Symbolic dynamics

The page follows to "Fractals, Chaos, Power Laws" by Manfred Schroeder.

Symbolic dynamics are used to extract orbit properties based on topology alone before considering metric properties. Instead of listing the sequence of iterates xn , it often suffices to state whether they fall to the left (L) or the right (R) or on the extremum or center (C) of the map. The sequence of symbols L, R, C is then called the symbol dynamics (or pattern) for a given orbit. Symbols L, R, C determine a partition i.e. separation of phase space into disjoint regions.

Period doubled superstable orbits

Thus the superstable period-2 orbit has the symbol dynamics (or kneading sequence) CLCL... Restricting the notation to a single period we write simply CL. You can check it easy in the right applet window (note, that L means low too).

We change L and R in comparison with the Schroeder's book as since x2 + c map is turned over with respect to the logistic map.

To obtain the period-4 orbit one writes two periods of the period-2 orbit CLCL, and then changes the second C to R if the number of L to the left of it is odd. Otherwise the second C is changed to L.

I.e. we get CLRL and so on.

The period-8 orbit CLRLLLRL can be rewriten as CLRL3RL .
The algorithm of counting the previous L's and checking whether it is even or odd is related to the fact that the slope of the quadratic map is negative for the left half of the map. Each iteration when orbit falls to the left, a small difference in xn changes its sign. For an odd number of sign changes (L) the final difference changes its sign (while for an even number does not).

This is one of the most important properties of all unimodal (one-extremum) maps. Therefore these maps have universal ordering of their symbolic dynamics as the control parameter is changed.

You can easy recall that you should change the central C to compliment initial substring to odd L-parity as since period-2 orbit CL is obtained as period doubling of period-1 orbit C.

The parenting of new orbits

This algorithm can be applied to any superstable period-n orbit. E.g. the period-3 critical orbit below with symbol dynamics CLR is doubled as CLRRLR.

More generally one can derive period-km orbit from K and M orbits with periods k and m. Copy m times the symbolic dynamics of K and replace each of the (m-1) C's (exept the first C) by one after another of the (m-1) symbols of M, interchanging L and R if the number of L in K is odd.
For example, the 2-orbit CL is tripled by means of the 3-orbit CLR by copying the 2-orbit three times CLCLCL and by replacing the second and the third C's by the complement of the second and the third simbols of CLR. This yields CLRLLL.
Symbolic dynamics determine ordering of orbits too. If the number of L's in the initial equal portion of the strings is odd (even), then the orbit with C or R (L) as the first distinct symbol follows the other. Thus the 6-orbit CLRRLR (with c = -1.772886) follows the orbit CLRLLL (c = -1.4759).

Harmonic and antiharmonic

Another algorithm interpolates a new orbit between two known orbits P and Q by taking the intersection of harmonic H(P) of P and antiharmonic A(Q) of Q. The harmonic of an orbit is formed as before by period doubling. The antiharmonic of an orbit Q, which in general is not a possible periodic orbit, is defined just like the harmonic exept that R and L are interchanged in the replacement of the second C. The intersection means the string with initial equal symbols of harmonic and antiharmonic strings.

As since we only illustrate (but not explain) these rules, the page looks like Harry Potter's magic :)

I found in the Net only Symbolic Dynamics in Mathematics, Physics, and Engineering and Symbolic Dynamics by Nicholas B. Tufillaro. He wrote:
Symbolic dynamics allow to name uniquely the orbits in the quadratic map. It turns out that for the quadratic map, the kneading sequence increases as c increases. By combining the kneading theory with an additional property of the quadratic map (namely that it has a negative Schwarzian derivative), we obtain a detailed description of how periodic orbits arise as c (and hence the kneading sequence) increases. This theory explains the qualitative features of the bifurcation diagram.
A nice introduction to the theory is given by R.L.Devaney in "An introduction to chaotic dynamical systems".

Kai T. Hansen Symbolic dynamics in chaotic systems (1993)


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updated 2 July 2002