Unnested islands of period-doublings in an injected semiconductor laser

We present a theoretical study of unnested period-doubling islands in three-dimensional rate equations modelling a semiconductor laser subject to external optical injection. In this phenomenon successive curves of period-doublings are not arranged in nicely nested islands, but they intersect each other. This overall structure is globally organized by several codimension-two bifurcations. As a consequence, the chaotic region existing inside an unnested island of period-doublings can be entered not only via a period-doubling cascade but also via the break-up of a torus, and even via the sudden appearance of a chaotic attractor. In order to fully understand these different chaotic transitions we reveal underlying global bifurcations and we show how they are connected to codimension-two bifurcation points. Unnested islands of period-doublings appear to be generic and, hence, must be expected in a large class of dynamical systems.