Stability and Bifurcation in the Two-Dimensional Stochastic Zeeman Heartbeat Model.

Comparison of some experimental data and deterministic dynamical models of heartbeat show that it is essential to consider stochastic mathematical models. The Zeeman heartbeat model is one of the main heartbeat models whose stochastic dynamics is less studied. Especially, investigating bifurcations in stochastic dynamical models can be useful for identifying abnormal cardiac rhythms. This paper is concerned with two essential features of the two dimensional stochastic Zeeman heartbeat model i.e., stability and bifurcation. To achieve this approach, Taylor expansion, polar coordinate transformation, and stochastic averaging procedure will be used to convert the classical system into an Ito averaging diffusion system. Furthermore, we consider several theorems which provide sufficient conditions of drift and diffusion coefficients to establish stochastic stability, D-bifurcation and phenomenological bifurcation of the model. In the end, numerical simulation plays an important role to show the influences of the noise severity and confirm our theoretical results.