🚀 go-pugleaf

starke@RHRK.UNI-KL.DE (Gerolf Starke) writes:
: Dear reader!
: I wonder if there exist a definition of local lyapunov exponents,
: or if the terminus exist.

Yes.

: The lyapunov exponent characterise the long time behavior of a
: dynamical system, so, what is about the short time behavior?
: 
: A definition of local lyapunov exponents maybe could simply be
: found if adapted from the definition of (global) lyapunov ex-
: ponents.
: Does anybody know of some reference? Or do I talk through one's
: hat?

Nope.  It's a perfectly well-defined concept, and people are starting
to use it.

Specifically, take a look at (shamless plug)

H D I Abarbanel, B Brown, M B Kennel
"Variation of Lyapunov Exponents on a Strange Attractor"
Journal of Nonlinear Science, 1: 175-199 (1991).

though there are other papers in various fields who are starting
to use the same concepts.

Brief rundown:  short time exponents can have a very wide variation.
As this variation is in the _exponent_ it means a wide variation in the
amount of short term predictability (which is all you can do anyway) 
depending on where you are in phase space.  

The "average" local finite-time exponents (ergodic average for finite-time)
appear to approach the global one as

exponent(L) = exponent(infinity) + C/ L^p  (L being the composition
length)

where p is between 0.5 and 1. I.e. a slow power law decay.  The
variation around the mean also approaches 0 with a slow power law.

Finally, the average finite-time largest exponent always approaches the
asymptotic largest exponent from above.  Therefore average short term
predictability is worse than what you'd expect from using the largest
lyapunov exponent.

The global exponents are invariant under a smooth invertible change of
coordinates, whereas the local exponents will vary with a factor that
goes as 1/L, so the particular representation used can have a substantial
effect on predictability.

: Thanks for your response in advance.
: Gerolf

hope it's useful...

Matt Kennel
mbk@inls1.ucsd.edu
Institute for Nonlinear Science, UCSD

: ---
: Gerolf Starke                                 phone: +49/631/205-2128
: Institute of Applied Mechanics                internet: starke@rhrk.uni-kl.de
: University of Kaiserslautern, 6750            compuserve: 100010,3451
: Germany                                       fax: +49/631/205-3055