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.
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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
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Next, the time history responses were determined (Figure 2) as well as the Poincare plots (Figure 3).
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Figure 1: 3D Phase Plots of Periodic Solution to the Lorentz Model
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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
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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
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Figure 3: Poincare Plots of Periodic Solution to the Lorentz Model
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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.
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Cross correlations for the periodic solution
cross correlation xy
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Figure 4: Cross Correlations of the x-y, y-z, and x-z signals for the Periodic Solution to the Lorentz Model
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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
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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
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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
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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.
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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
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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
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Figure 9: Poincare Plots for the Chaotic Solution to the Lorentz Model
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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.
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cross correlation xy
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Cross correlations for the chaotic solution
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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
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Figure 11: Autocorrelations of the x, y, and z signals for the Chaotic Solution to the Lorentz Model
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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
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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.
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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
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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
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Figure 14: Time History Responses of Periodic Solution to the Lorentz Model with the Addition of White Noise
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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.
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cross correlation xy
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Cross correlations for the periodic solution
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cross correlation yz
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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
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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
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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
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Figure 19: 3D Phase Plots of Chaotic Solution to the Lorentz Model with the Addition of White Noise
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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
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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
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Figure 21: Poincare Plots of Chaotic Solution to the Lorentz Model with the Addition of White Noise
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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
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cross correlation xy
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Cross correlations for the chaotic solution
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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.
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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
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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
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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
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Figure 25: Periodic Solutions with and without White Noise
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Figure 26: Chaotic Solutions with and without White Noise
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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)
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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
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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.
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Figure 28: Percent of Mean Square Energy for the Eigenvalues
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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)
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mode shape, 3(t) i
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Figure 29: First Three Eigenfunctions for the x, y, and z directions
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Figure 30: Reconstruction up to Mode 1
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Figure 31: Reconstruction up to Mode 5
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Figure 32: Reconstruction up to Mode 10
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Figure 33: Reconstruction up to Mode 45
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Figure 34: Reconstruction up to Mode 50
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Figure 35: Reconstruction up to Mode 55
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Figure 36: Reconstruction up to Mode 100
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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.
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30 reconstructed x original x 25
40 reconstructed y original y 30
90 reconstructed z original z 80
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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
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Figure 39: Reconstruction up to Mode 40 with r = 40
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30 reconstructed x original x 25
50 reconstructed y original y 40
90 reconstructed z original z 80
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Figure 40: Reconstruction up to Mode 50 with r = 40
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Figure 41: Coefcients for r = 40
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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.
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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...