The origin of localized patterns with a spatiotemporal oscillatory background state

The localized patterns observed with a spatiotemporal oscillatory background in the experiment are believed to emerge due to the bistability of supercritical Turing–Hopf modes. However, the branching origin of these patterns remains unclear. In this paper, we explore the formation of local patterns near the subcritical Turing–Hopf bifurcation point using the Gray–Scott model as an example. By employing the multiple scales analysis method, we derive the amplitude equation coupling both time and space, demonstrating that this special localized pattern can persist even under a subcritical bifurcation. Through numerical continuation and bifurcation analysis, we reveal that the patterns originate from a new branch on the homoclinic snaking. Our findings provide new insights into the formation of complex spatiotemporal patterns and offer a reasonable explanation for the origin of oscillatory localized patterns from the perspective of higher-order bifurcations.