We propose a simple one-dimensional non-autonomous map, in which some novel bursting patterns (e.g., "fold/double inverse flip" bursting, "fold/multiple inverse flip" bursting, and "fold/a cascade of inverse flip" bursting) can be observed. Typically, these bursting patterns exhibit complex structures containing a chain of inverse period-doubling bifurcations. The active states related to these bursting can be period-2(n) (n = 1, 2, 3,…) attractors or chaotic attractors, which may evolve to quiescence by a chain of inverse period-doubling bifurcations when the slow excitation decreases through period-doubling bifurcation points of the map. This accounts for the complex inverse period-doubling bifurcation structures observed in bursting patterns. Our findings enrich the possible routes to bursting as well as the underlying mechanisms of bursting.