Hopf dances near the tips of Busse balloons

In this paper we introduce a novel generic destabilization mechanism for (reversible)spatially periodic patterns in reaction-diffusion equations in one spatial dimension.This Hopf dance mechanism occurs for long wavelength patterns near the homoclinic tipof the associated Busse balloon($=$ the region in (wave number, parameter space) for which stable periodic patterns exist).It shows that the boundary of the Busse balloon locally has a fine-structure of two intertwining'dancing' (or 'snaking') Hopf destabilization curves (or manifolds) that limit on the Hopfbifurcation value of the associatedhomoclinic limit pulse and that have infinitely many, accumulating, intersections.The Hopf dance is first recovered by a detailed numerical analysis of the full Busse balloon inan explicit Gray-Scott model. The structure, and its generic nature, is confirmed by a rigorousanalysis of singular long wave length patterns in a normal form model for pulse-type solutionsin two component, singularly perturbed, reaction-diffusion equations.