Communication delays are common in many complex systems. It has been shown that these delays cannot be neglected when they are long enough compared to other timescales in the system. In systems of coupled phase oscillators discrete delays in the coupling give rise to effects such as multistability of steady states. However, variability in the communication times inherent to many processes suggests that the description with discrete delays maybe insufficient to capture all effects of delays. An interesting example of the effects of communication delays is found during embryonic development of vertebrates. A clock based on biochemical reactions inside cells provides the periodicity for the successive and robust formation of somites, the embryonic precursors of vertebrae, ribs and some skeletal muscle.
Experiments show that these cellular clocks communicate in order to synchronize their behavior. However, in cellular systems, fluctuations and stochastic processes introduce a variability in the communication times. Here we account for such variability by considering the effects of distributed delays. Our approach takes into account entire intervals of past states, and weights them according to a delay distribution. We find that the stability of the fully synchronized steady state with zero phase lag does not depend on the shape of the delay distribution, but the dynamics when responding to small perturbations about this steady state do. Depending on the mean of the delay distribution, a change in its shape can enhance or reduce the ability of these systems to respond to small perturbations about the phase-locked steady state, as compared to a discrete delay with a value equal to this mean. For synchronized steady states with non-zero phase lag we find that the stability of the steady state can be altered by changing the shape of the delay distribution.
We conclude that the response to a perturbation in systems of phase oscillators coupled with discrete delays has a sharper functional dependence on the mean delay than in systems with distributed delays in the coupling. The strong dependence of the coupling on the mean delay time is partially averaged out by distributed delays that take into account intervals of the past.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa.de:bsz:14-qucosa-98456 |
| Date | 08 March 2013 |
| Creators | Wetzel, Lucas |
| Contributors | Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften, Prof. Dr. Frank Jülicher, Dr. Saul Ares, Prof. Dr. Frank Jülicher, Prof. Dr. Lutz Schimansky-Geier |
| Publisher | Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis |
| Format | application/pdf |
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