|Bursting Oscillation Mixed-Mode oscillation Slow Manifold Quasi-Integral slow-fast analysis QSSA|
The present thesis deals with two different aspects of nonlinear dynamical systems. On the one hand, it is concerned with model reduction of complex (bio-)chemical reaction networks in order to identify the essential dynamical degrees of freedom in such networks. On the other hand, the emergence of complex bursting oscillations in a particular model system, the hemin-hydrogen peroxide-sulfite system, is investigated. The hemin system is an enzyme model system. It belongs to a family of pH oscillators which may cause periodic changes of the pH value in their reaction medium. Since these changes, in turn, can induce changes of physiological parameters such as the permeability of membranes or the activity of other enzymes, the hemin system as a pH oscillator is of high potential relevance. In the first Part of the thesis, I systematically develop the idea of quasi-integrals using the hemin system as an example. This numerical method can be used to detect slow invariant manifolds in a given ODE system without any a priori knowledge about the time scales in the system. Quasi-Integrals are defined as nonlinear functions of the phase space variables that are approximately constant along the numerically obtained trajectories of an ODE system. In order to find such functions, I systematically test whether certain ratios of components of the reaction rate vector equilibrate along the trajectories. Each such function defines a quasi-integral which can be used to eliminate one dynamical degree of freedom. Since the reduction is initially valid only at one point in parameter space, I, afterwards, compare the dynamical properties of the original system with those of the reduced system based on a local 2-parameter bifurcation diagram. Using this method, I can reduce the originally 6-dimensional hemin system to a 3-dimensional one while keeping its generic dynamical properties. In the second Part of the thesis, I investigate the origin of the bursting oscillations in the hemin system. To this purpose, I use one of the dynamical variables of the reduced 3-dimensional ODE system as a quasi-static bifurcation parameter for the remaining 2-dimensional subsystem, since it evolves on a much slower time scale than 2-dimensional subsystem. Using this method, I can classify the bursting behavior of the hemin system to be of subHopf/fold-cycle type in a broad range of parameters. However, a systematic 2-parameter bifurcation analysis of the fast subsystem reveales a transition of the bursting behavior from subHopf/fold-cycle to fold/subHopf type. Finally, the slow-fast analysis of the fast subsystem near a Neimark-Sacker bifurcation of the 3-dimensional system also explains the origin of quasi-periodic behavior in the hemin system.