Isolating Block Validation in Matlab
This web page contains supplemental material for the article Rigorous
validation of isolating blocks for flows and their Conley indices by
Thomas Stephens and Thomas Wanner, which has been published in the
SIAM Journal on Applied Dynamical Systems.
The Matlab code for validating
three-dimensional isolating blocks using the interval arithmetic package
Intlab can be
downloaded here as a gzipped tar archive.
The software contains a few sample scripts, which are described below.
- The script scriptExitSet.m demonstrates the basic usage of the
nodal domain validation algorithm at the heart of our method.
Upon running the script, the nodal domains of the exit set test function
u are validated for the isolating block C1 in Section 4.3 of
the paper. The script also generates numerically computed images of the
nodal domains of u, as well as of the associated modified tangency
test function w. After its successful completion, the code will produce
the three images below. Notice that the algorithm is a randomized method,
so the specific adaptive grid might be different.
- The script scriptIsoBlock.m uses the isolating block algorithm
to rigorously validate the set C1 in Section 4.3 of the paper
as an isolating block for the flow.
Upon running the script, it first produces an adaptive grid which
can be used to validate the nodal domains of the exit set test function u,
and thus furnishes an image similar to the first one below. It then moves
through the boxes in the grid in a number of sweeps to ensure the positivity
of the modified tangency test function w on the nodal lines of u. This
usually requires additional box refinements and should lead to an image
to the second one below. Finally, it overlays the final adaptive grid on
numerically computed nodal domains of u, as well as the black zero set
of w. Since the algorithm is a randomized method, the specific structure
of the produced grids might be different.
- The Mathematica
notebook createUWfiles.nb produces the Matlab m-files for the exit
set test function u, the modified tangency test function w, as well as for
their partial derivatives with respect to x and y.
Thomas Wanner, April 30, 2023.
