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∼ 102−103, 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.