A Hamiltonian Analogue of the Meandering Transition

In this paper a Hamiltonian analogue of the well-known meandering transition from rotating waves to modulated rotating and modulated travelling waves in systems with the Euclidean symmetry of the plane is presented. In non-Hamiltonian systems, for example in spiral wave dynamics, this transition is a Hopf bifurcation in a corotating frame, as external parameters are varied, and modulated traveling waves only occur at certain resonances. In Hamiltonian systems, for example in systems of point vortices in the plane, the conserved quantities of the system, angular and linear momentum, are natural bifurcation parameters. Depending on the symmetry properties of the momentum map, either modulated traveling waves do not occur, or, in contrast to the dissipative case, modulated traveling waves are the typical scenario near rotating waves, as momentum is varied. Systems with the symmetry group of a sphere and with the Euclidean symmetry group of three space are also treated.