On the interpretation of Dirac δ pulses in differential equations for phase oscillators

Vladimir Klinshov; Leonhard Lücken; Petro Feketa

doi:10.1063/5.0040995

On the interpretation of Dirac δ pulses in differential equations for phase oscillators

Chaos. 2021 Mar;31(3):031102. doi: 10.1063/5.0040995.

Authors

Vladimir Klinshov¹, Leonhard Lücken², Petro Feketa³

Affiliations

¹ Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950 Nizhny Novgorod, Russia.
² Institute for Chemistry and Biology of the Marine Environment, University of Oldenburg, 26111 Oldenburg, Germany.
³ Chair of Automatic Control, Kiel University, Kaiserstraße 2, 24143 Kiel, Germany.

PMID: 33810720
DOI: 10.1063/5.0040995

Abstract

In this note, we discuss the usage of the Dirac δ function in models of phase oscillators with pulsatile inputs. Many authors use a product of the delta function and the phase response curve in the right-hand side of an ordinary differential equation to describe the discontinuous phase dynamics in such systems. We point out that this notation has to be treated with care as it is ambiguous. We argue that the presumably most canonical interpretation does not lead to the intended behavior in many cases.