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Bifurcations of the Logistic Map

parameter r
2.9
initial condition
x
0
0.1
The sequence
x
n+1
=r
x
n
(1-
x
n
)
exhibits complicated behavior for certain values of the parameter
r
. When
r<3
, the sequence converges to a fixed point, but around
r=3
this fixed point bifurcates into an attracting two-cycle. As
r
increases further, the attractors continue to bifurcate until the sequence displays chaotic behavior around
r=3.57
.
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