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Real-world dynamical systems with retardation effects are described ingeneral not by a single, precisely defined time delay, but by a range of delaytimes. An exact mapping onto a set of $N+1$ ordinary differential equationsexists when the respective delay distribution is given in terms of a gammadistribution with discrete exponents. The number of auxiliary variables oneneeds to introduce, $N$, is inversely proportional to the variance of the delaydistribution. The case of a single delay is therefore recovered when$N\to\infty$. Using this approach, denoted here the `kernel series framework',we examine systematically how the bifurcation phase diagram of the Mackey-Glasssystem changes under the influence of distributed delays. We find that localproperties, f.i.\ the locus of a Hopf bifurcation, are robust against theintroduction of broadened memory kernels. Period-doubling transitions and theonset of chaos, which involve non-local properties of the flow, are found incontrast to be more sensitive to distributed delays. In general, the observedeffects are found to scale as $1/N$. Furthermore, we consider time-delayedsystems exhibiting chaotic diffusion, which is present in particular forsinusoidal flows. We find that chaotic diffusion is substantially morepronounced for distributed delays. Our results indicate in consequence thatmodeling approaches of real-world processes should take the effects ofdistributed delay times into account.

Mapping dynamical systems with distributed time delays to sets of ordinary differential equations