Oscillations, Waves, and Interactions - GWDG

2.3 Hybrid systems
Complex dynamics of nonlinear systems 415
The third example of a dynamical system exhibiting chaotic dynamics is a hybrid
system consisting of a set of ordinary differential equations (ODEs) coupled to an
automaton (i. e., a finite state machine). This kind of systems occurs, for example,
whenever the dynamics is governed by some switching rules. With Karsten Peters we
studied simple hybrid systems in order to understand irregular behaviour in production
systems as used in large factories [23]. Of course, such systems are perturbed by
many stochastic influences (e. g., human behaviour, accidents, supply problems, ...)
but one may ask the question whether irregular behaviour in a production line may
also originate from its internal rules and operating conditions. In other words, does
an ideal production process without any stochastic influences always operate properly
with some constant or periodic output? To address this question we investigated
a simple hybrid system representing some unit in a larger production line. As illustrated
in Fig. 8 this unit consists of a server S processing three types of workpieces
coming from previous production units P1, P2, and P3. Since at a given instant of
time the server can process the input from one of the production units, only, some
switching rules are required and the outputs of the Pi have to be stored in some
buffers. Let x1, x2, and x3 denote the contents of the buffers belonging to P1, P2,
and P3, respectively. If each of the production units Pi provides constant output with
some rate fi the buffer contents xi will increase with ˙xi = fi. On the other hand,
the buffers are emptied by the server which formally can be described by emptying
rates ei. The ODE-system describing the buffer contents is thus given by
˙x1 = f1 − e1
˙x2 = f2 − e2 (9)
˙x3 = f3 − e3
where the filling rates fi and the emptying rates ei should fulfill a balancing condition
f1 + f2 + f3 = c = e1 + e2 + e3 to avoid overflow or complete emptying of buffers.
The total input rate c is constant if all rates fi are constant and without loss of
generality we shall set c = 1 in the following. As already mentioned at a given time
the server can process only input from one of the production units Pi. Therefore,
the emptying rates ei are actually functions of time and only one of them is larger
than zero corresponding to the Pi which is currently emptied by the server S. This
is the point where some switching rules have to be defined that state which Pi has
to be served (ei(t) > 0) at a given time. A possible set of rules is the following:
• If a buffer content xk reaches some maximal value b then switch server S to
that buffer to empty it (→ ek = 1).
• If the buffer that is currently processed by S is empty (xk(t) = 0) then switch
server S to the next buffer (→ ek+1 = 1) in a cyclic order.
Here we introduced the maximal buffer content b which is an important parameter
from the practical as well as the dynamical point of view. Of course, any manufacturer
will try to keep buffers small. But it turned out that the dynamics very strongly