A Hamiltonian Analogue of the Meandering Transition

Abstract

In this pap er a Hamiltonian analogue of the well-known meandering transition from rotating waves to modulated rotating and modulated traveling 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 occur only at certain resonances. In Hamiltonian systems, for example, in systems of p oint vortices in the plane, the conserved quantities of the system, angular and linear momentum, are natural bifurcation parameters. Dep ending on the symmetry prop erties 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-dimensional space are also treated.

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