Biomedical Engineering Reference
In-Depth Information
ful theorem shows that the important topological features of strange attractors
can be reconstructed from a time series measurement of a single variable (30).
One can construct the state space structure of the attractor using time-delayed
values of that variable rather than synchronous measurements of all of the sys-
tem variables. The reconstruction is said to be
embedded
in a space of dimen-
sion equal to the number of time delays used and the theorem says that as long
as the embedding dimension is large enough, the topological features of the tra-
jectory will be accurately reconstructed. (See (24) for a discussion of embed-
ding.) This has led to the development of a number of computational tools for
analyzing time series data to determine whether a system can be modeled using
a small number dynamical variables or not, though prohibitive difficulties are
almost always encountered if more than about 10 variables are required. (As-
pects of this sort of time-series analysis are discussed in the preceding chapter
by Shalizi.)
The second item of interest is the characterization of the strength of the
chaos (its degree of unpredictability) via the
Lyapunov exponents
. Suppose a
system is following a trajectory in state space that is a long time solution to the
equations of motion. We imagine an almost exact copy of the system at time
t
=
0. The copy has exactly the same parameter values as the original—it is the
same system—but the variable values at
t
= 0 differ by a tiny amount from the
original. In a chaotic system, the difference between the variable values in the
copy and the original will grow (on average) with time. Ignoring short-time
scale fluctuations, the difference between a given variable, say
x
, in the two sys-
tems will grow exponentially: E
x
= E
x
0
exp(M
t
). The quantity M, with dimensions
of 1/time, is called the Lyapunov exponent.
4
A large M indicates rapid divergence of nearby trajectories, which implies
that prediction of future values of the variables requires extremely precise
knowledge of the present values. The consequence of exponential divergence is
that accurate prediction becomes prohibitively difficult over times larger than a
few times M
-1
. This is only a quantitative issue: chaos does
not
imply some mys-
terious new source of randomness of the type, say, that is found in measure-
ments on quantum systems. Nevertheless, the mathematics of exponential
growth makes a qualitative difference in practice for would-be predictors of the
motion. The increase in precision of measurement required to make accurate
predictions is so rapid within the desired time interval covered that useful long-
time prediction is impossible.
The third item of interest is the nature of the transition to chaos as a parame-
ter is varied, i.e., the type of bifurcation that leads to the emergence of a strange
attractor. Perhaps the most celebrated result in chaos theory is the proof by Fei-
genbaum that all discrete maps in a broad class go through a quantitatively iden-
tical transition, dubbed the
period-doubling route to chaos
(9). In a period-
doubling bifurcation, a periodic orbit undergoes a change in which only every
other cycle is identical. One then still has a periodic orbit, but its period is twice
