Homework 1:

1) Create different two dimensional phase portraits of a time series of the Henon map (Example 3.2) using delay times tau = 1, 2, 3, ... (here, the sampling time is 1). Which picture gives the clearest information?  Rewrite the map in delay coordinates with unit delay.

2) Create a scalar time series of flow data numerically by integration of the Lorenz system described by the set of differential equations:
dx/dt = sigma*(y-x) 
dy/dt = r*x - y - y*z
dz/dt = -b*z +x*y
with parameters sigma = 10, r = 28, b = 8/3. Use the Runge-Kutta(**) numerical integration technique to solve these set of equations to obtain one scalar observable (e.g., x).
Plot the reconstructed attractor in 2-D delay coordinates with different time lags. Convince yourself (by the plots) that the reconstruction is best when the lag is about one-quarter of the mean cycle time (note that if a time series has a strongly periodic component, an optimal time delay is approximately a quarter of the cycle time).
Compute the autocorrelation function(as required in problem 4 below) as a function of tau for this time series and confirm this impression.

3) Numerically integrate the Lorenz system and perform a Poincare section. Record (y,z) every time the x-coordinate equals zero and it's derivative is negative.

4) Explain the concept of Auto Mutual Information. Why is it a better measure(instead of the first minima of the Autocorrelation function) of finding the optimal time-delay for embedding a time series?
Estimate the optimal time delay using the first minima of the autocorrelation function from the following time series:
- sin(2*pi*100*t)
- Lorenz series generated in problem 2
- Gaussian noise, with 0 mean and unit variance (Caution: does phase space reconstruction of such a series hold any meaning if the series was generated by an IID stochastic process?)
Also plot their frequency spectra. Are these plots what you expect of deterministic, chaotic and stochastic signals respectively?


** A review of the Runge-Kutta method for numerical integration can be found at:
http://mathworld.wolfram.com/Runge-KuttaMethod.html