These files accompany the paper

"Towards computational Morse-Floer homology:
forcing results for connecting orbits 
by computing relative indices of critical points"

Authors: 
Jan Bouwe van den Berg
Marcio Gameiro
Jean-Philippe Lessard
Robert van der Vorst

Year: 2022

This directory contains the code for Theorem 1.2, Figure 2 and Section 5,
i.e. the traveling waves problem on a cylinder.

Additional details of the estimates implemented can be found in
estimates-2D.pdf

To run the proofs and produce the graphs 
you need to have intlab installed and have it started.
Then run the script 
proveall12.m 
to run the proofs for parameter value lambda=12.

Note that not all proof attempts will be successful.
The subdirectory data12 contains the initial guesses for solutions
(these were obtained by manual branch-following/switching)
The subdirectory figures contains the figures (pdf)

The continuation and branch switching can be redone by running the script
runall.m 
which both generates the data and performs the proofs for two values
of the parameter lambda.
The results (proveall15.m) for parameter value lambda=15 
are not included in the paper.

--------------------------
 Below is a list of files 
--------------------------

altfftn.m
% Fourier transform for intervals and floats

altifftn.m
% inverse Fourier transform for intervals and floats

altsup.m
% supremum for intervals (just returns floats)

altzeros.m
% construct matrices consisting of the zero interval (or floats)

conv2cos.m
% convert cosine series to Fourier series and perform convolution

convtensor.m
% convolution of two tensors of the same size

countunstable.m
% determine number of eigenvalues with positive real part

FDF.m
% computes F(a) and DF(a) 

multiplicity.m
% determine the number of symmetry-related solutions

newtoniterate.m
% perform Newton iterations

plotcos2d.m
% plot the graphs of the numerical solutions

proveit.m
% perform the computer-assisted proof

trystructured12.m
% follow branches bifurcating from the trivial state, lambda=12

trystructured15.m
% follow branches bifurcating from the trivial state, lambda=15

trybifcont02aQ1a.m
trybifcont02aQ2a.m
trybifcont03bQ1a.m
trybifcont12aQ1a.m
trybifcont12bQ1a.m
% do numerical branch switching
