Localized structures as spatial hosts for unstable modes

We study spatially extended systems undergoing Hopf-Turing instabilities to temporal oscillations and periodic spatial patterns, focusing on mono-stability regimes where one mode nonlinearly damps the other. Using the pertinent normal-form equations, we identify a new type of instability beyond which localized structures of the dominant mode host the unstable, nonlinearly damped mode. Thus, stationary localized structures of the Turing mode can lose stability to breathing structures that host the Hopf mode, and propagating localized structures of the Hopf mode can lose stability to stationary structures hosting the Turing mode. Hosting instabilities of this kind are expected to be found in other multi-mode systems as well. Potential applications include self-organized waveguides, and data storage.