Nichtbeschattbarkeit durch Dimensionsvariabilität in dynamischen Systemen
Sinde
Erik
Sinde, Erik
aut
2001-01-30
2001-02-16
2014-08-12
de
Numerical simulations of dynamical systems often rely inexplicitely on the hypothesis
that the simulated pseudo-trajectory represents a true trajectory of the system in the sense
that both the pseudo-trajectory and the true trajectory stay close to each other for arbitra-rily
long time. This is guaranteed to hold for hyperbolic systems by the shadowing lemma
(Anosov 1967, Bowen 1975). However, this is not the case in general. Virtually, all real
systems which physicists encounter, are nonhyperbolic, and in most systems the shado-wing
property does not hold. Dynamical systems with unstable dimension variability
have recently gained interest as a source both of nonhyperbolicity and nonshadowability.
Some recent publications[7] claim that the nonshadowability due to unstable dimension
variability is particularly severe.
<br><br>In this work for a very simple dynamical system driven by one out of three input systems
is investigated. Numerical evidence is presented, that all the combined systems are prac-tically
unshadowable due to the same mechanism, although only two of them can have
unstable dimension variability, the third being a quasiperiodic map which has no periodic
orbits.
<br><br>In order to show numerically for the first and second input system that both singly and
doubly unstable periodic orbits are embedded into the attractor, a new method is applied
to one of the systems to show that the attractor fills a region of the phase space densely.
This method consists of iterating particular lines which are known to be subsets of the
attractor instead of points and consecutively applying a Poincaré surface-of-section-like
technique. For one of the studied maps this can reduce the dimensionality of the system
by one.
<br><br>For the three model systems, the probability distribution of the shadowing times is inve-stigated
through simulation. Additionally, a new deterministic measure to quantify the
severeness of nonshadowability due to unstable dimension variability is introduced and
applied to the model systems. This quantity estimates by how many digits the calculus
precision has to exceed the required closeness to a true trajectory on average for a given
trajectory length. It is argued and verified numerically that for the case of unstable dimen-sion
variability the number of additionally needed calculus precision digits is proportional
to the logarithm of the trajectory length. Thereby even pseudo-trajectories with arbitrarily
small one step errors cannot be shadowed by true trajectories for arbitrarily long time.
<br><br>The nonshadowability can however in the same time be rigorous and very small, so that
for some practical purposes this system can be regarded as almost shadowable. On the
other hand, the same mechanism can produce similar results for systems where unstable
dimension variability does not occur. In those systems there is an upper limit to the addi-tionally
needed calculus precision. This limit can be so large that for practical purposes
this system behaves as a nonshadowable one although it is shadowable in the sense of the
shadowing lemma.
urn:nbn:de:hbz:468-20010124
2014-08-12T08:33:36.861Z
2014-08-12T09:36:45.095Z
published
Diss
fb08/diss2000/sinde