Bifurcation Structure of the Two-Dimensional Aliev–Panfilov Model

This article presents a study on a model of autowave processes based on a parabolic system of partial differential equations, the Aliev–Panfilov (AP) model. The model is widely used for studying the characteristics of myocardial excitation waves during cardiac arrhythmias. Our previous studies, carried out for a two-dimensional version of this model with variation of one of its parameters in a wide range of values, resulted in detecting specific transient processes for systems near the bifurcation points. The specific transient processes were interpreted as the stability loss delay (bifurcation memory (BM)), which has been studied quite well for systems of ordinary differential equations. This paper presents a study of the specified model when two of its parameters are varied. Using the quantitative analysis of the behavior of a two-dimensional autowave vortex in a homogeneous isotropic model medium, we construct a parametric portrait of the two-dimensional version of the AP model, indicating the position of the bifurcation boundary (BB) and the bifurcation spot (the parameter region in which BM phenomena are observed). The difference between this model and classical models of autowave processes is demonstrated. Some special cases of autowave vortex behavior are presented, which have not been previously described in the literature. Previously, a relationship was described between the features of the electrocardiogram and the behavior of the autowave vortex in a model myocardium, taking into account the BM. Although the presented results are obtained theoretically, they may be useful in the clinical practice of cardiologists. The publication is intended primarily for specialists in the fields of mathematical biology, mathematical modeling, and biophysics.