# Image Entropy for the Identification of Chimera States of Spatiotemporal Divergence in Complex Coupled Maps of Matrices

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## Abstract

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## 1. Introduction

## 2. Preliminary Notes and the Objective

#### 2.1. A Network of Coupled Maps

#### 2.2. A Network of Coupled Map of Matrices

## 3. Chimera States of Spatiotemporal Divergence in Regular NCMMs

#### 3.1. Spatiotemporal Divergence in a Regular NCMM

#### 3.2. Chimera States of Spatiotemporal Divergence in a Regular Feed-Forward NCMM

## 4. Chimera States of Spatiotemporal Divergence in a Complex NCMM

#### 4.1. Chimera States of Spatiotemporal Divergence in the Erdős-Rényi NCMM

#### 4.2. Chimera States of Spatiotemporal Divergence in the Small-World NCMM

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The transient dynamics of a regular NCMM comprising 200 nodes ($a=3.699956$; $\epsilon =0.4$; ${\lambda}^{\left(0\right)}\left(i\right)$; $i=1,2,\dots ,200$ are randomly distributed in the interval $\left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$) represented by the variation of ${\mu}^{\left(t\right)}\left(i\right)$. The network diverges at $r=0.02$ (part (

**a**)); generates complex patterns at $r=0.025$ (part (

**b**)); and calms down at $r=0.03$ (part (

**c**)). Numerical values of ${\mu}^{\left(t\right)}\left(i\right)$ are truncated to 5 for the clarity of presentation.

**Figure 2.**The evolution of ${\mu}^{\left(t\right)}\left(1\right)$, ${\mu}^{\left(t\right)}\left(50\right)$, ${\mu}^{\left(t\right)}\left(100\right)$, ${\mu}^{\left(t\right)}\left(150\right)$ and ${\mu}^{\left(t\right)}\left(200\right)$ in time interval $500\le t\le 1000$ at the set of system parameters corresponding to Figure 1b. The numerical values of ${\mu}^{\left(t\right)}\left(i\right)$ are cropped to 5 in part (

**a**) and are shown uncropped in part (

**b**).

**Figure 3.**Image entropy of patterns is calculated for regular network when parameter r is set to $0.05$. Network coupling parameter $\epsilon $ is set to $0.2$, $0.366$, $0.6$, $0.823$ and $0.9$ in parts (

**a**), (

**b**), (

**c**), (

**d**) and (

**e**). Image entropy is equal to $2.29$ and $0.785$ in parts (

**b**) and (

**d**) respectively.

**Figure 4.**The visualization of chimera states of spatiotemporal divergence for networks of different structure: a regular NCMM (part (

**a**), parameter plane $\epsilon -r$); a regular unidirectional NCMM (part (

**b**), parameter plane $\epsilon -r$); the Erdős-Rényi NCMM (part (

**c**), parameter plane $\epsilon -d$); the small-world NCMM (part (

**d**), parameter plane $\epsilon -r$). The colorbar denotes numerical values of the entropy computed for steady-state evolution of the networks.

**Figure 5.**The transient dynamics of a regular directional NCMM comprising 200 nodes ($a=3.699956$; $\epsilon =0.4$; ${\lambda}^{\left(0\right)}\left(i\right)$; $i=1,2,\dots ,200$ are randomly distributed in the interval $\left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$) represented by the variation of ${\mu}^{\left(t\right)}\left(i\right)$. The network diverges at $r=0.025$ (part (

**a**)); generates complex fractal-type patterns at $r=0.035$ (part (

**b**)); and calms down at $r=0.045$ (part (

**c**)). Numerical values of ${\mu}^{\left(t\right)}\left(i\right)$ are truncated to 5 for the clarity of presentation.

**Figure 6.**The transient dynamics of the Erdős-Rényi NCMM comprising 200 nodes ($a=3.699956$; $\epsilon =0.4$; ${\lambda}^{\left(0\right)}\left(i\right)$; $i=1,2,\dots ,200$ are randomly distributed in the interval $\left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$) represented by the variation of ${\mu}^{\left(t\right)}\left(i\right)$. The network diverges at $d=0.031$ (part (

**a**)); generates complex fractal-type patterns at $d=0.033$ (part (

**b**)); and calms down at $d=0.035$ (part (

**c**)). Numerical values of ${\mu}^{\left(t\right)}\left(i\right)$ are truncated to 5 for the clarity of presentation.

**Figure 7.**The transient dynamics of the small-world network NCMM comprising 200 nodes ($a=3.699956$; $\epsilon =0.4$; ${\lambda}^{\left(0\right)}\left(i\right)$; $i=1,2,\dots ,200$ are randomly distributed in the interval $\left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$; $\beta =0.2$) represented by the variation of ${\mu}^{\left(t\right)}\left(i\right)$. The network diverges at $r=0.015$ (part (

**a**)); generates complex patterns of spatiotemporal divergence at $r=0.02$ (part (

**b**)); and calms down at $r=0.025$ (part (

**c**)). Numerical values of ${\mu}^{\left(t\right)}\left(i\right)$ are truncated to 5 for the clarity of presentation.

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**MDPI and ACS Style**

Smidtaite, R.; Lu, G.; Ragulskis, M.
Image Entropy for the Identification of Chimera States of Spatiotemporal Divergence in Complex Coupled Maps of Matrices. *Entropy* **2019**, *21*, 523.
https://doi.org/10.3390/e21050523

**AMA Style**

Smidtaite R, Lu G, Ragulskis M.
Image Entropy for the Identification of Chimera States of Spatiotemporal Divergence in Complex Coupled Maps of Matrices. *Entropy*. 2019; 21(5):523.
https://doi.org/10.3390/e21050523

**Chicago/Turabian Style**

Smidtaite, Rasa, Guangqing Lu, and Minvydas Ragulskis.
2019. "Image Entropy for the Identification of Chimera States of Spatiotemporal Divergence in Complex Coupled Maps of Matrices" *Entropy* 21, no. 5: 523.
https://doi.org/10.3390/e21050523