DOI: 10.34808/z8bc-jr94
title: Morse decompositions for a two-patch vaccination model

description: This dataset contains selected results of rigorous numerical computations described in Section 5 of the paper "Rich bifurcation structure in a two-patch vaccination model" by D.H. Knipl, P. Pilarczyk, G. Röst, published in SIAM Journal on Applied Dynamical Systems (SIADS), Vol. 14, No. 2 (2015), pp. 980–1017, doi: 10.1137/140993934.
The computations followed the general scheme explained in the paper "A database schema for the analysis of global dynamics of multiparameter systems" by Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, P. Pilarczyk, published in SIAM Journal on Applied Dynamical Systems (SIADS), Vol. 8, No. 3 (2009), pp. 757–789, doi: 10.1137/080734935.
The parameter space [0,0.5]✕[0.48,0.53] was sampled at the resolution of 200✕200. Note that in the paper, the computations restricted to [0,0.2]✕[0.48,0.53] were shown, and thus the smaller parameter space was effectively sampled at the resolution of 80✕200. The current dataset contains the results of the entire original computation. The phase space [0,100]✕[0,100]✕[0,100]✕[0,100] was sampled at the resolution of 512✕512✕512✕512. A collection of isolating neighborhoods that enclose Morse sets in a Morse decomposition was computed for each box of parameters, and a Conley-Morse graph was determined, with the Conley indices of the Morse sets computed where feasible. Clutching graphs between Morse decompositions found for adjacent boxes were also computed, and the parameter space was subdivided into classes of equivalent Morse decompositions, as described in the paper. The parallelization framework introduced in the paper "Parallelization method for a continuous property" by P. Pilarczyk was used in the computations, as published in Foundations of Computational Mathematics, Vol. 10, No. 1 (2010), 93–114, doi: 10.1007/s10208-009-9050-8.
The dataset contains projections of the Morse decompositions found for all the parameter boxes. A collection of projections of combinatorial Morse sets, that is, isolating neighborhoods of the actual Morse sets in each Morse decomposition, is encoded in terms of a PNG image. In this image, a single pixel corresponds to a square in the projection of the phase space. All the images are compressed together in the zipped archive file. The name of each file is in the format "pn_m.png", where n,m are the integer coordinates of the parameter box, both in the range [0,199]. Each combinatorial Morse set corresponds to a collection of pixels drawn in a specific color. The colors used for the consecutive sets are listed at the bottom of the image as squares: black, blue, red, green, etc. The order is the same as in the corresponding Conley-Morse graphs available in a separate dataset. Note that there is often a very small neighborhood of the origin that is hardly seen as a set of one ore slightly more pixels at the bottom left corner of the image.
An interactive browser of all the Conley-Morse graphs and phase space portraits of the Morse decompositions provided in the current series of datasets is available at the address http://www.pawelpilarczyk.com/infmodel/.