Global Bifurcation Map of the Homogeneous States in the Gray–Scott Model

Abstract

We study the spatially homogeneous time-dependent solutions and their bifurcations of the Gray-Scott model. We find the global map of bifurcations by a combination of rigorous verification of the existence of Takens-Bogdanov and a Bautin bifurcations, in the space of two parameters k-F . With the aid of numerical continuation of local bifurcation curves we give a global description of all the possible bifurcations.

Phase portraits and bifurcation diagram of the Gray-Scott model

“…The Gray-Scott model [13,18] is a cubic autocatalysis system that exhibits many interesting patterns, see for instance the papers [3,10,13,16,18] and the references quoted there. This model has been studied with slightly distinct differential equations, here we consider the model given by the differential equations ( 1)…”

Analyzing Pattern Formation in the Gray--Scott Model: An XPPAUT Tutorial

“…Then, for subsequent increases in k, the Hopf bifurcation is subcritical. The change in the criticality of the Hopf bifurcation at (k, F ) = (9/256, 3/256) is known as a Bautin bifurcation [2,8]. (Note that, in the phase planes of Figure 7 below, crossing the Bautin bifurcation corresponds to moving between the phanes planes of Figure 7(c) and Figure 7(d).)…”

“…This involves calculation of the model's steady states and their stability, and understanding how these change as parameters are varied, following the standard procedure outlined in a typical undergraduate dynamical systems course. Various authors have performed this type of analysis for the Gray-Scott model [2,10,14,15,16], to identify a rich structure of bifurcations, some of which an undergraduate would be familiar with, and some of which they typically would not. We explore this in more detail below.…”

Modeling ore generation in a magmatic context

“…9, 10) (Hale et al, 1999). Indeed, it presents two eigenvalues, namely -F and -(F+k), that determine a trivial critical point (Delgado et al, 2017). A mapping of the types of solutions is constructed (Fig.…”