Conclusions

The dynamics of the data transfer network are generally complex, despite the fact that this system is discrete. In particular, even for rather small networks (a array is typical) aperiodic behaviour is prevalent --- it is periodic behaviour that is atypical. Only for very lightly loaded networks (typically, those in which fewer than around 12% of the 256 sites contain a packet) is the behaviour periodic. Of course, behaviour in an autonomous, discrete system of finite order must eventually be periodic, but the length of possible cycles and transients here is so large ( e.g. ) as to be `effectively infinite', and it would take many Universe lifetimes to compute all of them.

This work should be contrasted with that discussed in [4] in which it was shown that chaos in the true sense was observed in a discrete system in the limit that the size of the discrete steps tended to zero.

We have shown that:

• there are at least two senses in which the behaviour of our system can be said to be governed by attractors
• that two nearby states of the system diverge exponentially initially, which is indicative of the presence of at least one positive Liapunov exponent

Both these properties are characteristic of continuous dynamical systems that display chaotic behaviour. We therefore believe that chaotic dynamics as observed in continuous systems describes features observed in coarsely discrete systems with a sufficiently large number of states.