Nonlinear Analysis of BVAM Model using Perturbation Techniques

Abstract

The bio-electric activity of the heart can be modeled by Barrio-Varea-Aragon-Maini (BVAM) model. This model covers normal rhythm and four arrhythmia that lie in the chaotic region. This model exhibits several bifurcations, starting from fixed point bifurcation, Hopf bifurcation, period doubling and ultimately leading to chaos. An analytical solution to the BVAM model is developed in the local region of Hopf bifurcation. Center manifold reduction is applied to the main governing equations to reduce the order of the system to limit cycle oscillations of center manifolds. Application of the method of multiple scales is used on the center manifolds to develop a normal form of Hopf bifurcation, which is then transformed back into the original coordinates. Comparison with the numerical solution shows that the analytical solution matches well with the numerical solution in the local region of the bifurcation. The value of the control parameter for period doubling bifurcation is identified using the Floquet multipliers of the monodromy matrix for the corresponding periodic solution. Shooting method is employed to get monodromy matrix and form periodic and period doubling cycle. It identifies periodic solution by iterating through the time period and initial conditions. The results of this research can be used for sensitivity and parametric analysis. Such solutions also allow simpler and low cost simulators for training and research purposes.