Rich bifurcation structure in a two-patch vaccination model.
Results of computations

This website contains additional interactive information and software referred to in the paper Rich bifurcation structure in a two-patch vaccination model by Díana H. Knipl, Gergely Röst and Paweł Pilarczyk.

Continuation diagrams

The continuation diagram computed for the following settings is available for browsing; please, click the small diagram on the left. Please, note that the diagram of 80x200 boxes shown in the paper was actually cut from the computed 200x200 one in order to expose the range of the parameters at which interesting bifurcations were found.

The parameter space [0,0.5] x [0.48,0.53] was sampled at 200 x 200.

The phase space [0,100]4 was sampled at 29 = 512 units in each direction.

Each continuation diagram shows α in the horizontal axis, and β in the vertical axis.

The phase space shows I1 in the horizontal axis, and I2 in the vertical axis.

You are welcome to browse the numerical Morse decompositions and the Conley-Morse diagrams to discover the information obtaind about the dynamics using this method.

Data

The data that can be browsed by clicking the diagram above, is available freely here for download:

  • inf18c.zip – the Conley-Morse graphs encoded in the text format compatible with the dot program from the Graphviz Graph Visualization Software package (https://graphviz.org/).
  • inf18p.zip – the computed Morse decompositions encoded in terms of a PNG image in which a single pixel corresponds to a square in the phase space.

Software

The computations were conducted using an expanded version of the Conley-Morse Graphs software that was published with the original paper about the database approach to cataloging dynamics in 2009. Although the software is still at a development stage and using it requires considerable amount of patience, the source code has been made available here for the benefit of the academic community. It is expected that those who are interested in doing similar computations for different systems or to verify our computations may benefit from this software.

This software is written in C++ and its compilation requires a relatively recent version of the CAPD software library and the Original CHomP library, as well as several other standard software libraries: boost, zlib, libpng, libbz2, and LAPACK. All of them must be prsent in the system before the software can be compiled. A relatively recent GNU C++ compiler is also necessary, e.g., 4.7.2, as in the current stable distribution of Debian GNU/Linux 7 (codename "wheezy").

Acknowledgment. The research whose results are described here was supported in part from the European Union as well as Portuguese and Hungarian national funds, as stated in the paper. The computations were carried out at UMinho and ISTA.