From: hsg@lem.phy.duke.edu (Henry S. Greenside)
Subject: Re: lyapunov stability in ODE and difference systems
Date: 16 Jan 2000 11:56:55 -0500
Newsgroups: sci.math.num-analysis
Summary: [missing]

Lou,

There is a substantial literature, more in the numerical
analysis than nonlinear dynamics community, of when the
asymptotic dynamics of a discretized approximation will be
similar to that of the exact equations. A key player is
Andrew Stuart of Stanford who has summarized much of what is
known in the following book

   @Book{Stuart96,
     author = 	 {A. M. Stuart and A. R. Humphries},
     title = 	 {Dynamical Systems and Numerical Analysis},
     publisher = {Cambridge U. Press},
     year = 	 {1996},
     address =	 {New York}
   }

The proofs mainly work for fixed points and limit cycles, I
don't know of any work regarding when a numerical method
produces the right details of a strange attractor (since
these are so hard to characterize anyhow).

I believe there are straightforward proofs that any
convergent numerical approximation such as forward Euler or
Runge Kutta (of some order, there are actually many
different Runge Kutta schemes with many different
properties) will give the correct qualitative dynamics in
the limit that some specified truncation error goes to
zero. The proof is topological: if one can show that the
vector field of the discretized system becomes close enough
to the vector field of the true equations, then they become
topologically equivalent and one gets the same dynamics
(e.g., if all arrows point inwards towards in some
neighborhood of a fixed point you get stability of the fixed
point.

Hope this helps.

	Henry
==============================================================================

From: Giuseppe Andrea Paleologo <gappy@stanford.edu>
Subject: Re: lyapunov stability in ODE and difference systems
Date Tue, 18 Jan 2000 11:11:58 -0800
Newsgroups: sci.math.num-analysis
Summary: [missing]

Thanks for the reference. For those interested, I found another
self-contained reference exactly on the issue I was interested in:

P.E.Kloeden and J.Lorenz. (1986) "Stable Attracting Sets in Dynamical
Systems and in their One-Step Discretization". Siam Journal of Numerical
Analysis, 23, 986-95.

Giuseppe

[quoted of previous article deleted --djr]
-- 
::Giuseppe A Paleologo::                  http://www.stanford.edu/~gappy
________________________________________________________________________
"If triangles had a God, he would have 3 sides." 
                                             - Montesquieu -
==============================================================================

From: David Eyre <eyre@math.utah.edu>
Subject: Re: lyapunov stability in ODE and difference systems
Date: Tue, 18 Jan 2000 13:17:39 -0700
Newsgroups: sci.math.num-analysis

Giuseppe Andrea Paleologo wrote:
> 
> Here it is: consider a smooth nonlinear ODE with an equilibrium 
> point whose stability has been succesfully proved via Lyapunov
> function methods. Now consider the difference system obtained by
> applying Euler's (or Runge-Kutta) method to the ODE. Provided 
> that the step is small enough, a reasonable conjecture is that
> the equilibrium point is stable for the difference system as well.

This is true, see the paper of A. Stuart and A. R. Humphries
in the SIAM Review, 36, 1994 and the paper of C. M. Elliott
and A. Stuart on numerical methods for dissapative systems
in SIAM Numerical Analysis (or SIAM Scientific Computing) in
about '93.  You might also be interested in the following
preprint.

http://www.math.utah.edu/~eyre/research/methods/stable.ps

-- 
David Eyre 
eyre@math.utah.edu