Computing correlation between two time series: confusion regarding nonlinear relationship and nonlinear data

Question

I am trying to understand if correlation can be computed between two time series generated from two different initial conditions for chaotic dynamical systems. In general, correlation is applicable for linear dependence found between two random variables. If correlated then value returned is 1, otherwise zero. For chaotic dynamical systems, if the initial condition is changed slightly we get two completely different time series. Can we say that the two time series are uncorrelated with each other? In other words, does sensitivity to initial conditions imply no correlation? I tried to test this by calculating the correlation between two time series generated by slightly changing the initial conditions for a chaotic logistic map. I got the value 1 which means that the two time series, x and y are correlated. Moreover, correlation is applicable for linear data whereas chaotic time series is nonlinear. Or is it that linear relationship and linear data are two different stuff?

To summarize my questions are: (a) Can I apply correlation for chaotic nonlinear dynamical systems to find if the two chaotic time series are related to each other or not? (b) Are chaotic time series linearly dependent on each other? I mean is linear dependence concept applicable here?

Below is the code in Matlab where I applied correlation for the chaotic logistic map

clear all
% The logistics map is a classic example of transition from stable to chaotic behavior as a single parameter changes value.
%  x(n+1) = r*x(n)*(1-x(n)) as r=4 generates chaotic time series.

M =500;
x(1) = 0.51; % initial condition (can be anything from 0 to 1)
r = 4;
for i = 2:M % iterate
    x(i) = r*x(i-1)*(1-x(i-1));
end

M =500;
y(1) = 0.52; % initial condition (can be anything from 0 to 1)
r = 4;
for i = 2:M % iterate
    y(i) = r*y(i-1)*(1-y(i-1));
end

subplot(1,2,1)
plot(1:length(x),x,'r')
subplot(1,2,2)
plot(1:length(y),y,'k')

>> R = corrcoef(x,y)

R =

    1.0000   -0.0359
   -0.0359    1.0000

Answer 1

First things first:

I got the value 1 which means that the two time series, $x$ and $y$ are correlated

No, you actually didn't. The 1's in R's diagonal are the correlations of $x$ with $x$ and of $y$ with $y$ - this just expresses that a variable is perfectly correlated with itself.

The correlation between $x$ and $y$ you measured is the small value -0.0359.

(a) Can I apply correlation for chaotic nonlinear dynamical systems to find if the two chaotic time series are related to each other or not?

Yes, you can. People often investigate the decay of these correlations (e.g., with time). You might want to check this paper.

(b) Are chaotic time series linearly dependent on each other?

For random initial conditions $x$ and $y$ won't correlate.

I mean is linear dependence concept applicable here?

The correlation might be linear, if you're using Pearson's (as corrcoef does), but that does not mean it's only applicable to linear stuff -- after all, you're precisely trying to assess to which degree those two series are linearly correlated.