Description
In the 1650's Christiaan Huygens invented the pendulum clock as a way of accurately measuring
time, in the hope that this would solve the problem of determining longitude for seafarers.
Incidentally, he noticed that two clocks suspended from a common frame, no matter how they were
originally set up, would always end up beating in time, a phenomenon he described as an `odd kind of sympathy'.
He realised that it was caused by the clocks interacting through minute deformations of the frame. A more recent
experimental realisation can be seen on youtube or if you prefer
mass production there is this.
This is the first historical record of synchronisation, where the dynamics of a number of oscillators which
are weakly coupled to each other eventually drives them to be in phase with each other. The subject has been applied in a
bewildering variety of different areas from why fireflies flash together, to pacemaker cells in the heart, to the control
of electronic circuits and Josephson junctions in physics. Some idea of the range of topics this applies to can be seen in
in a TED talk by Steven Strogatz
The basic idea of the project will be to examine some simple mathematical models of synchronisation. This will involve
using some basic mathematical techniques in dynamical systems, but potentially also touches on chaos and if desired
could also involve playing around with some models numerically.
Group project
The idea will be to study some basic mathematical models of synchronisation, including at least two of the following:
- Metronomes synchronisation on a common base. This is essentially the problem of Huygen's clock. This is essentially a problem in dynamics, which although cannot be solved exactlly, can be analysed using `fast time-slow time' techniques.
- The Kuramoto model. This system has a very neat mathematical description in terms of particles with a sine interaction between them. It has been widely studied in the literature.
- Integrate and Fire models. These can be used to describe such phenomena as the synchronisation of fireflies or pacemaker cells. Unlike the two models above which are described by the smooth evolution of ODEs, the dynamics here involves the system moving in discrete `jumps'
Mode of operation and evidence of learning for the group project
Students will develop an understanding of the problem through reading relevant papers and making their own numerical and analytic calculations, and then communicate this in written and oral formats.
Individual projects
There are many directions in which the ideas learnt in Michaelmas can be taken further. Some examples might include
- Multi-metronome models, more advanced models including the effects of damping or better modelling of the escapement mechanism
- Planetary locking. Recently Artemis II photographed the dark side of the moon. But why is this moon's rotation locked to the period of its orbit so that we always see the dark side? One can also explore synchronisation phenomena in systems of moons such as around Saturn.
- Generalisations of the simple Kuramoto model. Examples include oscillator dependent coupling, and solving the Kuramoto model on networks where only some oscillators
are coupled to each other
Mode of operation and evidence of learning for the individual project
Students will investigate their chosen aspect of the problem through reading relevant papers and making their own calculations, and then communicate their results in written and oral formats.
Prerequisites
None really. A nice way to describe the metronome problem uses Lagrangian mechanics so it may be useful to have taken 2H Mathematical Physics, but this is not essential. The project shares some ideas with the module `Mathematical Biology III' so this may have some synergy but is certainly not an essential corequisite
Resources
The following are some useful papers for getting into the three topics
- Metronomes: Pantaleone,J "Synchronization of metronomes" Am. J. Phys 70 No. 10 PDF
- Metronomes: Bennett, Schatz, Rockwood, Wiesenfeld "Huygen's Clocks" Proc. Roy. Soc. A458(No.2019) p. 563 access via Library
- Kuramoto model: S.H. Strogatz "From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators" Physica D 143 (2000) p. 1 access via Library
- Integrate and Fire model: Mirollo, R. E. and Strogatz, S. H. "Synchronization of pulse-coupled biological oscillators SIAM Journal on Applied Mathematics, 50 (6), pp. 1645 access via library
There are also number of books on synchronisation:
The first of these is a popular book with fascinating insights into the history and applications of synchronisation. The last book is
a self-contained but rather thorough treatment of the mathematics which goes somewhat beyond the scope of the project- the middle book is
somewhere in between.
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