Re: Local Lyapunov Exponents

From: Luca Mihai (luca_at_univ-brest.fr)
Date: 09/16/04 

Date: 16 Sep 2004 09:50:23 -0700

Lou Pecora <pecora@anvil.nrl.navy.mil> wrote in message news:<pecora-093097.12013315092004@ra.nrl.navy.mil>...
> In article <9baf5e12.0409141443.13d317a0@posting.google.com>,
> luca@univ-brest.fr (Luca Mihai) wrote:
>
> > Sorry I should have them specified the first time.
> > The logistic map I use is given by the recurence:
> > x(n+1) = 1-2*x(n)^2
> > Now the global Lyapunov exponent is ln(2)
> > The first local one I calculated using
> > L(1) = int(log(abs(4*x)*ro(x)),x=-1..1)
> > where ro(x)=1/pi/sqrt(1-x^2)
> > I obtain L(1) = ln(2)
> > For the others with a numerical integration method is the same
> > Thank you
>
> I just looked at Abarbanel's book and the local exponents are defined at
> the starting point x on the attractor and are labeled as such along with
> the time interval L over which they are determined. Thus, lambda(x,L)
> is a local exponent. As L->infinity the local exponents approach the
> global exponents. If we average the local exponents over the measure of
> the attractor (its probability density), then we also get the global
> expoenents (this is a result of ergodicity of an attractor).
>
> Having said that, when I look at your formulas I see that you are
> averaging the local exponent over the measure of the attractor (that''s
> what the ro(x) function is. So it seems to me that you are really not
> calculating the local exponents, but the global exponents, again.
>
> Hope I didn't miss anything.
>
> -- Lou Pecora (my views are my own)

I think you're right.
Actually I wanted to measure an average growth over the time interval
L for two orbits corresponding to close initial conditions.
When I have looked at Abarbanel's paper, he defines an exponent
labda(L) which actually takes in the itegration formula the invariant
density. So I have thought that this is my local Lyapunov exponent.
But is not. The name of local exponent, I now realize, that it is
specific to a particular initial condition, not only to a time
interval.
Thank you very much for showing me where I have made the error.
I have to think now, how it modifies my calculus, and maybe I will put
another question, or maybe another error. :)
Syncerely
Mihai Luca

Relevant Pages

  • Re: Local Lyapunov Exponents
    ... I just looked at Abarbanel's book and the local exponents are defined at ... the attractor, then we also get the global ... calculating the local exponents, but the global exponents, again. ...
    (sci.nonlinear)
  • Re: Local Lyapunov Exponents
    ... >>Sorry I should have them specified the first time. ... > I just looked at Abarbanel's book and the local exponents are defined at ... > the attractor, then we also get the global ... > calculating the local exponents, but the global exponents, again. ...
    (sci.nonlinear)