Entropy Estimates for Dynamical Systems

We apply a method proposed recently for estimat ing entropie s of sy mbol sequences to two ensembles of binary seq uences obtained from dynamical sys tems. T he first is th e ensemble of (0,1)sequences in a generating part it ion of the Henon map ; th e second is th e ensemb le of spat ial st rings in cellular automaton 22 in t he stat ist ica lly stat ionary state. In both cases , t he ent ropy est ima tes ag ree with prev ious estimates. In the latter case , we confirm a previous claim th at th e ent ropy of spatial strings in ru le 22 converges to zero, alt hough extremely slowly. One of the most important characterist ics of low dimensional chaot ic dynamical systems is t heir entropy. In spatially extended systems, we have in addition spatial strings whose non-vanishing entropy is an imp ortan t observable . In the first case , the ent ropy is most eas ily obtained indi rectly as the sum over the positi ve Lyapunov exponents [1,2) [for attractors; for repellers , see reference [3]), provided the equations of moti on are given analytically. If not, measuring the ent ropy can be non-tri vial if block-entrop ies (defined below) converge slowly. The latter is, in particular, the case for some cellular automata, where eit her spat ial st rings at fixed tim e or tempora l st rings at fixed space point s can have extremely slowly converg ing block ent rop ies [4]. Given an ensemb le of st rings made up of symbols Sj E {O,l} , with probabi lit ies p[51' 5 2 ... 5 n ] for finding blocks (S t5 2 ' " 5 n ) E {OJ l ] " at arbit rarily chosen posit ions, block entropies Hn are defined as Hn= L pIs, ... snJ log, pis , ...snJ· &, = 0, 1 The ent ropy h is defined as @ 1987 Complex Systems Publications, Inc. (1.1)