Biomedical Engineering Reference
In-Depth Information
of logical structures associated with the connections between them, including
complicated networks of causal relationships among variables, time delays be-
tween cause and effect, arbitrarily complex spatial inhomogeneities, or even
externally imposed noise. The way to recognize a linear dynamical system is
that its equations of motion will involve only polynomial functions of degree
one in the system variables; there will be no products of different system vari-
ables or nontrivial functions of any individual variable. (Examples of nontrivial
functions include squares or square roots, threshold functions that specify dis-
continuous switching of parameters as the system variables change, or quantities
that have simple geometric interpretations but turn out to be complicated func-
tions of the fundamental variables.) Gradients of any order may appear, how-
ever, as well as coefficients that are nontrivial functions of spatial position and
time.
2
For all types of linear systems, the constraint that the sum of any two solu-
tions also must be a solution has profound consequences. Simply put, the full
range of behavior of a linear system is understood as soon as its behavior in an
infinitesimal region of its state space is understood. In the absence of an external
driving force, there is one special solution to any linear system where the vari-
ables are time-independent—everything just sits still. This is called the
fixed
point
. A trivial example is the equilibrium point of a weight hanging from an
ideal spring in a perfectly uniform gravitational field. Here the system variables
are the position and velocity of the weight, which can both be defined to be zero
at the fixed point. Another example is the surface of a liquid that may have rip-
ples governed by surface tension (capillary waves) described by a linear theory.
The system variables here are a field representing the height of the liquid at all
points in space and the time derivative of that field. Again, the system variables
can be defined to be zero at the fixed point corresponding to a quiescent, flat
surface.
When variables are defined so as to be zero at the fixed point, linearity im-
plies that every solution can be multiplied by an arbitrary factor to yield another
solution. Thus solutions with arbitrarily large amplitudes can be multiplied by an
arbitrarily small factor to yield solutions infinitesimally close to the fixed point,
indicating that the nature of solutions very near the fixed point determines all of
the possible solutions. The situation is further simplified by the fact that solu-
tions in the vicinity of the fixed point come in only three types—stable, unsta-
ble, and marginal.
In a linear
stable system
, all solutions asymptotically approach the fixed
point as time progresses. The typical case is that beginning from any initial point
in state space, the variables decay toward the fixed point by first rapidly ap-
proaching a particular line in state space and then relaxing exponentially along
that line toward the origin.
3
In an
unstable system
, all solutions that do not start
exactly on the fixed point diverge from it exponentially at long times. The
mar-
ginal
case, in which the variables neither decay to zero nor diverge, occurs pri-
