Symbolic dynamics are used to extract orbit properties
based on topology alone before considering metric properties.
Instead of listing the sequence of iterates *x _{n}* , it
often suffices to state whether they fall to the

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 | |

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 | |

The period-8 orbit CLRL can be rewriten as
LLRLCLRL .
^{3}RL |

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*.

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 CL.
RLLL |

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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