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

Daniel Henrik Nevermann, Claudius Gros

Published 2023-03-26Version 1

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. It is shown that 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 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 sensible to distributed delays. Our results indicate that modeling approaches of real-world processes should take the effects of distributed delay times into account.