures and hands-on computational labs. There will be one midterm (worth 21%), 5 on-line quizzes (15%), 5 computational assignments (30%), 3 reviews of papers from the scientific literature (9%) and a major computatio
Wayne State University - CHM - 6440