Topological-numerical analysis of global dynamics in the discrete-time two-gene Andrecut-Kauffman model

We conduct a topological-numerical analysis of global dynamics in a discrete-time two-gene Andrecut-Kauffman model. This model describes gene expression regulation through nonlinear interactions. We use a numerical method to construct Morse decomposition of the system across a wide range of parameters at a fixed finite resolution in the state space and in the parameter space, both spaces split into uniform rectangular grids (a technique also called "pixelation"). We obtain qualitative results by effectively computing the Conley indices of the constructed isolating neighborhoods that form the Morse decomposition. We represent the Morse decomposition and connecting orbits by a directed acyclic graph. We introduce pictograms to convey the information provided by the Conley index in an easy to understand schematic way. We show and analyze bifurcations captured using this technique. We call this method CMAD for short (Conley-Morse graphs for the Analysis of Dynamics). The main advantage of our method is that it finds isolating neighborhoods of both stable and unstable invariant sets and that it provides validated (rigorous) numerical results: We actually obtain computer-assisted proof that the constructed sets are indeed isolating neighborhoods of Morse sets in a certain Morse decomposition of the system. In particular, this means that we have captured all the interesting dynamics within the analyzed range of the phase space perceived at the given finite resolution. We also conduct numerical simulations aimed at showing the location of attractors in the isolating neighborhoods found. The results demonstrate the usefulness of topological methods in understanding the global structure of dynamics at finite (coarse) resolution in an applied dynamical system depending on a few parameters, like the gene regulatory model that we analyze.