Curvature effects and radial homoclinic snaking

Gomila, Damià; Knobloch, Edgar
IMA Journal of Applied Mathematics , hxab028 (2021)

In this work, we revisit some general results on the dynamics of circular fronts between homogeneous
states and the formation of localized structures in two dimensions (2D). We show how the bifurcation
diagram of axisymmetric structures localized in radius fits within the framework of collapsed homoclinic
snaking. In 2D, owing to curvature effects, the collapse of the snaking structure follows a different
scaling that is determined by the so-called nucleation radius. Moreover, in the case of fronts between
two symmetry-related states, the precise point in parameter space to which radial snaking collapses
is not a ‘Maxwell’ point but is determined by the curvature-driven dynamics only. In this case, the
snaking collapses to a ‘zero surface tension’ point. Near this point, the breaking of symmetry between
the homogeneous states tilts the snaking diagram. A different scaling law is found for the collapse
of the snaking curve in each case. Curvature effects on axisymmetric localized states with internal
structure are also discussed, as are cellular structures separated from a homogeneous state by a circular
front. While some of these results are well understood in terms of curvature-driven dynamics and front
interactions, a proper mathematical description in terms of homoclinic trajectories in a radial spatial
dynamics description is lacking.


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