Closure Measures and the Tent Map

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

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Submission date: 22. Nov. 2013
Pages: 15
published in: Chaos, 24 (2014), art-no. 013136 
DOI number (of the published article): 10.1063/1.4869075
Bibtex
with the following different title: Closure measures for coarse-graining of the tent map
MSC-Numbers: 37-XX, 94A17, 34C28
Keywords and phrases: chaotic dynamics, coarse graining, Markov partition
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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 α ∈ and lumping together the points in the intervals or , 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 α, with α = 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.