# SKM 2023 – wissenschaftliches Programm

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# DY: Fachverband Dynamik und Statistische Physik

## DY 45: Poster: Nonlinear Dynamics, Pattern Formation and Networks

### DY 45.1: Poster

### Donnerstag, 30. März 2023, 13:00–16:00, P1

A systematic approximation scheme mapping systems with time delays to sets of ordinary differential equations — •Daniel Henrik Nevermann and Claudius Gros — Institut für Theoretische Physik, Goethe-Universität Frankfurt, Deutschland

Mathematically, delayed differential equations evolve in infinite dimensional state spaces. It is hence conceivable that time-delayed systems can be approximated by a set of N+1 ordinary differential equations, with the trajectory of the primary variable converging to the solution of the original time-delayed system when N→∞. We show that this program can be carried out using sequences of time-delay kernels related to discrete gamma distributions.

We present several analytical and numerical results for the
proposed approximation scheme, finding that the instability
of fixed points due to increasing time delays is captured
accurately already for N∼ 10. For the Mackey-Glass
system we find that the locus of a limit-cycle doubling
are recovered in good approximation only for substantially
larger N∼ 10^{2}−10^{3}, with the transition to chaos
requiring an even larger state space.
In general, we find that relative approximation errors
scale as 1/N. In addition, we discuss how the approximation
proposed can be applied to the case of distributed time delays.

It is in general an approximation to model a given experimental protocol by a dynamical system characterized by a single time delay T. Using a distribution of time delays peaked at T, with width ∼1/N, can hence be argued to provide a more accurate description of real-world non-Markovian processes.