Re: More Neural Network musings..

[H. I. SALEH]

On Thu, 30 Nov 1995, John Utz wrote:

> Hi gang;
> 
>       As i mentioned in my previous letter, i am trying to use octave 
> to solve dynamical systems and neural network problems.
> 
>       Both of these items are variations on systems of differential 
> equations. 
> 
>       One of the key techniques in looking at simple dymanical systems 
> is the use of the phase plane. The phase plane display of a dynamical 
> system will include locations called singularities. Singularities seem to 
> present a problem for a numerical solver such as lsode and dassl.
> 
>       The problem with singularities is that they are the points in 
> which a function under analysis will equal zero in the numerator ( not 
> unusual in any way ) and 0 in the *denominator* ( this is usually 
> construed as a Bad Thing (tm) ).
> 
> the following is exerted from octave's online manual:  
> 
>  {
> 
>  The function `dassl' can be used Solve DAEs of the form
> 
>      0 = f (x-dot, x, t),    x(t=0) = x_0, x-dot(t=0) = x-dot_0
> 
>      dassl (FCN, X_0, XDOT_0, T_OUT, T_CRIT)
>  ...
> 
>    The fifth argument is optional, and may be used to specify a set of
> times that the DAE solver should not integrate past.  It is useful for
> avoiding difficulties with singularities and points where there is a
> discontinuity in the derivative.
> 
>   }
> 
>       Well, heck. In my case, i wish to eagerly seek out 
> discontinuities and singularities! Worse yet. I want to plot them!
> 
>       Has anybody had any experience with this that they would like to 
> share with me?
> 
> *******************************************************************************
>  John Utz     address@hidden
>       idiocy is the impulse function in the convolution of life
> 

Consider a nonlinear 2nd order system (such as the one you are interested 
in). Now this can generally be put in the form:

  xdot = p(x,y)
  ydot = q(x,y)

singular points exist where

 p(x,y) = q(x,y) = 0

The behavior/stability of trajectories in the neighborhood of a singular 
point can be found from a linearized version of the above 1st order ODEs 
about the singular point.

i.e.

xdot = del p / del x | * (x-xs) + del p / del y | * (y-ys)
                     |                          |
                     | x = xs                   | x = xs
                     | y = ys                   | y = ys

ydot = del q / del x | * (x-xs) + del q / del y | * (y-ys)
                     | x = xs                   | x = xs
                     | y = ys                   | y = ys

where (xs,ys) is a singular point.

These can be written as

xdot = a x + b y + c1
ydot = c x + d y + c2     , c1 & c2 are constants

Singular points unaffected by c1 and c2. Hence it is sufficient to 
consider the autonomous system

zdot = Az, z = [ x y ]';

The behavoir of the singular points is determined by the eigenvalues of A 
i.e. det(A-Lam*I) = 0.

A = [ a b ]
    [ c d ]


The phase plane equation (slope) is

ydot    dy    c x + d y
---- = ---- = ---------
xdot    dx    a x + b y


I hope this helps

H. I. SALEH