Stability pockets of a periodically forced oscillator in a model for seasonality

A periodically forced oscillator in a model for seasonality shows stability pockets and chains thereof in the parameter plane. The frequency of the oscillator and the season indicated by a value between zero and one are the two parameters. The present study is intended as a theoretical complement to the numerical study of Schmal et al. in 2015 of stability pockets or Arnol'd onions in their terminology. We construct the Poincar\'e map of the forced oscillator and show that the Arnol'd tongues are taken into stability pockets by a map with a number of folds. Stability pockets are already observed in an article by van der Pol \& Strutt in 1928 and later explained by Broer \& Levi in 1995

Differentiability of the volume of a region enclosed by level sets

The level of a function f on an n-dimensional space encloses a region. The volume of a region between two such levels depends on both levels. Fixing one of them the volume becomes a function of the remaining level. We show that if the function f is smooth, the volume function is again smooth for regular values of f. For critical values of f the volume function is only finitely differentiable. The initial motivation for this study comes from Radiotherapy, where such volume functions are used in an optimization process. Thus their differentiability properties become important.Comment: 11 pages, 1 figur

A reversible bifurcation analysis of the inverted pendulum

The inverted pendulum with a periodic parametric forcing is considered as a bifurcation problem in the reversible setting. Parameters are given by the size of the forcing and the frequency ratio. Normal form theory provides an integrable approximation of the Poincaré map generated by a planar vector field. Genericity of the model is studied by a perturbation analysis, where the spatial symmetry is optional. Here equivariant singularity theory is used.

Critical points at infinity in charged N-body systems

We define the notion of critical points at infinity for the charged N-body problem, following the approach of Albouy (1993). We give a characterisation of such points and show how they can be found in the charged 3-body problem. The symmetry group of the N-body problem and accompanying integrals play a key role. In fact critical points at infinity are indispensible in understanding the bifurcations of the integral map. Together with the critical points at infinity in the charged 3-body problem, we present the bifurcation values.</p