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

Abstract

Real-world dynamical systems with retardation effects are described in general not by a single, precisely defined time delay, but by a range of delay times. An exact mapping onto a set of $N+1$ ordinary differential equations exists when the respective delay distribution is given in terms of a gamma distribution with discrete exponents. The number of auxiliary variables one needs to introduce, $N$, is inversely proportional to the variance of the delay distribution. 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-Glass system changes under the influence of distributed delays. We find that local properties, f.i.\ the locus of a Hopf bifurcation, are robust against the introduction of broadened memory kernels. Period-doubling transitions and the onset of chaos, which involve non-local properties of the flow, are found in contrast to be more sensitive to distributed delays. In general, the observed effects are found to scale as $1/N$. Furthermore, we consider time-delayed systems exhibiting chaotic diffusion, which is present in particular for sinusoidal flows. We find that chaotic diffusion is substantially more pronounced for distributed delays. Our results indicate in consequence that modeling approaches of real-world processes should take the effects of distributed delay times into account.