Chaotic dynamics have been found in a single mode semiconductor laser subject to optical injection experimentally or by numerical simulation. In this paper we study this laser system by means of rate equations, which mathematically are a three-dimensional vector field. To study different routes to chaos we start from the knowledge of bifurcation curves in the plane of injection strength and detuning in Ref.~ of this issue. Our main tool is combining the continuation of bifurcation curves with computing the respective phase space objects. In this way, we obtain detailed knowledge of regions in parameter space of different types of chaos, and what transitions can be found at the boundaries of such regions. This gives new insight into chaotic output found in experiments. Furthermore, it allows relatively easy access to chaotic dynamics for applications such as chaotic data encryption schemes.
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