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Name: Fan Zhang EAS6490-HW1 August 28, 2012

1 Solutions of Lorenz Attractor

After increasing the initial condition of X slightly (0.2), both X and Y diverge after sometime.

Although we have exact Y conditions, it still diverges as X initial condition changes.This is because the systme is non-linear.

Small changes in X initial conditions result in big differences in the future state.

Variable Y stuidied in isolation of X appears to develop random fluctuations.

The system is chaotic.

Fig. 1.Time series of X, Y and Z. Blue: initial conditions (X = 10, Y = 10, Z = 10); red: smallchanges on X initial condition (X = 10 + 0.2, Y = 10, Z = 10).

2 Phase space diagram

The 3-D phase space diagram of X, Y and Z shows a clear pattern (butterfly pattern),which demonstrates that the trajectory is predictable.

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If a small error introduced into X initial condtion, the butterfly pattern remains: thereare small differences between trajectories with different initial conditions. We can still predictthe genral behavior of the system.

However, in the 2-D phase space diagram of X and Z, it seems that multiple trajec-tories pass through the same point (in fact, in 3-D, trajectory never intersect). The visible

intersection of trajectories indicates apparent randomness of the system, due to the lack ofobservations of Y.

In this context, X and Z are signal, and Y is the source of noise.

Fig. 2.Upper: The 3-D phase space diagram of X, Y and Z; lower: the 2-D phase spacediagram of X and Z.

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Name: Fan Zhang EAS6490-HW1 August 28, 2012

3 Probability Density Function (PDF), Joint and Con-

ditional PDF

3.1 PDF for X and Z

The Probability Distribution Function (PDF) of X and Z offer a global view of statisticsof X and Z. We could tell the occurence probability of X/Z at a certain value.

Fig. 3.Probability Distribution Function (PDF) of X and Z

3.2 Joint PDF of XZ

The Joint PDF of X and Z tells the probability of the system at certain values of X andZ. It is shown (Fig. 4) that the Joint PDF of X and Z has higher intensity in the center,from which we could infer that trajectories slow down in the center. Joint PDF adds anotherconstraint to the system.

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Name: Fan Zhang EAS6490-HW1 August 28, 2012

Fig. 4.The Joint PDF of X and Z.

3.3 Joint PDF of XZ with Y conditions

a) The Conditional Joint PDF of X and Z tells the occurence probability of X and Z oncertain values of Y. When Y has certain values, the Joint PDF of X and Z are constrainedto a specific pattern, which is part of the original Joint PDF of X and Z.

b) Compared to the 2-D phase spcae diagram of X and Z, the Joint PDF not only

shows the potential values and pattern of X and Z, but also reveals the possibility of X andZ at certain values. The Joint PDF provides more insightful information about the system.

Fig. 5.Conditional Joint PDF of X and Z on1 < Y < 1 and 8 < Y < 10.

P.S. There is probably a minor mistake in the matlab script ( EXAMPLE lorenz.m)from the class website, in which the line 115: tmp1 = [tmp1; xout2(in, 2)]; should be tmp1 =

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Name: Fan Zhang EAS6490-HW1 August 28, 2012

[tmp1; xout2(in, 1)];. This has impact on computing the Conditional Joint PDF of X and Z;consequently the appearence of the figures.

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