Description of BIFDD -                        B. Hassard  04/23/96
---------------------

File BIFDD.FOR contains a FORTRAN 77 program.
The program demonstrates application of subroutine BIFDD, which locates and
analyzes Hopf bifurcations in systems of autonomous delay-differential 
equations.

The file includes the following -
       - driver program and application subroutines
       - subroutine BIFDD and subroutines called by BIFDD, except
            for LINPACK subroutines
       - the required LINPACK subroutines

If LINPACK is available on your system library, use the versions
on the system rather than the subroutines provided.  

The code provided is double precision VAX FORTRAN 77.  An attempt was made
to adhere to the FORTRAN 77 standard, ANSI X3.9 (1978).  However, double
precision complex (COMPLEX*16) variables, not part of the standard, are used.

This double precision code also compiles and runs under IBM VM/CMS FORTVS, but
because of the limited dynamic range of IBM floating point variables, exponent
underflow occurs.  The error messages should be surpressed and the
results set to zero with CALL ERRSET(208,1000,-1,0).

You may ignore error messages issued by subroutine CHKHIM, provided the
subsequent countour integration gives the correct result (2, or close
to 2) for the number of eigenvalues with non-negative real parts.  The purpose 
of CHKHIM is to avoid a potential division by zero in the contour 
integration if additional eigenvalues should happen to lie exactly on the 
imaginary axis.  

For analysis of the Hutchinson-Wright and Jones delay-differential equations
used as test examples, see Hassard, Kazarinoff and Wan,  Theory and
Applications of Hopf Bifurcation, London Math. Soc. Lecture Note Series #41,
Cambridge Univ. Press 1981, Chapter 4.  For the centrifugal governor and
Hodgkin-Huxley systems used as test examples, see Chapter 3.  These are
actually ordinary differential systems, but can be analyzed by treating them as
delay-differential systems with one "delay" T(1) = 0.  The Mahaffy systems are
from two of his articles in J. Theo. Biology: see the comments in the code for
the references. 

The part of the code that checks the hypothesis that at the critical value of
the bifurcation parameter, all eigenvalues other than the complex conjugate
pair responsible for the bifurcation lie in the negative half plane,  will
eventually be replaced with a more efficient version.  When there are
eigenvalues in addition to the c.c. pair near the imaginary axis, the
quadrature performed in the present code can  require a large number of steps
to converge.  If this occurs in a problem you are solving, let me know. I have
an improved version, but it hasn't been as well tested as the present version. 
Send me the program for a system that is giving BIFDD problems, and I'll try to
get it to run well: the way the code improves is through refinement to handle
more difficult systems. 

The best way to reach me is via e-mail
hassard@acsu.buffalo.edu

Also -
Brian Hassard,  
Dept. of Mathematics,
106 Diefendorf Hall,
SUNY at Buffalo,
Buffalo, N.Y. 14214-3093

(716) 829-2144 extn. 123