Pfante Oliver, Olbrich Eckehard, Bertschinger Nils, Ay Nihat, Jost Jürgen
Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany.
Chaos. 2014 Mar;24(1):013136. doi: 10.1063/1.4869075.
We quantify the relationship between the dynamics of a time-discrete dynamical system, the tent map T and its iterations T(m), and the induced dynamics at a symbolical level in information theoretical terms. The symbol dynamics, given by a binary string s of length m, is obtained by choosing a partition point [Formula: see text] and lumping together the points [Formula: see text] s.t. T(i)(x) concurs with the i - 1th digit of s-i.e., we apply a so called threshold crossing technique. 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. As main benefit of this new information theoretical approach, we get all Markov partitions with full support of the time-discrete dynamical system induced by the tent map. Furthermore, we could derive an example of a Markovian symbol dynamics whose underlying partition is not Markovian at all, and even a whole hierarchy of Markovian symbol dynamics.
我们用信息论术语量化了时间离散动力系统、帐篷映射(T)及其迭代(T^{(m)})的动力学与符号层面上诱导动力学之间的关系。通过选择一个划分点([公式：见正文])并将满足(T^{(i)}(x))与(s)的第(i - 1)位数字一致的点归为一组，得到由长度为(m)的二进制字符串(s)给出的符号动力学，即我们应用了一种所谓的阈值穿越技术。将原始动力学和符号动力学解释为不同层面，这使我们能够定量评估和比较为识别动力系统涌现的宏观层面而提出的各种封闭度量。特别地，我们可以看到这些度量如何依赖于划分点(\alpha)的选择。作为这种新的信息论方法的主要优点，我们得到了由帐篷映射诱导的时间离散动力系统的所有具有完全支撑的马尔可夫划分。此外，我们可以推导出一个马尔可夫符号动力学的例子，其基础划分根本不是马尔可夫的，甚至可以推导出一整个马尔可夫符号动力学层次结构。
