Bifurcation Of Two Classes Of Dynamical Systems With Application To Cardiac Dynamics

The phenomenon of bifurcation exists widely in mathematical models that describe the problems in medical biology and physical engineering. And noise perturbation is ubiquitous in these actual models, hence the bifurcation theory under noise perturbation is a topic of great significance in dynamical system. Since the work of L. Arnold [8], a lot of attention has been paid to the research on the bifurcation with unbounded normally distributed noise. However, in many cases the noise is bounded, such as the temperature fluctuations and the measurement error. Recently A. Homburg and T. Young et al. provided a framework of bifurcation with bounded random perturbation, and they studied the Hopf bifurcation for planar differential system with bounded noise [15]. The present Ph. D. dissertation is denoted to the analysis on the bifurcation of nonhyperblic periodic orbits in planar differential system with bounded noise, and the bifurcation of periodic points in discrete mathematical model for cardiac cells, where the dynamical behavior of the model under bounded noise perturbation and their biological interpretation in cardiac cells are discussed.The thesis comprises three main parts. The first part is related to the bifur-cation of nonhyperblic periodic orbit in planar differential system with bounded noise. Based on the framework of A. Homburg and T. Young et al.[15,34,35], we discuss the variation of minimal forward invariant (abbreviated as MFI) sets to study the bifurcation. In such system with bounded noise, discontinuous change in MFI sets with respect to Hausdorff metric or the change in the number of MFI sets is referred to as hard bifurcation [15]. To start with, we recall three kinds of bi-furcation of a nonhyperblic periodic orbit for planar differential system:pitchfork bifurcation, saddle-node bifurcation and transcritical bifurcation. The direction and the stability of the periodic orbits bifurcating from the nonhyperblic periodic orbit are investigated. Under noise perturbations, we prove that the number of MFI sets of the perturbed system changes and the hard bifurcation occurs. Some examples are provided to illustrate the theoretical results.Secondly, we study the dynamics of cardiac cells by the analysis on a two-dimensional discrete mathematical model. Plenty of medical and experimental researches have shown that the genesis of cardiac arrhythmias in the whole heart scale has been linked to dynamical instabilities at the cellular level, where action potential duration (APD) and calcium transient are two important objects of study. Y. Shiferaw et al.[69,71,72] proposed a series of abstract mathematical models with respect to APD and calcium transient. Most of these models have no explicit expressions, which leads to the lack of strict theoretical analysis. In the present part, based on previous experimental results and mathematical models, we obtain a concrete discrete model as well as its parameter region. By strict mathemat-ical analysis, we prove that the model can undergo period-doubling bifurcation and Neimark-Sacker bifurcation in the permitted parameter region, and it can not further go to chaos through period-doubling bifurcation. These results provide the-oretical support on some experimental observations, such as the alternans of APD and calcium transient, and quasi-periodic oscillations between APD and calcium transient in paced cardiac cells.Finally, considering the stochastic factors for calcium cycling in cardiac dy-namics, we propose a discrete model with the impact of calcium-channel noise. Without noise perturbations, bifurcation analysis on the model shows that period-doubling bifurcation and Neimark-Sacker bifurcation can occur for the unperturbed system. Under the noise perturbation, periodic orbit from period-doubling bifurca-tion is perturbed to fluctuation, and the invariant curve from Neimark-Sacker bifur-cation is perturbed to an annulus on the plane. Moreover, power spectrum analysis is first introduced to discuss the correspondence between power spectrum and bi-furcation behavior in cardiac cells. Numerical analysis shows that the existence of high-frequency peak in the power spectra links to the period-doubling orbits, while the existence of low-frequency peak corresponds to quasi-periodic orbit. The result may provide some theoretical support on future medical and experimental research.