The deterministic logistic map is the well known difference equation
xn+1 = µ xn (1 - xn)
where μ is a bifurcation parameter.
This deterministic equation can be obtained from
the stochastic logistic map by choosing
(the ergodic processes)
= µ and
= 0.
For each fixed µ in [0,4], the interval [0,1] is forward invariant
and therefore the long-run behavior of the system takes place in
this set for all initial values from [0,1]. The figure below
depicts this long-run behavior. It can be checked analytically that
at µ = 1 a transcritical bifurcation happens and that at
µ = 3 a period-doubling bifurcation occurs.
However, these two bifurcations can also be found through a numerical
study only. Noting that zero is a fixed point for all µ's, the
figures below clearly indicate the occurrence of these bifurcations.
Bifurcation diagram of the logistic map
Lyapunov exponent of the logistic map
(large version)
Bifurcation diagram for the second iterate of the logistic map
= µ and
= 0.
