phase synchronization time (color coded) in the control parameter plane of coupling delay $\tau$ and self-feedback delay $\tau_K$. Other parameters: $a=1.3$, $\epsilon=0.01$, $C=0.5$, $K=0.5$. ](fig13.eps){width=".5\textwidth"}

In Fig. \[fig:hauplot\] the regime of oscillations is shown in the parameter plane of the coupling strength $C$ and coupling delay $\tau$. The oscillation period is color coded. The boundary of this colored region is given by the minimum coupling delay $\tau_{min}$ as a function of $C$. For large coupling strength, $\tau_{min}$ is almost independent of $C$; with decreasing $C$ it sharply increases, and at some small minimum $C$ no oscillations exist at all. At the boundary, the oscillation sets in with finite frequency and amplitude as can be seen in the insets of Fig. \[fig:hauplot\] which show a cut of the parameter plane at $C=0.8$. The oscillation period increases linearly with $\tau$. The mechanism that generates the oscillation is a saddle-node bifurcation of limit cycles (see inset (b) of Fig. \[fig:hauplot\]), creating a pair of a stable and an unstable limit cycle. The unstable limit cycle separates the two attractor basins of the stable limit cycle and the stable fixed point.

Delayed self-feedback and delayed coupling
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In this section we consider the simultaneous 