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.

Pyramiding of Ryd2 and Ryd3 conferring tolerance to a German isolate of Barley yellow dwarf virus-PAV (BYDV-PAV-ASL-1) leads to quantitative resistance against this isolate

Barley yellow dwarf virus (BYDV) is an economically important pathogen of barley, which may become even more important due to global warming. In barley, several loci conferring tolerance to BYDV-PAV-ASL-1 are known, e.g. Ryd2, Ryd3 and a quantitative trait locus (QTL) on chromosome 2H. The aim of the present study was to get information whether the level of tolerance against this isolate of BYDV in barley can be improved by combining these loci. Therefore, a winter and a spring barley population of doubled haploid (DH) lines were genotyped by molecular markers for the presence of the susceptibility or the resistance encoding allele at respective loci (Ryd2, Ryd3, QTL on chromosome 2H) and were tested for their level of BYDV-tolerance after inoculation with viruliferous (BYDV-PAV-ASL-1) aphids in field trials. In DH-lines carrying the combination Ryd2 and Ryd3, a significant reduction of the virus titre was detected as compared to lines carrying only one of these genes. Furthermore, spring barley DH-lines with this allele combination also showed a significantly higher relative grain yield as compared to lines carrying only Ryd2 or Ryd3. The QTL on chromosome 2H had only a small effect on the level of tolerance in those lines carrying only Ryd2, or Ryd3 or a combination of both, but the effect in comparison to lines carrying no tolerance allele was significant. Overall, these results show that the combination of Ryd2 and Ryd3 leads to quantitative resistance against BYDV-PAV instead of tolerance

Exponentially small heteroclinic breakdown in the generic Hopf-zero singularity

In this paper we prove the breakdown of an heteroclinic connection in the analytic versal unfoldings of the generic Hopf-Zero singularity in an open set of the parameter space. This heteroclinic orbit appears at any order if one performs the normal form around the origin, therefore it is a phenomenon "beyond all orders". In this paper we provide a formula for the distance between the corresponding stable and unstable one dimensional manifolds which is given by an exponentially small function in the perturbation parameter. Our result applies both for conservative and dissipative unfoldings

Resonances in a spring-pendulum: algorithms for equivariant singularity theory

A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is handled by equivariant singularity theory with a distinguished parameter, yielding an integrable approximation of the Poincaré map. This makes a concise description of certain bifurcations possible. The computation of reparametrizations from normal form to the actual system is performed by Gröbner basis techniques.