**1.** Compose a program to locate unstable cycles of
period 2, 4, 8,...2* ^{k}* for
logistic map

Hint: Use a
well convergent recurrent schema _{
}
, where *f*(*x*) designates a result of 2* ^{k}* -th iteration of the
logistic map starting at

**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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