The are discrete variables, i.e. for each is or ...
... aren't they?...
First, a warm "Hello to all!"...and excuse my English, pls
I need very much to calculate the following limit (it's a Lyapunov exponent):
After some numerical estimations, it seems to me that L is a periodic function for r<10 and a linear function for r>10. In both casese I obtained L < r*ln(2).
But I need very much something more exact...
Thank you all!
For the ease of reading, I rewrite the original limit in Latex:
Thank you very much again chisigma for your help!
As they are coordinates of points from an attractor of a chaotic dynamical system which seems to me to cover very dense a rectangle (and I made NIST tests for the randomness and the results were very good), I think that you can assume that these are uniformly distributed.
The proposed problem is very interesting... in order to avoid misundestanding however I have again a question [I do hope the last question...] : what is r?... an integer o a real number?... and if it is a real number, what is its range?...
Very well!... we can start writing the identity...
The first term in (1) doesn't depends from k and the second term [containing the 'summation'...] for the central limit theorem has as limit the expected value of where y is uniformely distributed in , so that is...
The next step is of course the computation of the integral in (2) and it seems to me that that requires some little effort ...
Wooow...you're a fabulous man chisigma...thank you a billion times!!!
I hope I will be able to estimate the integrale...as I remeber, it cann't be calculated in an explicit manner...but I will do more tries...even an approximation it's OK!
Best regards and infinite thanks!
I made some numerical estimations and it seems to me that the integral is between -1 and 0 for values of r between 0 and 10...which it's in accord with my visual estimates...so, the work continues!
Thank you again chisigma!
I made more approximations using the trapezoidal method and it seems that the integral is around -0.0007
Looking at the graphic of the function f(x)=ln(abs(cos(2^r*x))), it seems to me to be obvlious that for a big r the area becomes smaller...
I wrote this program in Matlab to approximate the integral (the results are in the file "integrala.txt"):
fid = fopen('integrala.txt', 'w');
x = -0.5:0.001:0.5;
y = log(abs(cos(2^r*x)));
z = 0.001*trapz(x,y);
Rearranging the terms in more tractable form we arrive to write the function as...
The integral in (1) can be 'attacked' numerically and using the Simpson rule with 10000 points we arrive to the following results...
For low values of r [ ...] the oscillates but for higher values it tends to -.693 so that for is with good approximation...
... as illustrated in the figure where the (2) is the 'red line' and the (1) the 'black line'...