Distributed Delays Facilitate Amplitude Death of Coupled Oscillators

Coupled oscillators are shown to experience amplitude death for a much larger set of parameter values when they are connected with time delays distributed over an interval rather than concentrated at a point. Distributed delays enlarge and merge death islands in the parameter space. Furthermore, when the variance of the distribution is larger than a threshold, the death region becomes unbounded and amplitude death can occur for any average value of delay. These phenomena are observed even with a small spread of delays, for different distribution functions, and an arbitrary number of oscillators.

Figure 1

Enlargement of the stability region as the spread of the delay distribution increases: (a) $\alpha =0$, (b) $\alpha =0.007$, (c) $\alpha =0.008$, and (d) $\alpha =0.02$. The value of ${\omega }_0$ is 30.

Figure 2

Minimum value of ${\omega }_0$ for which amplitude death is possible decreases as the spread of the delay distribution increases.

Figure 3

Amplitudes of the limit cycles of individual oscillators versus the standard deviation of the delay distribution, for common distribution functions with the same mean value $\tau =0.5$. Other parameters are $K=30$ and ${\omega }_0=30$.

Figure 4

Stability regions for coupled oscillators with different frequencies when the delay is (a) discrete at $\tau =0.5$, and (b) uniformly distributed over $0.5\pm 0.02$. The mean frequency of the oscillators is fixed at 30.

Figure 5

The limiting shape of the stability region for an infinite number of globally coupled oscillators, calculated for delays uniformly distributed over $\tau \pm 0.07$. The area enclosed by the dashed line is the stability region for the discrete delay.