Intuitively, a system is unpredictable if a lot of information about its past state tends to yield only a little information about its future state. There are many different ways to make this precise. Here we shall consider four different definitions of unpredictability, three of them original.
Let us consider a discrete dynamical system (f,X), which consists of a "state space" X and a
function f mapping X into X. For the details to follow, it should be assumed that X is a finite
space, so that concepts of algorithmic complexity may be easily applied. But in fact, the ideas are much more general; they apply to any metric space X. A trajectory of the system (f,X) is a
sequence (x,f(x),f2(x),…), where fn(x)=f(f(…f(x)…)), the n’th iterate of f applied to x.
In this notation, we may define the Liapunov sensitivity, or L.-sensitivity, of a dynamical
system as follows:
Definition 4.1: The L.-sensitivity K(a,n) of a dynamical system (f,X) at a point x in X is
defined as the average over all y so that d(x,y)<a of d(fn(x),fn(y)).
The function K tells you, if you know x to within accuracy a, how well you can estimate fn(x).
Different choices of "averaging" function yield different definitions. The most common way
of averaging two entities A and B is the arithmetic mean (A+B)/2, but there are other common
formulas. For positive numbers such as we have here, there is the geometric mean (AB)1/2 and
the power mean (Ap + Bp)1/p. In general, a function A which takes in n real numbers and puts
out another is said to be an average if min(x1,…,xn) % A(x1,…,xn) % max(x1,…,xn) for all n-tuples of numbers x1 ,…,xn.
If the average of a set of n numbers is defined as the maximum element of the set, and X is not
a discrete space but a space of real numbers or vectors, then in many cases it is known that
K(a,n) is equal to a exp(L(x)n), where L(x) is called the "Liapunov exponent" of the dynamical
system (Collet and Eckmann, 1980). Often the Liapunov exponent is independent of x, i.e.
L(x)=L. This exponent has the advantage of being easily computable. But the maximum function
is not always a reasonable choice of average: if one is interested in making a guess as to what
fn(x) is probably going to be, then one wants an average which (like, say, the arithmetic mean)
does not give undue emphasis to unlikely situations.
To measure the sensitivity of a system, one merely averages the sensitivities at all points x in
X. Here again, there is a choice to be made: what sort of average? But since we are speaking
conceptually and not making explicit calculations, this need not bother us.
Next, let us consider a form of unpredictability which has not previously been identified:
structural sensitivity, or S-sensitivity.
Definition 4.2: The S-sensitivity K(a,n) of a dynamical system (f,X) at a point x in X is
defined as the average over all y so that d(x,y)<a of d#(xf(x)…fn(x),xf(x)…fn(x)).
This measures how sensitively the structure of a trajectory depends on its initial point.
Conversely, one may also consider reverse structural sensitivity, or R.S.-sensitivity — roughly, how sensitively the point a trajectory passes through at time n depends on the structure of the trajectory up to that point. To be precise:
Definition 4.3: The R.S.-sensitivity K of a dynamical system (f,X) at a point x in X is defined as the average over all y so that d#(xf(x)…fn(x),xf(x)…fn(x))<a of d(fn(x),fn(y)).
This is not so similar to L-sensitivity, but it has a simple intuitive interpretation: it measures how well, from observing patterns in the behavior of a system, one can determine its immediately future state.
Finally, let us define what might be called structural-structural sensitivity, or
Definition 4.4: The S.S.-sensitivity K(a,n,m) of a dynamical system (f,X) at a point x in X is
defined as the average, over all y so that
d#(xf(x)…fn(x),xf(x)…fn(x))< a, of
This measures how difficult it is to ascertain the future structure of the system from its past
What is essential here is that we are talking about the unpredictability of structure rather than the unpredictability of spe