Subcritical Hopf instabilities at finite wavenumber create localized spatiotemporal patterns — oscillating structures that exist in a region of space while the surrounding medium remains at rest. The instability is subcritical (the pattern appears discontinuously) and has finite wavenumber (the pattern oscillates at a specific spatial frequency), creating a rich zoo of localized states: spots, stripes, and spirals that oscillate in time while remaining confined in space.
The paper maps these structures on a line and a disk, classifying the localized solutions that branch from the subcritical Hopf point. On a line, the localized states are fronts connecting the oscillating pattern to the quiescent state. On a disk, the confinement by the boundary selects discrete families of localized modes — target patterns, spiral waves, and radially symmetric oscillations — each with a specific number of spatial oscillations within the disk.
The selection mechanism is geometric: the disk's boundary quantizes the allowed wavenumbers, and only those localized states whose spatial oscillation period is commensurate with the disk size are stable. Change the disk's radius and the selected pattern changes — the boundary controls the interior by constraining what fits.
The localized oscillating patterns exist in the bistable region where both the quiescent and oscillating states are stable. They are not transients or decaying disturbances but stable, persistent structures maintained by the nonlinear balance between the subcritical instability (which wants to grow) and the spatial localization (which constrains where the growth occurs). The pattern and its absence coexist, separated by fronts that neither advance nor retreat.