# Question about predicting number of independant variables from measure data.

1. Sep 21, 2009

### Edwin

Suppose one has a continuous complex valued function f:X->X, where X is some a non-empty compact subset of the complex numbers, and of the form Z_n+1 = f(Z_n; a1, a2,..., am), where a1, a2,..., am, with 0 <=|ai|<=1, for i =1, 2,...,m, are independant complex valued parameters, and n = 1,2,3 ... etc.

Suppose we consider some attractor set B, such as the set of all points z1 whose successive iterates under the mapping f converge to a fixed point of f in X.

Note: since f is a continuous mapping of a non-empty compact subset of complex numbers into itself, the existance of a fixed point is gauranteed from a theorem in mathematical analysis.

Since there are m independant parameters, then the attractor set can have at most m degrees of freedom.

Suppose we are not given f, but just different snap shots of the attractor set B for different values of the parameters ai, i = 1, ..., m: is there a statistical technique that can give us some information about the parameters, or number of parameters, controlling the dynamics of our iterative map, if all we have are the snap shots of the attractor set B for different unknown values of the parameters ai, for i=1,...,m?

In particular, can one predict the number of parameters controlling the dynamics of the iterative map if those parameter values are changing slowely simultaneously as the iterations are unfolding?

Is it different if the ai are each independant random variables that obey the uniform distribution?

Inquisitively,

Edwin Schasteen