### Abstract

Original language | English |
---|---|

Article number | 033611 |

Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Chaos |

Volume | 28 |

Issue number | 3 |

Early online date | 20 Mar 2018 |

DOIs | |

Publication status | Published - Mar 2018 |

### Fingerprint

### Keywords

- nlin.CD
- eess.SP
- physics.data-an
- Markov partitions
- Shannon entropy
- Information Theory
- Complex Systems

### Cite this

*Chaos*,

*28*(3), 1-11. [033611]. https://doi.org/10.1063/1.5002097

**Entropy-based Generating Markov Partitions for Complex Systems.** / Rubido, Nicolás; Grebogi, Celso; Baptista, Murilo S.

Research output: Contribution to journal › Article

*Chaos*, vol. 28, no. 3, 033611, pp. 1-11. https://doi.org/10.1063/1.5002097

}

TY - JOUR

T1 - Entropy-based Generating Markov Partitions for Complex Systems

AU - Rubido, Nicolás

AU - Grebogi, Celso

AU - Baptista, Murilo S.

N1 - Authors thank the Scottish University Physics Alliance (SUPA) support and NR also thanks PEDECIBA.

PY - 2018/3

Y1 - 2018/3

N2 - 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 real-world experiment, measurements are made with finite resolution and over a finite time-span. The task gets even more complicated if the system is a network composed of interacting dynamical units, namely, a high-dimensional complex system. Here, we tackle this task and solve it by defining a method to approximately construct GMPs for any complex system's finite-resolution and finite-time 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.

AB - 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 real-world experiment, measurements are made with finite resolution and over a finite time-span. The task gets even more complicated if the system is a network composed of interacting dynamical units, namely, a high-dimensional complex system. Here, we tackle this task and solve it by defining a method to approximately construct GMPs for any complex system's finite-resolution and finite-time 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.

KW - nlin.CD

KW - eess.SP

KW - physics.data-an

KW - Markov partitions

KW - Shannon entropy

KW - Information Theory

KW - Complex Systems

U2 - 10.1063/1.5002097

DO - 10.1063/1.5002097

M3 - Article

VL - 28

SP - 1

EP - 11

JO - Chaos

JF - Chaos

SN - 1054-1500

IS - 3

M1 - 033611

ER -