P- and D-Bifurcations of the Additively Perturbed Logistic Map
The ergodic noise process used is given by an i.i.d. sequence of random variables
which are uniformly distributed on some fixed interval [a,b]. We will fix a=0 and
study the dependence of the bifurcation behavior on the upper bound b of this interval
in the following.
The logistic map perturbed by additive noise is given by
xn+1 = µ xn (1 - xn) +
where
is the i.i.d. sequence of random variables introduced above.
Note that for non-trivial noise, zero is not a solution of this equation anymore.
The links given below lead to P- and D-bifurcation diagrams of the additively perturbed
logistic map where the noise processes take their values in the interval specified.
The dependence of the Lyapunov exponent on the upper bound of the interval
[0,b] is analyzed in the following figures.
In summary, the numerical results lead to the conjectures
- additive noise destroys the transcritical bifurcation at µ= 1 on the P- as well as on the D-level
- for small noise, the period-doubling bifurcation at µ = 3 remains on the P- but not on the D-level
and large noise destroys the bifurcation on both levels
- the noise stabilizes for &mu's less than or equal to 3.0
- the noise may destabilize, an effect that can be observed for many µ's > 3.0
To the
Introduction
Multiplicative noise case
Deterministic case
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