David Rackl

Abstract

We study transcritical and pitchfork bifurcations in scalar nonautonomous difference equations. In the nonautonomous setting, these bifurcations are no longer described by isolated equilibria, but by bounded solutions and their stability properties.

Using explicitly solvable model equations as prototypes, we show that classical bifurcation scenarios persist via comparison with these model systems under suitable spectral and nonlinear balance conditions. The approach is local and applies when the dichotomy spectrum reduces to a single point, which allows the relevant spectral quantities to depend smoothly on the parameter under additional summability assumptions.

The same scalar mechanism appears in the radial dynamics of planar systems and provides a route toward nonautonomous Neimark–Sacker bifurcations. For skew-product systems, this requires incorporating the angular dynamics into the base, leading to a minimal circle extension of the original base flow. Scalar bifurcation results can then be used to control radial transitions while the angular dynamics is treated as part of the extended driving system. I conclude by sketching this reduction and its connection to ongoing work in Jena.