Biomedical Engineering Reference
In-Depth Information
finite speeds of signal propagation cause
f
to depend also on values of
x
at times
earlier than
t
.
In
spatially extended systems
, each system variable is a continuous func-
tion of spatial position as well as time and the equations of motion take the form
of partial differential equations:
s
x
=
fxxx
( ,
,
2
,...;
p
,
, )
t
[2]
s
t
where / is the spatial gradient, /
2
is the Laplacian, and the dots represent higher-
order derivatives of
x
. The parameters
p
may also be externally imposed func-
tions of position, as would be the case for a system evolving in an inhomogene-
ous environment. (If the environment itself is affected by the system variables,
however, then variables representing the environment become system variables
rather than external parameters.)
A set of equations describing a discrete-time dynamical system takes the
form
x
(
t
+ 1) =
F
(
x
(
t
);
p
t
).
[3]
Here the function
F
directly gives the new
x
at the next time step, rather than the
derivative from which a new
x
can be calculated. The function
F
is often re-
ferred to as a
map
that takes the system from one time step to the next.
In all cases, the evolution of the system is described as a motion in
state
space
, the space of all possible values of the vector
x
. A
trajectory
is a directed
path through state space that shows the values of the system variables at succes-
sive times. The theory of dynamical systems is concerned with classifying the
types of trajectories that can occur, determining whether they are robust against
small variations in the system variables, categorizing the ways in which the pos-
sible trajectories change as parameters
p
are varied, and developing techniques
both for simulating trajectories numerically and inferring the structure of trajec-
tories from incomplete sets of observations of the system variables. The most
basic structures arising in the classification of state space trajectories—fixed
points, limit cycles, transients, basins of attraction, and stability—will be ex-
plained below as they arise in the context of some simple examples.
3.
LINEAR SYSTEMS AND SOME BASIC VOCABULARY
A
linear system
is one for which any two solutions of the equations of mo-
tion can be combined through simple addition to generate a third solution, given
appropriate definitions of the zeros of the variables. The system of equations can
be extremely complicated, representing large numbers of variables with all sorts
