Biomedical Engineering Reference
In-Depth Information
system variables are defined.
All physical systems describable in terms of classi-
cal equations of motion are nonlinear
. (The quantum mechanical theory of at-
oms and molecules is a linear theory: the connection between it and the
comparatively macroscopic processes at the cellular level and larger is beyond
the scope of the present discussion.) In all real systems, deviations of large
enough amplitude require nonlinear terms in the relevant model. There is no
such thing as a truly linear spring or a waves on a fluid obeying a perfectly lin-
ear equation of motion. This is why the study of nonlinear dynamics has such
broad relevance.
The consequences of nonlinearity are profound. Most importantly, nonlin-
ear systems may contain multiple attractors, each with its own basin of attrac-
tion. Thus the fate of a nonlinear dynamical system may depend on its initial
state, and a whole new set of phenomena arises associated with the way in
which basins of attraction shift as parameters are varied.
Nonlinearity can also give rise to an entirely new type of attractor. Limit
cycles in nonlinear systems may be quite complicated, circling around in a
bounded region of state space many times before finally closing on themselves.
It is even possible (and quite common) for a trajectory to be confined to a region
of state space where there are no stable limit cycles or fixed points. The system
then appears to follow an irregular trajectory that is said to lie on a
strange at-
tractor
. The trajectory comes arbitrarily close to closing on itself, but never
quite does, and two identical systems that come arbitrarily close to each other in
state space diverge rapidly thereafter. A strange attractor is the state space struc-
ture associated with the phenomenon known as
chaos
.
Finally, in spatially extended systems nonlinearities can give rise to
pattern
formation
, the spontaneous creation of attractors with nontrivial spatial struc-
ture in a system with no externally imposed inhomogeneities. (See (33,8,20) for
textbook treatments of pattern formation and spatiotemporal structures in large
systems. For more technical treatments from a physics perspective, see (17,6,4)).
Examples include the formation of stripes or spots in chemical reaction-
diffusion systems and excitable media, which find applications in such processes
as butterfly wing coloration and cardiac electrodynamics.
The essential features of nonlinear systems can be illustrated with the sim-
plest of examples, the driven, damped oscillator. Figure 1 shows a picture of the
system. We assume the spring is nonlinear: it gets stiffer under compression
and softer under extension. With an appropriate definition of the zero of
x
,
the position of the mass, the equation of motion can be written as
2
a
where
m
, H,
k
, and
k
' are constants and
h
is
the deviation of the ceiling from its average height. Defining
z
=
x
, the equation
of motion can be written as two coupled equations in a two-variable state space:
2
mx
=
H
x
+
k h
(
x
)
+
k
(
h
x
) ,
a
and
x
=
v
. We will consider cases in which the
ceiling oscillates according to
h
= B sin(X
t
).
v
=
H
v
k h
(
x
)
+
k
(
h
x
)
