Predicting the bifurcation structure of localized snaking patterns

We expand upon a general framework for studying the bifurcation diagrams of localized spatially oscillatory structures. Building on work by Beck et al., the present work provides rigorous analytical results on the effects of perturbations to systems exhibiting snaking behavior. Starting with a reversible variational system possessing an additional Z 2 symmetry, we elucidate the distinct effects of breaking symmetry and breaking variational structure, and characterize the resulting changes in both the bifurcation diagram and the solutions themselves. We show how to predict the branch reorganization and drift speeds induced by any particular given perturbative term, and illustrate our results via numerical continuation. We further demonstrate the utility of our methods in understanding the effects of particular perturbations breaking reversibility. Our approach yields an analytical explanation for previous numerical results on the effects of perturbations in the one-dimensional cubic–quintic Swift–Hohenberg model and allows us to make predictions on the effects of perturbations in more general settings, including planar systems. While our numerical results involve the Swift–Hohenberg model system, we emphasize the general applicability of the analytical results.