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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.
