Why be afraid of state-dependent delays?

Bernd Krauskopf
Department of Mathematics, University of Auckland

Delay differential equations (DDEs) are the mathematical models of choice in applications where delays arise naturally, for example, due to the time it takes different subsystems to communicate, process information and finally react. It is a good approximation in many cases to model such delay as constant. On the other hand, communication/processing times may well depend on the state of the system in a significant way. What does this mean for the dynamics of the governing DDE?

After a brief introduction to systems with delays, we present as a test-case example a prototypical conceptual DDE model for the El Niño Southern Oscillation climate system. We present physical arguments for the state-dependence of relevant delays and conduct a bifurcation analysis that demonstrates its effects for the overall dynamics of the system.

Apart from the specific interest for climate dynamics, this example also serves as a 'health warning' that state dependence is capable of generating additional dynamical phenomena. Hence, it must be taken seriously in applications. On the other hand, as this talk will also demonstrate, tools from bifurcation theory and associated numerical methods are now available to deal effectively with state-dependent delays. This means that there is no need to avoid/disregard state dependence in DDE models.

This is joint work with Andre Keane (Auckland) and Henk Dijkstra (Utrecht).