Biomedical Engineering Reference
In-Depth Information
the labial motion stops. Now the bird is ready to perform a mini-inspiration
before the execution of a new syllable.
5.1.2 Bifurcations
In our description, an increase in pressure has a remarkable consequence for
the state of the system. As the pressure grows beyond a critical value, the
labia begin to oscillate. This qualitative change in the behavior of a physical
system as a parameter is changed is called a
bifurcation
[Solari et al. 1996].
Bifurcation is a concept that refers to any qualitative change in the solutions
of a nonlinear problem, as a control parameter is varied. The transition from
a stationary state to an oscillatory state is one example. But not the only
one.
Some systems show a bifurcation from one static state to another static
state. As an example, let us analyze briefly what happens when a pair of forces
is exerted on a metal plate (as illustrated in Fig. 5.2a). If the magnitude of
the force on the plate is below some critical value, the undeformed plate is
stable
. We can deform the plate a little, and after the perturbation is removed,
the system will evolve back towards the undeformed state. The situation is
different if the magnitude of the force exceeds the critical value. Then, the
(a)
(b)
Fig. 5.2.
A bifurcation is a qualitative change in the dynamics of a system as a
parameter is changed. In this figure, two examples are illustrated. In (
a
), a plate
is subjected to a force (hold a credit card between your fingers and try the ex-
periment). If the force is high enough, the plate (card) will bend. The symmetric
solution becomes unstable. The initial conditions will determine whether the bend-
ing is to the right or to the left, if the plate is perfectly symmetric. In (
b
), we show
what happens if you blow air between a pair of paper sheets. If the airflow exceeds
a threshold, then the sheets begin to oscillate
