Homoclinic Snakes Bounded by a Saddle-Center Periodic Orbit

We describe a new variant of the so-called homoclinic snaking mechanism for the generation of infinitely many distinct localized patterns in spatially reversible partial differential equations on the real line. In standard snaking a branch of localized states undergoes infinitely many folds as the pattern grows in length by adding cells at either side. In the cases studied here the localized states have a defect or hump in the middle corresponding to an additional orbit homoclinic to the underlying spatially periodic orbit, and the folds accumulate on a parameter value where the periodic orbit undergoes a saddle-center transition. By analyzing an appropriate normal form in a spatial dynamics approach, it is shown that convergence of the folds is algebraic rather than exponential. Specifically the parameter value of the $n$th fold scales like $n^{-4}$. The transition from this saddle-center mediated snaking to regular snaking is described by a codimension-two bifurcation that is also analyzed. The results are compared with numerical computations on two distinct complex Ginzburg--Landau models, one of which is variational and so represents a conservative system in space, while the other is nonvariational. Good agreement with the theory is found in both cases, and the connection between the theory and the recently identified defect-mediated snaking is established.