Numerical Methods for Delay Differential Equations with Threshold State-Dependent Delay. Issue 36 (2022) - Wang, Zhao; Humphries, Antony R.
Abstract: Threshold delays arise naturally in a wide variety of dynamical systems. Standard routines for performing numerical analysis of delay differential equations (DDEs) typically handle discrete constant or state-dependent delays, but are not directly applicable to distributed delay problems. We show how to reformulate the threshold delay problem as a discrete delay DDE, with the delay as an extra variable, by differentiating the threshold condition. This allows initial value problems to be solved in Matlab using ddesd . We also describe two implementations of the threshold delay DDE in DDE-BIFTOOL to perform numerical bifurcation analysis. One approach enables us to determine steady state bifurcations via a correction of characteristic values from a modified but related problem with discrete delays. The second approach involves introducing dummy constant delays which are used to discretize the integral in the threshold condition. This allows the threshold delay to be solved for directly within DDE-BIFTOOL, and enables the full functionality of DDE-BIFTOOL, so we can compute periodic orbits as well as steady states, along with their stability and bifurcations. We illustrate these techniques on a gene regulation model where the delay, driven by a transport process, is implicitly defined by a threshold condition. The speed of the transport process depends on the state of the system making the delay both state-dependent and distributed.