Global organization of axisymmetric snaking: isolas and the upper branch

Explain the occurrence of the intermediate stack of isolas separating the lower and upper snaking branches in the bifurcation diagrams of axisymmetric localized solutions of the planar and three‑dimensional Swift–Hohenberg equation, and determine whether the upper snaking branch is connected and whether it is in fact the uppermost branch.

Background

Numerical continuation of axisymmetric localized states in the planar and 3D Swift–Hohenberg equation reveals a fragmented bifurcation structure: a lower snaking branch, a stack of isolas, and an upper snaking branch that appears to condense near the Maxwell point.

While partial asymptotic theory explains the condensation of the upper branch, the origin of the isolas and the global connectivity and ordering of branches remain unresolved.

References

Primary among them is that there is still no explanation for the intermediate stack of isolas that splits the lower and upper snaking branches. Similarly, it is not clear why the upper branch forms a connected curve or if it is even the uppermost branch in the bifurcation diagram.

— Localized Patterns (2404.14987 - Bramburger et al., 23 Apr 2024) in Subsection “Axisymmetric Snaking Branches” (Section 4.2)