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We give a survey on local and semi-local bifurcations of divergence-free vector fields. These differ for low dimensions from ‘generic’ bifurcations of structure-less ‘dissipative’ vector fields, up to a dimension-threshold that grows with the co-dimension of the bifurcation.
We investigate the relation between broken time-reversal symmetry and localization of the electronic states, in the explicitly tractable case of the Landau model. We first review, for the reader’s convenience, the symmetries of the Landau Hamiltonian and the relation of the latter with the...
In a number of models for coupled oscillators and nonlinear wave equations primary resonances dominate the phase-space phenomena. A new feature is that in a Hamiltonian framework, the interaction of primary and higher order resonances is shown to be important and can be signaled by using...
The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether’s conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called ‘collective’. Here,...
In this article we give a brief survey on the physics and mathematics of the phenomenon of conductivity in metals under a strong magnetic field.
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