Project III (MATH3382) 2019-20


Synchronisation

Peter Bowcock

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.

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.

Prerequisites

None really but it may be useful to have taken 2H Mathematical Physics. Also some of the ideas are related with Dynamical Systems III.

Resources

There are number of books on synchronisation:
  • "SYNC- The emerging science of spontaneous order" Strogatz,S
  • "Synchronisation: A Universal Concept in Nonlinear Sciences", Pikovsky, Rosenblum, Kurths
  • "Synchronisation: From Simple to Complex", Balanov, Janson, Postnov, Sosnovtseva
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.

There are also some shorter articles considering specific models of interest which will be useful:


email: Peter Bowcock

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