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From: beer@cthulhu.ces.cwru.edu (Randall D. Beer)
Newsgroups: sci.math,comp.theory.dynamic-sys
Subject: Dynamical Systems Question
Message-ID: <1990Jul16.140630.10110@usenet.ins.cwru.edu>
Date: 16 Jul 90 14:06:30 GMT
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Suppose we are given a system of N differential equations of the form 

                       N
Ci * dyi/dt = -yi + F(sum{Wji * yi})     where F(x) = 1/(1 + e^(-x))
                      j=1

My question is this: Can this N dimensional dynamical system be made
to approximate the dynamics of a given M dimensional dynamical system
(M <= N) on an M dimensional subspace arbitrarily well by appropriate
adjustment of the time constants (Ci) and coupling strengths (Wij) as
N -> infinity?  By "approximate the dynamics" I mean that all
trajectories on the M dimensional subspace of the N dimensional system
can be made arbitrarily close to those of the system we are
approximating.  One way of expressing this is that the following term
should approach 0 for all initial conditions:

                 M   infinity
		sum{integral{(yi - di)^2 dt}}
		i=1     0

where di(t) is the value of the ith state variable of the system we
are approximating at time t and yi(t) is the value of the
corresponding state variable in our approximation.

What I have in mind here is a dynamical analogue to function
approximation.  If the answer to my question is yes, can any bounds be
placed on how the accuracy of the approximation will scale with N?  If
the answer is no, do any such "basis systems" exist?  Can the above
system at least approximate an arbitrary M dimensional attractor,
rather than the entire phase space?

Any help on these questions would be greatly appreciated.

Thanks, 

R. Beer 
(beer@cthulhu.ces.cwru.edu)