How do small differences in nonidentical pulse-coupled oscillators induce great changes in their synchronous behavior?
1 Instituto de Investigaciones Físicas, Casilla 8635, Universidad Mayor de San Andrés, La Paz, Bolivia
2 Institut für Physik, Humboldt-Universität zu Berlin, Robert-Koch-Platz 4, 10115 Berlin, Germany
3 Potsdam Institut für Klimafolgenforschung, PO Box 60 12 03, 14412 Potsdam, Germany
4 Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3FX, UK
5 Interdisciplinary Center for Nonlinear Phenomena and Complex Systems & Unité d'Ecologie Sociale, CP. 231, Université Libre de Bruxelles, Campus de la Plaine, Bld. du Triomphe, Brussels, Belgium
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Received: 25 June 2014
Revised: 17 October 2014
Published online: 10 December 2014
We studied synchronization and clustering in two types of pulse-coupled oscillators, namely, integrate-and-fire and light-controlled oscillators. We considered for the analysis globally coupled oscillators, either by a mean-field type coupling or a distance-dependent one. Using statistically diverse measures such as the transient, probability of total synchronization, fraction of clustered oscillators, mean size, and mean number of clusters, we describe clustering and synchronous behavior for populations of nonidentical oscillators and perform a comparative analysis of the behavioral differences and similitudes among these types of oscillators. Considering a mean-field approach, we found high probability of total synchronization in all cases for integrate-and-fire oscillators; on the other hand, in a more realistic situation, for light-controlled oscillators, i.e., when oscillators do not fire instantaneously, the probability of total synchronization decreases drastically for small differences among the oscillators and subsequently, for larger differences, it slightly increases. When the coupling strength depends on the distance, the probability of total synchronization plummets dramatically with the number of oscillators especially in the case of integrate-and-fire oscillators. The latter constitutes an interesting result because it indicates that in realistic situations, the probability of total synchronization is not very high for a population of pulse-coupled oscillators; this entails that its utilization as a paradigmatic model of total synchronization does not suit well, especially when the coupling depends on the distance. This article is dedicated to our good friend and colleague Hilda Cerdeira as a tribute to the scientific work developed over her career.
© EDP Sciences, Springer-Verlag, 2014