Let
be a Rational Number in the Closed Interval
, and generate a Sequence using the
Map
![\begin{displaymath}
x_{n+1}\equiv 2x_n{\rm\ (mod\ 1)}.
\end{displaymath}](0_11.gif) |
(1) |
Then the number of periodic Orbits of period
(for
Prime) is given by
![\begin{displaymath}
N_p={2^p-2\over p}
\end{displaymath}](0_13.gif) |
(2) |
(i.e, the number of period-
repeating bit strings, modulo shifts). Since a typical Orbit visits
each point with equal probability, the Natural Invariant is given by
![\begin{displaymath}
\rho(x)=1.
\end{displaymath}](0_14.gif) |
(3) |
See also Tent Map
References
Ott, E. Chaos in Dynamical Systems. Cambridge: Cambridge University Press, pp. 26-31, 1993.
© 1996-9 Eric W. Weisstein
1999-05-25