Stability analysis of one-dimensional dynamical systems applied to an isolated beating heart

In this paper we propose a new model of an isolated beating heart. The model is described by a one-dimensional non-linear discrete dynamical system which depends on several parameters. Applying stability analysis we investigate the dynamic properties of the non-linear system. We find those domains in the parameter space in which the equilibrium point of the system (the fixed point) and the periodic orbits are attractors and in which they are unstable. These domains correspond to a normal and abnormal beating heart, i.e. when the end diastolic volumes are stable time invariant and time variant, respectively. On transition between these domains there is a bifurcation which gives rise to a pair of attracting points of period 2. This case corresponds to the simplest type of period doubling behavior of an abnormal beating heart, called mechanical alternans. Our results provide qualitative and quantitative predictions which can be used for comprehensive experimental design.