MCE 591
Assignment #5
Fall 2013
Due Date: December 5, 2013
The main purpose of this last assignment is to make sure you all get some
practice with the fundamental nonlinear data analysis procedures we have been
and will be talking about in class.
You can choose your own data from online sources or from what I will post for
the project data. However, for the purposes of this assignment I can suggest
using periodic/chaotic time series generated for the Assignment # 4. If you use
this assignment data, use both continuously sampled (say 32 points per each
forcing/fundamental period) or Poncaré section data (e.g., for Duffing oscillator
sample once per forcing period, and for Lorenz sample when z crosses zero going
from positive to negative value—can use events option in Matlab ode routines
as shown bellow). If you do so, be sure to also add a reasonable amount of
noise to it, say on the order of 0.5 or 1% of the signals root-mean-square (RMS)
value.
function [t, y, te, ye, ie] = lorenz_poincare
options = odeset('Events', @events);
[t, y, te, ye] = ode45(@lorenz, [0 5000], ones(3,1), options);
figure
plot(ye(:,1), ye(:,2), '.', 'markersize',3)
xlabel('x')
ylabel('y')
title('Poincare Section for z = 0 & z'' < 0')
function [value,isterminal,direction] = events(t,y)
% Locate the time when z passes through zero in a
% decreasing direction and record x and y.
value = y(3);
% Detect z crossing 0
isterminal = 0;
% Do not stop the integration
direction = -1;
% Negative direction only
end
function dy = lorenz(t, y)
% lorenz equations for a chaotic orbit
dy = [-8*y(1)/3 + y(2)*y(3);
-10*(y(2) - y(3));
-y(1)*y(2) + 28*y(2) - y(3)];
end
end
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Exercise 1: Given a periodic solution (continuously sampled), show that D2 ≈
1. If you happen to have a quasiperiodic time series (alternatively it easy to
generate on your own), also show that D2 ≈ n, where n is the number of
incommensurate frequencies. Discuss how closely you are able to get to the
expected results.
Remember that the estimation of D2 requires that you choose an embedding
dimension d (false nearest neighbors), fix a delay m (average mutual information), and generate C2 (correlation sum). All estimates should be presented in
b 2 ± E, where E is the standard error. You should also always
the form, D2 = D
look at PSD, S(f ), and autocorrelation, R(τ ).
Exercise 2: Take a chaotic, or otherwise “complex” time series, and estimate
D2 for both continuously sampled and Poicaré section data. Discuss the quality
of your result. Does the data appear to be “fractal?” Over what range of length
scales does your data appear to have a scaling region? Do temporal correlations
have any effect on the results?
Exercise 3: Using the same time series of Exercise 2, find the largest Lyapunov
exponent of a candidate chaotic time series. Can you conclude in fact that the
system is chaotic?
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