Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle.

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 suitable blow-up transformations.

Saddle-node bifurcation of viscous profiles.

Traveling wave solutions of viscous conservation laws, that are associated to Lax shocks of the inviscid equation, have generically a transversal viscous profile. In the case of a non-transversal viscous profile we show by using Melnikov theory that a parametrized perturbation of the profile equation leads generically to a saddle-node bifurcation of these solutions. An example of this bifurcation in the context of magnetohydrodynamics is given. The spectral stability of the traveling waves generated in the saddle-node bifurcation is studied via an Evans function approach. It is shown that generically one real eigenvalue of the linearization of the viscous conservation law around the parametrized family of traveling waves changes its sign at the bifurcation point. Hence this bifurcation describes the basic mechanism of a stable traveling wave which becomes unstable in a saddle-node bifurcation.