Re: Neato chaotic equations for analog computers to display?

From: Nicholas O. Lindan (see_at_sig.com)
Date: 12/22/04 

Date: Wed, 22 Dec 2004 17:27:25 GMT

"Lou Pecora" <pecora@anvil.nrl.navy.mil> wrote

> There are a lot of systems that behavior according to the Shadowing
> Theorem (see Yorke, et. al and probably others) wherein, roughly, for
> any calculated trajectory there is a real trajectory that follows it
> within some epsilon for some specified length of time (arbitrarily
> long).

There are three cases: Sometimes you can; Sometimes rounding error
gets in the way; Sometimes you can't no matter what.

Quoting from: Younghae et. al., 2003, "Universal
and nonuniversal features in shadowing dynamics...", Arizona
State University

   An understanding of the shadowing dynamics relies on
   the mathematical notion of hyperbolicity. Roughly, the dynamics
   is hyperbolic on a chaotic set if at each point of the
   trajectory, the tangent space can be split into expanding and
   contracting subspaces and the angle between them is
   bounded away from zero. Furthermore, the expanding subspace
   evolves into the expanding one along the trajectory
   and the same holds for the contracting subspace. Otherwise,
   the set is nonhyperbolic. The following results have been
   established.

   1 Hyperbolic chaotic systems permit infinite shadowing
     of numerical trajectories.

   2 For nonhyperbolic chaotic systems with tangencies
     (i.e., points at which the expanding and contracting directions
     coincide), shadowing can be expected for a finite
     amount of time that depends on the computer roundoff error.

   3 If the dimensions of the expanding and contracting
     subspaces are not constant on different parts of the invariant
     set, i.e., if there is unstable dimension variability, then shadowing
     of numerical trajectories for relatively long time is
     impossible. The severe obstruction to shadowing in the
     presence of unstable-dimension variability appears to be
     common in high-dimensional chaotic systems, i.e., those
     with multiple positive Lyapunov exponents.

What's needed is hyperbolic weather. With the way the snow is
coming down I think I will have to settle for hypobaric. 

-- 
Nicholas O. Lindan, Cleveland, Ohio
Consulting Engineer:  Electronics; Informatics; Photonics.
Remove spaces etc. to reply: n o lindan at net com dot com
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