MURPHYS-HSFS-2014 - 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems, April 7-11, 2014 - Abstract

Kuznetsov, Alexey

Relaxation oscillations and chaos in a delay differential equation model of a regulatory network.

One application of delay differential equations (DDEs) is modeling of oscillations in genetic regulatory networks. The delay simplifies modeling because it replaces many unknown reaction steps in the feedback loop that sustains oscillations. However, mathematical analysis of DDEs is difficult because they are infinitely-dimensional by construction and cannot be analyzed by methods common for finite systems of ordinary differential equations (ODEs). We simulate dynamics in a single delay differential equation and find that the oscillations remain periodic for growing delay if the equation includes only monotonic functions of the variable. Under this condition, we reduce the DDE to a three-dimensional system of ODEs. The resulting system is equivalent to a standard relaxation oscillator. We discuss implications of the similarity of the delay-induced oscillations and hysteresis-based relaxation oscillations.