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[Stephen C. Roe]

Greetings,

About the eigenfunction

The eigenfunction is a tool used to solve two-point boundary
value problems, which are a class of differential equations.
My exposure to eigenfunctions have been in the solution of 
linear differential equations but there is likely research, beyond my
experience, which applies the concept to the nonlinear differential
equations used to describe chaotic systems.

A typical two-point boundary value problem can be stated:
solve the equation  dy(x)^2/dx^2 + lambda* y(x) = 0
with the boundary conditions  ay(0) + by'(0) = 0 and
                              cy(l) + dy'(l) = 0.

The functions y(x) which solve the above system of equations are called
the Eigenfunctions.  The values of lambda which solve the equations
are called the eigenvalues.  For practical engineering problems the
solution is often in the form of an infinite series.  The series is composed
of harmonic funcitons such as sine, cosine, hyperbolic sine and cosine
and other creatures such as bessel functions or Chebychev polynomials.

Please note that in many applications neat ( although infinite series)  
analytic funcitions which solve the equations do not exist and we must
resort to brute calculation.  To perform the calculation we must integrate
twice.  Pynchon referred to this technique in "Gravity's Rainbow".  These
are the references to the double integral.  The double S's of the SS.
The double lightning bolts which the underground rocket factory Nordhausen
mimics when viewed from above.

In fact, much of the rocket's dynamics were characterized by these two-
point boundary value equations.  The heat transfer equations, the yaw
control equations, and the inertial guidance system used to calculate
Brennschluss.

I hope this helps.  I can provide more specific examples if you like and
if you are mathematically inclined you can refer to any advanced text
on differential equaitons.  I refered to "Differential equations and
Their Applications", M. Braun, Springer-Verlag, 1986.

Sincerly,
Stephen Roe
roe at enuxsa.eas.asu.edu