Oscillations, Waves, and Interactions - GWDG

416 U. Parlitz
P 1 P 2 P 3
S
P 1 P 2 P 3
Figure 8. Tank system describing a logistic process where three production units P1, P2,
and P3 deliver some inputs (e. g., workpieces) that are stored in buffers and processed by a
subsequent server S that switches between the buffers to empty them. The filling is governed
by Eqs. (9) with fi = 1/3 following the switching rule stated in the text. If the buffer size
b (indicated by the horizontal orange line) is large the resulting dynamics is periodic (left
column, b = 0.8). Too small buffers, however, lead to chaotic switching as illustrated with
the example shown in the right column for b = 0.5.
depends on b. This is illustrated in Fig. 8. In the left column a case is shown where
b = 0.8 is relatively large. The system started with a configuration shown in the top
row where the server processes the blue buffer P1. When P1 is emptied S switches to
the yellow buffer P3 etc. following the rules stated above. The switching process is
shown row by row in the lower left diagram starting in the upper left corner. As can
be seen a periodic state is reached after some short transient visible in the first six
rows. The right column of Fig. 8 shows what happens if the buffer size is reduced to
b = 0.5. Now the server doesn’t operate periodically anymore but switches chaotically
between the buffers of the production units.
This shows that irregular dynamics can occur in (production) logistic processes
even if they are not randomly perturbed. Here, the transition to chaos consists of
interesting (non-standard) bifurcation scenarios and of course the type of dynamics
has also some influence on the throughput of the whole unit [23,24]. An example
for a hybrid (or tank) system occuring in physics is discussed in Ref. [25] where this
formalism is used to describe front dynamics in semiconductors.
S