Bifurcations of the Randomly Perturbed Logistic Map
- Numerical Study and Visualizations -
Klaus
Reiner Schenk-Hoppé
December 1997
Thanks go to
Jörg
Nikutta
(for support in coloring and recoloring)
and
Marc
Müller
(for producing the 3d pictures)
This web site contains several numerical results for the randomly perturbed
logistic map. It complements my paper
Bifurcations of the Randomly Perturbed Logistic Map
in the way that it contains colored versions of all pictures given
there and many new numerical results which are partly presented as
movies. The paper can be downloaded in
postscript.gz,
pdf
and postscript.zip
format here.
We are aiming at a thorough study of the stochastic bifurcation behavior
of the discrete-time random dynamical system generated by the stochastically
perturbed logistic map. The numerical simulations
are therefore concerned with Lyapunov exponents, stationary measures, and
invariant measures. These concepts are explained briefly here, and
the reader is referred to the paper for detailed information.
The numerical analysis has been carried out using a beta-version of
the simulation toolkit MACRODYN. It is part of
the project Dynamic Macroeconomics supported by the Deutsche Forschungsgemeinschaft.
Let me repeat what I already said in the paper:
"Our presentation will follow a descriptive style due to the facts
that there is a lack of rigorous results and that this paper is
intended to motivate further research and applications. The reader is
requested to take a careful look at the pictures and to develop his
or her own interpretation. Let us point out that we consider this
paper as a first attempt to understand the effects which noise can
have on simple systems such as the logistic map. In our opinion,
everything which is observed has to be proved in a mathematical
rigorous way and has to be carefully analyzed in particular
applications."
The randomly perturbed logistic map is the random difference equation
xn+1 = 
xn (1 - xn) + 
where
and
are
ergodic processes entering multiplicatively resp. additively with respect to
the trivial solution of the deterministic
logistic map
xn+1 = µ xn (1 - xn).
This study focuses on the effect of noise on the bifurcations of the
logistic map, and it aims at an understanding of the stochastic
bifurcation behavior of the randomly perturbed logistic map.
The numerical results motivate particularly clear-cut
conjectures for the stochastic outcomes of the randomly perturbed
first two bifurcations of its complete bifurcation scenario:
the transcritical bifurcation at µ = 1
and the period-doubling bifurcation at µ = 3.
The mathematical framework used is provided by the theory of
random dynamical systems.
We consider the two most common approaches to stochastic bifurcation theory:
the P-bifurcation and the D-bifurcation approach.
The P-bifurcation
approach studies qualitative changes of stationary measures when parameters
are varied. Roughly speaking, one says that a P-bifurcation occurs if the
stationary measure of a (parameter dependent) Markov process changes its shape.
Well known scenarios are the transition of a one-peak to a two-peaked density.
This approach is limited to a Markovian setup, and it is not related to the
stability of the stationary measure in general.
The D-bifurcation approach deals with invariant measures and their
stability determined by the multiplicative ergodic theorem. A (stable) invariant
reference measure undergoes a D-bifurcation if it losses its stability and
a new invariant measure occurs. The notion of invariant measure is more general
than that of stationary measure, and the D approach covers all systems
which are subject to stationary perturbations.
Here, we give numerical results for the logistic map subject to
an ergodic noise process that enters either additively or
multiplicatively. There are many studies of this system from
different point of views manly carried out by physicists.
Some of which are cited in the references.
The next two links carry you to either of these cases.
Some analytical results on the bifurcation behavior
(mainly on the existence of stationary and invariant measures
and some calculations of Lyapunov exponents) of ths system considered
are provided in the
paper which can also be downloaded following this link.
To the
Introduction
Additive noise case
Multiplicative noise case
Deterministic case
List of all pages