Symbol sequence

The time evolution of the function f is thus replaced by a sequence of discrete symbols labeling the bins visited by each point of the orbit. Because of the coarse-graining of the phase space, however, detailed knowledge of the actual orbits is generally lost i.e. many different orbits may yield the same symbolic sequence. Different state-space partitionings also generally give rise to different symbolic representations. 

In this way, B preserves the essential dynamical properties of / by preserving orbits i.e., y oB x) = Bof [x). In particular, there is a one-to-one correspondence between repeating symbol sequences in F and periodic orbits of... 

The discretized symbol-sequence defined in equation 8.39 suggests that we might use two other familiar measures of complexity to characterize the various dynamical regimes of behavior namely, the pattern entropy and dynamical entropy. 

The symbol sequence in a formula (LaPb3 or Pb3La) is of course arbitrary and, in... 

The formulae of selected compounds and prototypes are listed in alphabetical order. The alphabetical order is also used to define the symbol sequences within all the formulae. Reference is made to the numbers of paragraphs where the specific compound or structure type is described. 

In particular, the periodic orbits are in correspondence with finite sequences such as )W2 Up of period p. The periodicity occurring in the symbol sequences translates into the periodicity of the corresponding trajectory crossing the Poincare surface of section. 

So, assume nt > 1. If nt = 1 for some i, then there is a unique way of closing the structure and we obtain a triple of 8-gons associated to the symbolic sequence... 

In the case b = 10, the only known symbolic sequences ( i, n,), such that the... 

Cantor s next step was the following. Suppose there is a set S. If a one-to-one correspondence between the elements of S and the elements of N can be estabhshed. Cantor called the two sets sets of the same cardinality . In symbols iV = 5 = Uq. The question now is whether there exist larger sets than the set N. Cantor proved this in the aflBr-mative in the following way. Suppose there are two symbols a and 6. Consider the set of semi-infinite symbol sequences S = abaabbbaab..., aababaabbbaaba..., ababaababaab.,... that are made up of the two sym-... 

Since our grammatical rule does not impose any restrictions on the length of a symbol sequence, we are obviously allowed to write down infinitely long sequences. There are two kinds periodic and aperiodic sequences. With the help of the mapping A —> 0, B —> 1 and C —> 2 the symbol sequences can be interpreted as real numbers in base-3 notation. On the basis of this analogy, we call finite (or infinite periodic) sequences rational and infinite aperiodic sequences irrational . Thus, the itinerary of the trajectory shown in Fig. 2.11(a) is rational. An ex-... 

Symbolic dynamics is a powerful, but qualitative, tool. We can easily imagine a system trajectory that is close to the trajectory shown in Fig. 2.11(a), but not quite identical. Both trajectories axe then characterized by the same word, ABC AC. Thus, symbol sequences do not specify system trajectories uniquely. In other words, there is no one-to-one correspondence between the impact parameters b and the words constructed from the alphabet A, B, C. ... 

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