A Hamiltonian analogue of the meandering transition
In this paper we present a Hamiltonian analogue of the well-known meandering transition from rotating waves to modulated rotating and modulated travelling waves in systems with Euclidean symmetry. In dissipative systems, in particular in spiral wave dynamics, this transition is caused by varying external parameters such that a Hopf bifurcation in a corotating frame occurs. In the Hamiltonian case, for example in models of point vortex dynamics, the conserved quantities of the system, angular and linear momentum, are bifurcation parameters. We prove that, depending on the symmetry properties of the momentum map, either modulated traveling waves do not occur at all, or that, in contrast to the dissipative case, modulated traveling waves are the typical scenario near rotating waves in momentum parameter space. We also treat systems with the symmetry group of a sphere and with the Euclidean symmetry group of rotations and translations in three-dimensional space.