# American Institute of Mathematical Sciences

December  2015, 20(10): 3507-3524. doi: 10.3934/dcdsb.2015.20.3507

## Entropy determination based on the ordinal structure of a dynamical system

 1 Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, 23562 Lübeck 2 Institute of Mathematics of NAS of Ukraine, Tereshchenkivs'ka str., 3, 01601 Kyiv 3 Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, Lübeck, 23562, Germany

Received  December 2014 Revised  February 2015 Published  September 2015

The ordinal approach to evaluate time series due to innovative works of Bandt and Pompe has increasingly established itself among other techniques of nonlinear time series analysis. In this paper, we summarize and generalize the theory of determining the Kolmogorov-Sinai entropy of a measure-preserving dynamical system via increasing sequences of order generated partitions of the state space. Our main focus are measuring processes without information loss. Particularly, we consider the question of the minimal necessary number of measurements related to the properties of a given dynamical system.
Citation: Karsten Keller, Sergiy Maksymenko, Inga Stolz. Entropy determination based on the ordinal structure of a dynamical system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3507-3524. doi: 10.3934/dcdsb.2015.20.3507
##### References:

show all references

##### References:
 [1] Karsten Keller. Permutations and the Kolmogorov-Sinai entropy. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 891-900. doi: 10.3934/dcds.2012.32.891 [2] Tim Gutjahr, Karsten Keller. Equality of Kolmogorov-Sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4207-4224. doi: 10.3934/dcds.2019170 [3] Alexandra Antoniouk, Karsten Keller, Sergiy Maksymenko. Kolmogorov-Sinai entropy via separation properties of order-generated $\sigma$-algebras. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1793-1809. doi: 10.3934/dcds.2014.34.1793 [4] Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018 [5] Min Qian, Jian-Sheng Xie. Entropy formula for endomorphisms: Relations between entropy, exponents and dimension. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 367-392. doi: 10.3934/dcds.2008.21.367 [6] Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288 [7] Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101 [8] Xiao-Qian Jiang, Lun-Chuan Zhang. Stock price fluctuation prediction method based on time series analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 915-927. doi: 10.3934/dcdss.2019061 [9] Michael Brandenbursky, Michał Marcinkowski. Entropy and quasimorphisms. Journal of Modern Dynamics, 2019, 15: 143-163. doi: 10.3934/jmd.2019017 [10] Wenxiang Sun, Cheng Zhang. Zero entropy versus infinite entropy. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1237-1242. doi: 10.3934/dcds.2011.30.1237 [11] Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701 [12] María Anguiano, Alain Haraux. The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors. Evolution Equations & Control Theory, 2017, 6 (3) : 345-356. doi: 10.3934/eect.2017018 [13] Annalisa Pascarella, Alberto Sorrentino, Cristina Campi, Michele Piana. Particle filtering, beamforming and multiple signal classification for the analysis of magnetoencephalography time series: a comparison of algorithms. Inverse Problems & Imaging, 2010, 4 (1) : 169-190. doi: 10.3934/ipi.2010.4.169 [14] José M. Amigó, Karsten Keller, Valentina A. Unakafova. On entropy, entropy-like quantities, and applications. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3301-3343. doi: 10.3934/dcdsb.2015.20.3301 [15] Ping Huang, Ercai Chen, Chenwei Wang. Entropy formulae of conditional entropy in mean metrics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5129-5144. doi: 10.3934/dcds.2018226 [16] François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275 [17] Boris Kruglikov, Martin Rypdal. Entropy via multiplicity. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 395-410. doi: 10.3934/dcds.2006.16.395 [18] Nicolas Bedaride. Entropy of polyhedral billiard. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 89-102. doi: 10.3934/dcds.2007.19.89 [19] Baolin He. Entropy of diffeomorphisms of line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4753-4766. doi: 10.3934/dcds.2017204 [20] Lluís Alsedà, David Juher, Deborah M. King, Francesc Mañosas. Maximizing entropy of cycles on trees. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3237-3276. doi: 10.3934/dcds.2013.33.3237

2018 Impact Factor: 1.008