We present a theoretical study of unnested period-doubling islands in three
-dimensional rate equations modeling a semiconductor laser subject to exter
nal optical injection. In this phenomenon successive curves of period doubl
ings are not arranged in nicely nested islands, but intersect each other. T
his overall structure is globally organized by several codimension-2 bifurc
ations, As a consequence. the chaotic region existing inside an unnested is
land of period doublings can be entered not only via a period-doubling casc
ade but also via the breakup of a torus, and even via the sudden appearance
of a chaotic attractor. In order to fully understand these different chaot
ic transitions we reveal underlying global bifurcations and we show how the
y are connected to codimension-2 bifurcation points. Unnested islands of pe
riod doublings appear to be generic and hence must be expected in a large c
lass of dynamical systems.