1. Compose a program to locate unstable cycles of
period 2, 4, 8,...2k for
logistic map
at the critical point
and calculate
multipliers (stability eigenvalue) for these cycles
(up to period 4096).
Hint: Use a
well convergent recurrent schema
, where f(x) designates a result of 2k -th iteration of the
logistic map starting at x.
2. For logistic map draw several levels of the Cantor-like construction of the critical attractor on the computer screen.
3. Compose a program to draw a plot of the Feigenbaum sigma-function.
4. Calculate the Hausdorff
dimension D of the critical
attractor. For this, compose a program to compute sums
and select D to make
sums
and
equal. How does the accuracy depends on the level k?
5. Compose a program to obtain spectrum of the
generalized dimensions of Rényi
and scaling-spectrum
. Draw plots of these functions estimated at different levels
k. Compare
the data with results of computations on a basis of the two-scale Cantor set
model of the critical attractor. Compute as accurate as possible the
information and correlation dimensions of the critical attractor.
1. Depict on computer screen a
bifurcation tree and Lyapunov exponent plot for
logistic map with additive noise:
, where
is a random sequence
generated by computer (with zero mean). Demonstrate that the noise destroys
subtle structure of the bifurcation tree with increasing power as we consider
higher levels of its organization. How does the noise influence onto position
of the border of chaos?
2. Demonstrate scaling on the
bifurcation tree and on the Lyapunov exponent plot.
For this redraw the plot several times with subsequent rescaling for x by factor
, for
by factor
(here
, for Lyapunov exponent by factor
2, and for the noise intensity
by factor
(the last constant has been
found first by Crutchfield et al.).
3. Invent a way to utilize the scaling
property to estimate the noise scaling constant for mappings
with extremum of order N=4,
6, and 8.
4. Draw the charts of Lyapunov exponents for cubic map
at different values of
the noise amplitude. How is the chaos border
transformed and how is changed location of regimes of maximal stability?
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