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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
108/2013

Closure Measures and the Tent Map

Oliver Pfante, Eckehard Olbrich, Nils Bertschinger, Nihat Ay and Jürgen Jost

Abstract

We quantify the relationship between the dynamics of a particular time-discrete dynamical system, the tent map, and the induced dynamics at a symbolical level in information theoretical terms. The symbol dynamics is obtained by choosing a partition point $\alpha \in \left[ 0, 1 \right]$ and lumping together the points in the intervals $ \left[ 0, \alpha \right]$ or $\left( \alpha ,1 \right]$, resp. Interpreting the original dynamics and the symbolic one as different levels, this allows us to quantitatively evaluate and compare various closure measures that have been proposed for identifying emergent macro-levels of a dynamical system. In particular, we can see how these measures depend on the choice of the partition point $\alpha$, with $\alpha =\frac{2}{3}$ yielding a minimum. Also, when we study the iterated tent map, i.e., probe more time steps, we get more refined insights into this relationship between the two levels, and even a whole hierarchy of mesoscopic levels.

Received:
Nov 22, 2013
Published:
Nov 25, 2013
MSC Codes:
37-XX, 94A17, 34C28
Keywords:
chaotic dynamics, coarse graining, Markov partition

Related publications

inJournal
2014 Repository Open Access
Oliver Pfante, Eckehard Olbrich, Nils Bertschinger, Nihat Ay and Jürgen Jost

Closure measures for coarse-graining of the tent map

In: Chaos, 24 (2014), p. 013136