We use singularity theory to classify forced symmetry-breaking bifurcation problems where f1 is 𝕆(2)-equivariant and f2 is 𝔻n-equivariant with the orthogonal group actions on z ∈ ℝ2. Forced symmetry breaking occurs when the symmetry of the equation changes when parameters are varied. We explicitly apply our results to the branching of subharmonic solutions in a model periodic perturbation of an autonomous equation and sketch further applications.

The connection between the discrete and the continuous coagulation–fragmentation models is investigated. A weak stability principle relying on a priori estimates and weak compactness in L1 is developed for the continuous model. We approximate the continuous model by a sequence of discrete models and, writing the discrete models as modified continuous ones, we prove the convergence of the latter towards the former with the help of the above-mentioned stability principle. Another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation-fragmentation model.

By means of a certain variational principle, where rank-one convexity enters in a fundamental way as part of feasibility, we construct a whole class of rank-one convex functions. We show the existence of optimal paths and motivate our analysis by the issue of the equivalence of quasiconvexity and rank-one convexity for 2 × 2 matrices.

This work is concerned with basic structural properties of first-order hyperbolic systems with source terms divided by a small parameter ε. We identify a relaxation criterion necessary for the solution sequences indexed with ε to have reasonable limits as ε goes to zero. This relaxation criterion is shown to imply hyperbolicity of the reduced systems governing the limits. Moreover, we introduce a so-called GC-stability theory and strengthen the hyperbolicity result. The latter shows that there are no linearly stable hyperbolic relaxation approximations for non-hyperbolic conservation laws.