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Model for the dynamics of bipolar disorders
We study a model for the dynamics of bipolar disorders. The model is four-dimensional and not in the standard form of slow-fast systems. The geometric analysis of the model is based on identifying and using hierarchies of local approximations based on various –- hidden -– forms of time scale separation. A central tool in the analysis is the blow-up method which allows the identification of a complicated singular cycle.
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Model for the mitotic oscillator
A minimal model describing the embryonic cell division cycle at the molecular level in eukaryotes is analyzed mathematically. It is known from numerical simulations that the corresponding three-dimensional system of ODEs has periodic solutions in certain parameter regimes. We prove the existence of a stable limit cycle and provide a detailed description on how the limit cycle is generated. The limit cycle corresponds to a relaxation oscillation of an auxiliary system, which is singularly perturbed and has the same orbits as the original model. The singular perturbation character of the auxiliary problem is caused by the occurrence of small Michaelis constants in the model. Essential pieces of the limit cycle of the auxiliary problem consist of segments of slow motion close to several branches of a two dimensional critical manifold which are connected by fast jumps. In addition, a new phenomenon of exchange of stability occurs at lines, where the branches of the two-dimensional critical manifold intersect. This novel type of relaxation oscillations is studied by combining standard results from geometric singular perturbation with several suit- able blow-up transformations.
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Model for glycolytic oscillations
A detailed geometric analysis of the Goldbeter–Lefever model of glycolytic oscillations is given. In suitably scaled variables the governing equations are a planar system of ordinary differential equations depending singularly on two small parameters ε and δ. In [L. Segel and A. Goldbeter, J. Math. Biol., 32 (1994), pp. 147–160] it was argued that a limit cycle of relaxation type exists for ε ≪ δ ≪ 1. The existence of this limit cycle is proved by analyzing the problem in the spirit of geometric singular perturbation theory. The degeneracies of the limiting problem corresponding to (ε, δ) = (0, 0) are resolved by a novel variant of the blow-up method. It is shown that repeated blow- ups lead to a clear geometric picture of this fairly complicated two-parameter multiscale problem.
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