Entropybased generating Markov partitions for complex systems
Abstract
Finding the correct encoding for a generic dynamical system's trajectory is a complicated task: the symbolic sequence needs to preserve the invariant properties from the system's trajectory. In theory, the solution to this problem is found when a Generating Markov Partition (GMP) is obtained, which is only defined once the unstable and stable manifolds are known with infinite precision and for all times. However, these manifolds usually form highly convoluted Euclidean sets, are a priori unknown, and, as it happens in any realworld experiment, measurements are made with finite resolution and over a finite timespan. The task gets even more complicated if the system is a network composed of interacting dynamical units, namely, a highdimensional complex system. Here, we tackle this task and solve it by defining a method to approximately construct GMPs for any complex system's finiteresolution and finitetime trajectory. We critically test our method on networks of coupled maps, encoding their trajectories into symbolic sequences. We show that these sequences are optimal because they minimise the information loss and also any spurious information added. Consequently, our method allows us to approximately calculate the invariant probability measures of complex systems from the observed data. Thus, we can efficiently define complexity measures that are applicable to a wide range of complex phenomena, such as the characterisation of brain activity from electroencephalogram signals measured at different brain regions or the characterisation of climate variability from temperature anomalies measured at different Earth regions.
 Publication:

Chaos
 Pub Date:
 March 2018
 DOI:
 10.1063/1.5002097
 arXiv:
 arXiv:1711.09072
 Bibcode:
 2018Chaos..28c3611R
 Keywords:

 Nonlinear Sciences  Chaotic Dynamics;
 Electrical Engineering and Systems Science  Signal Processing;
 Physics  Data Analysis;
 Statistics and Probability
 EPrint:
 10 pages, 10 figures